The Strength of Nonstandard Analysis

e0 : 8o E F} .

Definition 10.3.1

=

We prove in this paper that, except for a case studied in § 10.4.3 where

either \i8o E F, e0 = 1 and then
•

or 38o E F, e 0 = 0 and then

=

0 (if He0 is accepted for at least

At the beginning of § 10.4 we shall explain that a has t o b e infinitesimal. We shall prove in § 10.5 that this NSLRT is uniformly more powerful than any "regular" (this term will be defined later) nonstandard test based on X. Convention: if E is an external event, and P R(E) its external probability and if T) is an external number, "the probability of E is equal to T)" means P R(E) = T) and "the probability of E is exactly equal to TJ" means P R(E) TJ · =

10.3.4

Large deviations for

X

In exponential families, J( · , >.(Bo)) can be regarded as the Cramer transform of Pe0 • rviore generally, classical literature establishes that

1

-

n nlim - ln P (X n E A) = - xinf C(x) -+ oo n

EA

where A is a Borel set of �k such that A C A0 and X n the mean of n i.i.d. ran­ dom variables taking values in � k , C the Cramer transform of their distribution (see [16] , section 4: the proof is based on the minimax theorem for compact convex sets) . Even more generally, if the i.i.d. random variables take values in some topological vector space E , classical literature ( cf. [3] ) establishes the same result if A is a finite union of convex Borel sets of E. For exponential families, it is possible to establish specific proofs by using half-spaces (cf. [10] , chap. 7 and exercises 7.5.1 to 7.5.6). In a more general setting, half-spaces are also used in [3] . section 2. Our propositions 10.3.2 and 10.3.3 give nonstandard results which imply (by transfer) the classical results. If the first part of the proofs is based on the law of large numbers as in the classical literature, the second part, based on infinitesimal pavings, seems to be original.

10.3. Exponential families 10.3.5

151

n-regular sets

Definition 10.3.2 Let A is n-regular in a iff 3b E

be a subset of A and a E 8A. We shall say that A hal(a), 3p E J Jn, 0] such that P l l >- - 1 (b) I I 0 and B(b, p) C A (where B(b, p) is the open ball of centrum b and radius p) .

Proposition 10.3.1 Let A hal(8A) = hal( a 0A); then A

=-:

be a limited subset of A such that 0A = ( 0A )0 and is n-regular in any point a E a A.

We first claim that A \ hal(8A) = 0A \ hal(8 °A) . Indeed, if x E A \ hal(8A) then °x E 0A and 0x tf. hal(8A) hal( a 0A) and so 0x E 0A \ hal(8 °A); consequently, 0x E (0A)0 and so x E (0A)0; finally, as x tf. hal(8A) = hal( a 0A) , x E 0A \ hal( a 0A) . Conversely, if x E 0A \ hal( a 0A), there exists y E A such that x y; then x E A for if not x E hal(8A) = hal(8 °A) . If a tf. hal(8A), choose an infinitesimal p such that p > Jn · Then for each b E A \ hal(8A) , B (b, p) C A: it is obvious if b is standard and if b is nonstan­ dard, with 0b its shadow, there exists a standard po such that B(0b, po) C A and then B(b, p) C B(0b, Po) C A. So Proof.

=

=-:

Vc ?!:, 0, 3b EA, d(a, b) < c and B (b, p) C A. Then the internal set { c E � : 3b EA, d( a, b) < c 1\ B(b, p) C A} contains all appreciable c. Cauchy ' s principle yields that ::lc

=-:

0, 3b E A, d(a, b) < c and B(b, p) C A;

as a tf. hal(8A) then b tf. hal(8A) and so >. - 1 (b) is limited (because A is a standard diffeomorphism from 8 ° onto A0). Finally p 1 1 >- - 1 (b) l l ::: 0. If a E hal(8A) , fix p = n -1 14 ; if b E A\ (hal( 8A)Uhal( 8A)) then B (b, p) c A and, as >. - 1 (b) is limited p 1 1 >- - 1 (b) l l < n- 1 18 (for example). So the internal set

{ c E � : 3b EA. d(a., b) < c 1\ B(b, p) C A 1\ P l l >-- 1 (b) ll < n- 1 } 18

contains all appreciable c. Cauchy 's principle yields that

::lc ::: 0, 3b E A, d(a, b) < c 1\ B(b, p) C A 1\ p I I >- - 1 (b) I I < n- 1 18 .

D

For example, a standard set A such that A c A0 (e.g. an open standard set) is n-regular in any boundary point. It is possible to prove that a limited convex set A such that 0A has a nonempty interior is also n-regular in any boundary point (we shall not use this result in the following) .

10. Nonstandard likelihood ratio test in exponential families

152 Lemma 10.3.1

(i) Let (�o , 6) E A2 and let � E ]�o , 6 [ (the segment between �o and 6). Then J ( ( 6 ) < J ( �o , 6 ) . (ii) Let A be a nonempty subset of A and 6 E A0 \ A. Then J(A, 6) J(A, 6 ) = J (aA, 6 ) . (iii) Let E and F be subsets of A such that E is a compact set included in F 0 ; let � E A0 \ E. Then J(F, �) < J(E, 0 and J(�, F) < J (�, E) . =

Proof. (i) As A is convex, � E A. We set �o - � =: a(� - 6 ) where a > 0; denoting e = >, - 1 (�) and e 1 := >,- 1 (6 ) , we have ( using corollary 2.5 in [10]) (ii) Let �o E A. If �o tf. aA, let � E aA n ].;o , 6 [. Using (i), we can write J(�, 6 ) :::; J(�o, 6 ) . So J(aA, 6 ) :::; J(A, 6 ) . Conversely, using the continuity of J ( , 6), we have ·

J (aA, 6 )

�

J (A, 6 ) = J(A, 6 ) .

(iii) If � E F, then J ( P �) = J (�, F) = 0 and J (E, �) -/= 0 , J(�, E) -/= 0 because � t/. E. We suppose now that � tf. F. Let �E E §E be such that J(E, �) = J (�E , �) ; there exists at least one �F E aF n ]�E , �[ and then, using (i), J (�F, �) < J(�E, �) and so J(F, �) < J(E, �) . Let now �k E §E be such that J (�, E) J (�, �k) ; there exists at least one �� E §F n J �k , �[. We set B E := A - 1 (�k) and Bp := >. - 1 (�� ). Then =

J (�, �k) - J (�, �� ) = =

1/J (BE) - 1/J (B) + (e - e E 1 �) - 1/J (BF) + 1/J (B) - (e - eF 1 �) = '1/J (e E ) - 1/J (eF) + ( eF - e E 1 �) = 1/J (BE) - 1/J (BF) + (eF - e E 1 �� ) + (eF - eE 1 � - �� ) = 1 ( eF , e E) + ( eF - e E 1 .; - �� ) .

Then setting .; - �� = : (3( �� - �k) where (3 > 0, we can write according to corollary 2.5 in [10] .

D

10.3.

Exponential families

153

Proposition 10.3.2 Let A be a limited subset of A \ hal (8A) , 6 = >. (B1) be a limited point of A \ hal (8A) and �o E aA such that J(�o , 6 ) :-: J(A, 6 ) . If A is n-regular in �o , then

1

-

- ln Pl) (X E A) :-: -J(A, 6 ) , 1

n and if furthermore d(6 , A) 7: 0, then

1

-

-- ln Pl) (X E A) 1

n

=

J (A, 6) ( 1 + 0)

=

@.

Proof. For all (el , e2) E 8 2 ' denoting X (xl , . . . ' Xn ) and X (where Xj = (xj ,i hs i s k E � k ) we can write =

(

=

� 'L.j= l Xj

)

PlJ; ( dx) = exp n ( ( el - e2 I x) - 1/J ( el) + 1/J ( e2)) p� ( dx) .

Let eo := >. - 1 ( �o) and let 6 =: >. (e2 ) and p > Jn be such that 6 E hal ( �o ) and B( 6 , p) c A. As >. is a standard diffeomorphism from 8° onto A0 , Bo , B1 , B2 are limited. Then

PlJ; ( X E A) > PlJ; (X E B ( 6 , p)) > c exp n ((el - e2 I X) - 1/J(Bl ) + 1/J(B2)) dP� jXEB(b,p) > exp n c ((el - e2 I X) - '1/J (Bl ) + 1/J (B2)) dPlJ2 JXEB(b,p)

(

(

)

)

by Jensen ' s inequality. P� (Xi E 6 , - p/ 2, 6,; + p/2]) = + 0 for According to (8.4) in each i = . . . k. Then the nonstandard law of large numbers yields P� (X E B ( 6 , p)) = + 0 and we can write ( denoting by 1:k the external set of limited vectors of �k )

[ 4] ,

1

1 1 - ln P(j ( X E A) n

1

>

[

1

,

.!. ln( 1 + 0 )

(B 1 - e2 1 6 + J:k p ) - 1/J(BI ) + 1jJ(B2) + n 0 ( e l - e2 1 6) + J:p - 1/J(e 1 ) + 1/J(e2) + n - 1 (e2 , e1 ) + 0.

As

I (B2 , B1 ) - I(eo , B 1 ) = (eo - e1 1 6 - �o ) + I (B2 , Bo), and I (B2 , Bo) = @ I I B2 - Bo ll = @ 11 6 - �o il with I I Bo - B 1 ll limited, 11 6 - �o il we have ( cf. lemma 3. 2 . 2 in

[ 13] ),

:-:

0

- I (B2 , e1 ) = - I(eo , el ) + 0 = - J(A, 6) + 0 ,

10. Nonstandard likelihood ratio test in exponential families

154

and so

-

1 n

- ln P� (X E A)

2'

- J (A, 6 ) + 0.

Conversely, the limited set A is included in a hypercube [ -p, p] k where p is a standard integer; pave it with n k (2p) k hypercubes ( T1 ) with side 6 = � · Among these hypercubes, eliminate the ones which do not intersect A and for the others choose t1 E A n Tt . In the aim of simplicity, we shall denote again the selected hypercubes by ( Tt h< t < N · Let Bt : = >.- 1 ( tt) , Bt is limited because t1 is limited and t 1 tf. hal(8A) . F�om the relation N < nk (2p) k we deduce In N = In n £ . Then we can write ' n n N dPenl p� ( X E A ) :::: L r l=l Jp<E Tl } N L r X- E T1 } exp n( ( el - el I X) - 1/J (Bl ) + 1/J (Bt)) dP� l=l }{

(

< <

Thus

1

)

f:l=l exp (n ( (el - el I tt + �k ) - 1/J (Bl) + 1/J (Bt) )) J: k ax exp { n ( ( el - el I t l + ) - ?jJ(Bl ) + 1/J (Bt) ) } . N l ,!?l...N n

-

{(

1

k

)

J: - ln Pe (X E A) < - ln N + :m ax el - el I tt + - - 1/J (Bl ) + 1/J (Bt) l - l...N n n n J: 1 < - ln N + max { (Bl - Bt l tt) - 1/J (BI ) + ?/J (Bt) } + l=l...N n n 1

< <

Finally,

1

}

J:

max - I (Bt , Bl ) + -n ln N + l=l...N n

-J(A. 6 ) + 0. 1 n

-

- ln Pe (X E A ) 1

=

- J ( A, 6 ) + 0.

As we said previously, according to lemma 3.2.2 in [ 13] , if a and 6 are limited elements of A \ hal(A) , then J(a , 6 ) = @d(a, 6 ) . Consequently, if d ( A, 6 ) = @ then J ( A , 6) = @. Thus

1

-

- -:;:;, ln P� (X E A) = J(A, 6 ) ( 1 + 0) = @.

D

In fact. the continuity of J(·, 6) allows us to generalize this result to limited subsets of A:

10.3. Exponential families

155

Proposition 10.3.3 Let A be a limited subset of A, 6 be a limited point of A \ hal(8A) and let �o E aA such that J(�o, 6 ) J(A, 6 ) . ff A is n-regular in �o then 1 -n ln Pl)1 (X E A) :-: - J (A, 6). :-:

Proof. Let 6 =: >.(B2 ) E hal(�o) and p > Jn be such that B ( 6 , P) C A and p IIB2 II :-: 0. As in the proof of proposition 10.3.2, we can write 1 -n ln Pl)1 (X E A) > (B 1 - e2 1 6 + £kp) - 1jJ (B 1 ) + 'ljJ (B 2 ) + _!_n ln (1 + 0 ) (e l - e2 1 6 ) + £p + 0 - 1/J (B 1 ) + 1/J (B2 ) + 0 -.J( 6 , 6 ) + 0 = -J( �o , 6) + 0 = - J(A, 6 ) + 0 because 116 - �o il :-: 0 and .J( · , 6 ) is S-continuous. Conversely, as in the proof of proposition 10.3.2 , use a paving (T1h < l < N with side 6 = � where each T1 is closed and such that T1 n A -/=- 0 , but cho;se t1 in T1 (maybe outside A) such that: if B1 , z - el. z > 0 then t l , z is maximal in T1 if B1 , z - el. i � 0 then tl , i is minimal in T1 So, for X E Tl we can write (e l - el I X) :::.: (e l - el I tl) and then

Pe� ( X E A)

N

�

L Jrp<E Tz } dr'e 1

<

L Jr{x- E T1 } exp ( n((el - el I X) - 1/J (Bl ) + 1/J (Bl )) ) dP�

<

l=l N

l=l N :L: exp n((e l - Bl l tl ) - 1/J (B I ) + 'l/J (B I )) . l=l

(

Thus

1 -n ln Pl)1 (X E A)

)

1

<

- ln N + =maxN {(el - el I t l ) - 1/J (B l ) + 1/J ( Bl ) } l l... n

<

- ln N + =max l...N {-.J(t1 , 6 )} .

1 n

l As .J(· , 6 ) is S-continuous, J(tl , 6 ) = J (0tl , 6) + 0 and then , as 1 - ln Pe: (X E A) � - J( 0 A, 6 ) + 0. n

0tl

E

0A

10. Nonstandard likelihood ratio test in exponential families

156

We now claim that J(0A, 6) ::: J(A, 6). Indeed, if a E A, then °a E 0A and J(a, 6 ) ::: J(0a, 6); so J(0A, 6 ) :;, J(A, 6). Conversely, if a E 0A, there exists a' E A such that a ::: a' and thus such that J (a, 6 ) ::: J (a', 6); so J(A, 6) ;::_ J( 0A, 6 ) . D Finally � ln PJ: ( X E A ) :::; -J(A, 6) + 0. 10.3.6

n-regular sets defined by Kullback-Leibler information

Let 8o be an internal subset of 8, included in a standard compact K included in the interior of e. Let co := co (8o) := sup {I(B, 8o) : e E 8} :::; oo . For c E ]O, co[, let Ac : = {� E A : J(�, Ao) :::; c} where Ao : = >.(8o). We shall see in lemma 10.3.2 that J(· , Ao) is continuous; then if c < c' < co , Ac is a proper subset of Ac' (for if not, { � E A : J(�, Ao) :::; c1c' } = { � E A : J(�, Ao) < c1c' } is a connected component of A which is a connected set.) Lemma 10.3.2

If c :::; co is limited, then aAc is a limited compact set.

= { � E A : J ( �, Ao ) = c} is closed in A since the function � J(�, Ao) is continuous on A as we prove now. By transfer, we just have to prove this continuity for a standard Ao: if 6 and 6 are such that 11 6 - 6 11 ::: 0, if �o E Ao is such that J(6 , �o) ::: J(6 , Ao) then, as J(-, �o) is S-continuous, J( 6 , �o) ::: J(6 , �o) and so J( 6 , Ao);;,J(6 , Ao) ; similarly J(6 , Ao);;,J( 6 , Ao). We prove now that if � is unlimited, then J(�, A0 ) is unlimited. For each �o E Ao , �o belongs to the standard compact K; then II � - �o il ::: oo and so J(�, �o) ::: oo (cf. [10, p. 177]). Thus J(�, Ao) is unlimited. Therefore { � E A : J(�, Ao) = c} is limited. D ____,

Proof. a Ac

Proposition 10.3.4 Let c :::; co be limited and B1 be a limited point of 8' such that 6 := >.(B 1 ) tf. hal (8A) . (i) If 6 tf. Ac, then � ln PJ: (X E Ac) = � ln PJ: (X E A�) ::: - J( aAc, 6 ) .

(ii) If 6 E Ac, then � ln PJ: ( X E Ac)

:::

(iii) If 6 tf. Af , then � ln PJ: ( X E Af)

0.

=

� ln PJ: (X E Af) ::: -J( aA c , 6 ) .

(iv) If 6 E Af , then � ln PJ: ( X E Af) ::: 0.

10.3. Exponential families

157

Proof. We first prove the both similar results (i) and (iii), and then the both similar results (ii) and (iv) which are obtained in a same way. (i) If 8o and c are standard, Ac is a standard compact set such that Ac A2 . Indeed, A� {� E A : J (�, Ao) < c}. If � is such that J (�, Ao) c , let >.. o E Ao (a compact set) be such that J(�, Ao) = J (�, Ao) = J (�, >..o ) = c. According to lemma 10.3.1 (i) , for any ( E ] ( >.. o [, J ((, >..o ) < J(�, >..o ) and so J((. Ao) < c. Thus ]�, >..o [ E A� and then � E A2. So, according to proposition 10.3. 1 , Ac and A� arc n-regular in any point of 8Ac and then, ac­ cording to proposition 10.3.3, we can write � ln PlJ; (X E Ac) -::::: - J (Ac, 6 ) and � ln PlJ; (X E A�) -::::: -J(A� , 6 ) . Finally, lemma 10.3.1 (ii) yields J(Ac, 6 ) = J (A�. 6 ) J(aAc. 6 ) . Now, if c or 8o are not standard, the continuity of J (�, ) on >.. ( K) implies 0Ac(8o) Aoc(08o) and the hypothesis of proposition 10.3.1 is verified since =

=

=

=

·

=

hal (8Ac)

=

hal (aAc) U { � E 8A : J (�, Ao) � c}

and

hal( a 0Ac) = hal (a0Ac) u { � E aA : J (�, 0Ao) � °C} . Indeed, on one hand, the continuity of J (�, ) implies ·

and on the other hand, this continuity also implies hal( 8Ac)

{ � E A : J (�, >.. ( 8o)) -::::: c} { � E A : J(�, >.. ( 08o)) -::::: 0c}

=

hal (8 °Ac)

·

Then we can conclude as before, using proposition 10.3.1 , proposition 10.3.3 and lemma 10.3. 1. (iii) In the same way, Af is n-regular in any point of aAf = aAc, but Af is not always limited. Meanwhile, aAc is limited and so the proof (in proposition 10.3.3) of � ln PlJ; (X E Af ) ? - J (8Ac, 6 ) + 0 remains valid. Suppose now that 6 E Ac. Ac is included in a standard hypercube Up : = {x E �k : Xi > p} [ - p , p] k (where p is a standard integer) . If we set Sp,i and s�, 2 {X E �k : Xi < - p } then u:; u7= 1 (Sp,i u s�, i ) . I t i s clear that :=

=

:=

On one hand, as Up \ A c is limited and n-regular in any boundary point, we can write

10. Nonstandard likelihood ratio test in exponential families

158

Using lemma 10.3.1 (iii) with E = §uP and F = A( , we have J(aUPu aAc, 6 ) = � � 6). J(8Ac, 6) and so n1 ln Pe 1 ( X E Up \ Ac) -::::: -J(8Ac, O n the other hand, n

P8; ( X E Upc ) �

k

2._) P8; ( X E Sp,i) + P8; ( X E S�,i )) . i =1

We know that � ln P8; (X E Sp ,i) -::::: -J(Sp,2 , 6 ) and � ln P8; (X E 3� ,2) -::::: - J(S�,i ' 6) (it is a classical result concerning half-spaces: cf. chap. 7 in [10]); so 1

1 - E U C ) � max max( - J( SP 6 ) , -J( S , 6)) + 0 - ln P8; (X ·"' P p,; n 2=l... k which is less than - J(aAc, 6 ) according to lemma 10.3. 1(i). Then � ln P8; (X E � -J(8Ac, 6 ) + 0. 1 C � · Fmally, r;, ln Pe 1 ( X E Ac ) -::::: -J(8A c, 6 ). (ii) If 6 E hal(aAc) , then exists �o E §Ac such that J (�o , 6 ) -::::: J(A, 6 ) -::::: 0 and it remains possible to use proposition 10.3. 1 , proposition 10.3.3 and lemma 10.3.1 as for the proof of (i). If 6 tf. hal(aAc) , then 6 E 0Ac \ hal(aAc) according to the proof of (i) and so there is a standard number p such that the ball B (�, p) is included in 0Ac \ hal(§Ac) and consequently included in Ac. Then the nonstandard law of large numbers yields P8; ( X E Ac) = 1 + 0 and finally ln P8; ( X E Ac) -::::: 0. (iv) is obtained in a similar way. D A

A cc )

n

Definition 10.3.3 Let e E 8'. Let he be the function defined on [0, co [ by h e( c ) := J(A(, >.(B)) and ge the function defined on [0, co [ by ge ( c) : = J(Ac, >.(B) ) . ___,

As c Ac is increasing, ge is a non-increasing function and he is a non­ decreasing function. Let r(B) : = I(e, 8 0 ); >. ( e ) E Ac iff c ? r(B) and so ge ( c ) = 0 if and only if c E [r(B) , co [ and h e( c) = 0 if and only if c E ]o, ,(B)].

Proposition

10.3.5 ___,

___,

h e (c) are S-continuous on (i) The functions (e, c) ge (c) and ( e, c) {e E 8 \ hal(88) : e limited } x ( [O , co [ n £). (ii)

The function ge is decreasing on [0 , /( B)] and he is increasing on [r(B) , co] .

10.4. The nonstandard likelihood ratio test

159

Proof. Let e be standard and e ' ::: e , then .>.. ( e ) ::: .A ( e ' ) ; let c and c' be such that c' ::: c. Then °Ac' = 0Ac and Similarly, J(Ac' ' .>.. ( e' )) ::: J(0Ac' ' .>.. ( e)). The monotonicity of ge is deduced from lemma 10.3.1 (iii) which shows that if c' < c and .A ( e ) tf. Ac' then J(Ac' , .>.. ( e)) > J(Ac, .>.. ( e)). D The continuity and monotonicity of he are proved in the same way. 10.4

The nonstandard likelihood ratio test

Recall that F := { 8o internal : So c 8o c hal(So) } , that the NSLRT is defined by e 0 : 8o E F}, that e 0 is defined through Re 0 and that, for exponential families, Re 0 = J(X, Ao) where Ao : = .A(8o). So, the relation sup Pl) (Re0 ? d) sup Pl)(Re 0 > d) :S: a :S: eEe eEe o o (see §10.2.1) can be written : ==:

sup Pl) ( X E Ar (8o)) eEe o

:S:

a :S: sup Pl) ( X E Ar (8o)). eEeo

Consequently, Ar is the rejection set (relatively to X ) , A� the acceptation set and the test e 0 is randomized if X E aAc . As n is illimited, if d < co (8o) and if d is limited then, for e E 8o, proposition 10.3.4 yields

As supeE e o -J(aAd, .>.. ( e ))

=

-d by definition of Ad, we have d(a, 8o) ::: ln a . - -

n

This is the nonstandard form of a classical result of Bahadur ( [1] , [2] ) . We shall study the NSLRT according as these d( a, 8o) are infinitesimal or appre­ ciable. In this latter case, we shall have to distinguish the cases d( a, 8o) < co (8o) and d(a, 8o) ? co (8o). In the following, Zo stands for co (So), Ao for ).. (So) and Ro J (X, Ao) for R e 0 ; Ac will denote Ac ( Ao); the notations he (c) and ge (c) will also refer =

10. Nonstandard likelihood ratio test in exponential families

160

to these Ac implies

:=

K

Ac (Ao). Note that if

8o E

F, then the S-continuity of A on

�

Re0 Ro. The following lemmas will be used for the calculation of the risks of the first and the second kind. They are valid not only for exponential families, but also for any statistical family. Here 8" is a standard subset of 8. Lemma 10.4.1 Let T be a statistic, S a (perhaps external) subset of 811 and F : D ____, � + , ( e , c) ____, Fe (c) a standard continuous function defined on a domain D = {(B, c) E 8" x �+ : de � c � d� } such that e • •

the functions e ----+ de and e ----+ d� aTe continuous, for each e E S, Fe is decreasing on [ de , d�] , for each e in s and c nearly standard in l de ' d� [ ' 1 -n ln Pe(T � c) � Fe ( c) .

Then, for each e nearly standard in S and Co standard in [de, d� [ Proof. PRIJ (T � Co) = supw =@ Pe (T > Co + w). Now write 1 - ln Pe (T � Co + w) � Fe (Co + w) � o Fe (Co + w). n

So, as oFe = Fe0 ( where Bo we have

:=

oe) is a standard continuous decreasing function,

� ln PR(j (T� C0 ) = J -oo, ��� Fe (Co + w) [ = ] -oo, Fe (Co) + 0[ .

Similarly, 1 -n ln PR e (T � C0 ) = ] -oo, ° Fe (Co) + 0[ = ] -oo, Fe (Co) + 0 [ , -n

for if not, there exists T) < 0,

T)

�

0 such that

1

Vr:: E hal + ( o ) , -n ln Pe(T � Co + r:: ) > o Fe (Co) + TJ

where hal + ( o )

:=

{c: E hal ( O ) : � 0} . r::

10.4. The nonstandard likelihood ratio test

161

Then { E � + : � ln PlJ (T ? Co + ) > ° Fe (Co) + 7]} is an internal set in­ cluding hal + ( o ) and consequently ( by Cauchy ' s principle ) including an interval [0, wo] where wo is standard. But c:

c

� ln Pe (T ? Co + wo) :-: Fe (Co + wo) :-: ° Fe (Co + wo) n with °Fe(Co + wo) < °Fe (Co) which is a contradiction, because both these numbers are standard. Finally D Lemma 10.4.2 Let T be a statistic, S a (perhaps external) subset of 811 and G : E ----+ � + , ( e , c) ----+ Ge (c) a standard continuous function defined on a domain E = { (B , c) E 8" x � + : e e ::; c ::; e� } such that •

the functions e

____,

ee

and e

____,

e� are continuous,

•

for each e E S, Ge is increasing on [ee , e�] ,

•

for each e in s and c nearly standard in l ee ' e� [ ' 1 - ln PlJ (T ::; c) :-: Ge (c) . n

Then, for each e nearly standard in S and Co standard in [e e , e � [ PRIJ (T � Co) = L: e nGe ( Co ) + n0 . Proof. PR� (T � Co)

=

inf t =©: PlJ (T ::; Co + t). Now write

� ln PlJ (T ::; Co + t) :-: Ge (Co + t) :-: 0Ge(Co + t). n So, as 0Ge is a standard continuous increasing function, we have

1 -n ln -n PRe (T � Co) =

]

- oo ,

]

. Ge (Co + t) = ] - oo , Ge (Co) + ] . 0 tmf =@

As in lemma 10.4.1 , one establishes that PR� (T � Co) finally PRIJ (T � Co) = L:enGe ( Co ) + n0 .

=

PRIJ (T � Co) and D

10. Nonstandard likelihood ratio test in exponential families

162

10.4.1

In a

-n

Proposition (i) if Ro

�

infinitesimal

10.4.1

For l�a

0 then
=

�

0,

0 otherwise
=

1;

(ii) for e E hal(Go ) , the risk of the first kind is exactly equal to £e -n @ -nhe ( 0l ; (iii) for a limited e E 8 \ (hal( 88) u hal( Go))! the risk of the second kind is exactly equal to £ e - nge ( O ) + n° . Proof. (i) If Ro 7: 0 then \i8o E F, Re0 7: 0 and so \i8o E F, e0 = 1 because d( a , 8o) - 1�a 0 and finally .-1 (X) E Go U { >.-1 (X) } E F. Then, e0 0 for this 8o Go U p-1 (X) } . Finally,

�

�

=

=

=

=

:=

�

____,

:

p Rlf ( Ro � 0 )

=

£e -n@ -nhe ( O ) .

(iii) According to proposition 10.3.4 (i) and (ii) , for a limited c :::; co , we have � ln Pl) (Ro :S: c) -ge (c) . For any limited e E 8 \ (hal(88) U hal(Go)) , there exists a standard compact set 811 C 8 \ hal(88). such that e E 8". According to proposition 10.3.5, (e, c) -ge (c) is continuous on 8" [O, co (8")] and for e E 8", - ge is increasing on [o, , (e)] . We can now apply lemma 10.4.2 with D { (e, c) E 8" x � + : 0 :::; c :::; I ( e) } , S = 8" \ hal( Go) , T = Ro , Ge = -ge , Co = 0, t o obtain P RIJ (Ro � 0) �

____,

X

=

£e-nge ( O ) + n0 .

0

Remark 10.4. 1 Let e E hal( Go); the nonstandard version of the law of large numbers yields PRIJ (X E hal(>.(e)))

=

1 + 0 ==? PRIJ (X E hal(Ao))

=

1 + 0.

As the reject criterion is X tf. hal ( Ao), it is not logical to take a appreciable, but infinitesimal.

10.4. The nonstandard likelihood ratio test 10.4.2 0 �

1

n

163

l in a l � co

If co oo , we shall suppose that I !:a l is limited because some preliminary results (in § 10.3) are no longer available (for example Ac is no longer limited if c 00 ) . =

:::

Proposition 10.4.2 For o ; l l:a l ; zo and l l:a l limited, p1Lt C 0 ( - 1�a ) . (i) If Ro � C then
=

=

Proof. (i) If Ro � C then V8o E F, R e0 � - 1�a , so Re0 > d(a, 8o) and e 0 = 1 . Finally, e 0 = 0. If not, the external set F is included in the internal set { 8o : So C 8o C K 1\ e0 -/=- 0 } , and using the Cauchy ' s principle, there exists an internal set 8o � hal( So) such that e 0 -/=- 0, i.e. Re 0 2: d(a, 8o) ::: C. Denoting by So the shadow of this 8o we have R e0 ::: Re 0 � C. Go and So are two distinct relatively compact standard sets such that aso c eg and then such that 8Ao c :Ag (where Ao : = >.(So)) . According to lemma 10.3. 1 (iii) , J(�, Ao) < J(�, Ao) for each � tf. Ao and then Re = J(X, Ao) ::: J ( 0 X, Ao) < J( 0 X, Ao) ::: J (X, Ao) = Ro . o =

�

As J ( 0 X, Ao) and J ( 0 X, Ao) are both standard numbers, this contradicts R e � C and Ro :;, C. o (ii) The proof is similar to the proof of proposition 10.4. 1 (ii) , with C0 = C instead of Co = 0. We note that the risk of the first kind calculated in §10.4.1 is a particular case of this result. (iii) For the first result, the proof is similar to the proof of proposition 10.4.1 (iii) , with Co = C instead of Co = 0 and S = { e E 8" \ hal( So) : I ( e) � C} instead of S = 811 \ hal(So ) . For the second, we choose a standard compact set 8" c 8 \ hal( 8 8) . such that e E 8". In fact, the proof of (ii) is valid with S { e E 8" : I ( e) :;, C} and one obtains =

p R{f ( Ro

� C) = Le-n@-nhe ( C)

D

10. Nonstandard likelihood ratio test in exponential families

164

10.4.3

1 - l ln a l � co n

The number co is standard because 8o is standard and co -/=- oo ; A is a limited set for if not J(�, Ao) is unlimited because � is unlimited (see the proof of lemma 10.3.2) and then co (So ) � sup�E A J(�. Ao) is unlimited. We shall suppose here that for each subset 8o of e, the function e J(e, 8o) is continuous on 8. This assumption is verified by classical exponential families. Then for any 8o , co ( 8 o) = sup�E A J(�, A( 8 o)) . For any 8 o C K, if co ( 8 o) � � lln a l then d(a, 8 o) = co( 8 o) and so e0 < 1 because A d = A. If X E A0 ' then eo = 0 and if X E a A, either eo = 0, or 0 < e0 < 1 (there is a randomization) ; the problem is not simple, as the example of multinomial distributions shows (cf. [9]). We note that Kallenberg avoids this case in theorem 3.3.2 of [13] . ____,

Proposition

10.4.3

For � lln al ?; co,

(i) o < 1 , (ii) if X E A0 then o 0 . =

Proof. As

e0 < 1. If not, \i8 o E F, max e0 = 1 (which means that e0 takes value 1), and by Cauchy's principle 3So ::) hal(So), such that max S o = 1 . According to lemma 10.3. 1 (iii) , for any � E A , J(0�,0Ao) < J(0�,0Ao) because hal(Ao) C A o := >. (So). Both these numbers being standard, the relation implies J(�, Ao) � J(�, Ao) and then co (So) � co(So) . So co(So) � l lnna l which contradicts max e�- o = 1 . (ii) Let X E A0 ; if \i8 o E F, e0 -/=- 0, by Cauchy's principle 3So ::) hal( So) l l n l a such that s 0 -/=- 0. But co(8o) � co ::: ----;;;:- and so, as X E A0 , s 0 = 0, which D contradicts So -/=- 0. �

Remark 10.4.2 Let e be a limited element of e \ (hal(88) u (So)); the non­ standard version of the law of large numbers yields P RIJ ( X E hal( A( B))) = 1 +0 and thus PRIJ ( X E A0 ) = 1 + 0. So, for this e, the risk of the second kind is equal to 1 + 0: if � lln al � co, the NSLRT is not consistent (in the classical

10.5. Comparison with nonstandard tests based on X

165

theory, a sequence of tests ( (f?n)n ::>no {where n is the size of the sample} is con­ sistent if for any e E 81 , lim17_,= Ee ( l - (f?17 ) = 0: then, for n "" oo, the risk of the second kind is infinitesimal). As the example of multinomial distributions shows (cf. {9}), it seems impossible to give general results about the value of
Remark 10.4.3

=

10. 5

Comparison with nonstandard tests based on X

In the classical theory, for testing e E 8o against e E 81 , one fixes the level a and one tries to find a test that has maximum power in any e E 8 1 . However, such uniformly most powerful (U.M.P.) tests exist only in a few exceptional cases: for example for !-dimensional exponential families, if 8o : = { e E 8 : e :S: Bo } . If no U .M.P. exists, one studies an asymptotic approach, for sample sizes tending to infinity. One of the tools to compare asymptotically the power of two tests of the same hypothesis is the Bahadur relative efficiency. In a more general framework (not only for exponential families) Bahadur ([1] or [2] , see also [12]) has proved that the likelihood ratio test is at last as "efficient" as any test based on a rejection criterion of the form T > c (with randomization if T = c if necessary) , where T is a statistic built with the n-sampling. However, it may be that to get the same power in a point e E 8 \ 8o the size of the sampling for the likelihood ratio test has to be larger than for the test based on T: thus was introduced the notion of "Bahadur deficiency" (cf. [13]). We shall prove that the NSLRT is uniformly the most powerful in a large family of nonstandard tests if a � 1 and I�a � co . The standard likelihood ratio test is not as powerful in the corresponding standard family. Let ((f?e0 ) be a family of tests of level a (where (f?e 0 tests He0 : e E 8o against e tt 8o) . The corresponding level-a nonstandard test (f?O is defined by (f?O

=

: inf { (f?e 0

: 8o E F }

and when we shall refer to a "nonstandard test", it will be defined thusly. 10.5.1 R

Regular nonstandard tests

Suppose that for each 8o , R (a, 8o) such that

:=

y? e 0

is a test of size a defined by a rejection set

sup Pl) ( X E R) :::_: a :::_: sup Pl)( X E R u aR) .

liE8o

liE8o

10. Nonstandard likelihood ratio test in exponential families

166

If R is n-regular in any point of aR and if aR is limited , rpe0 will be called "regular". The most famous example of such a test is the chi-squared test since the relative frequency F is in fact the sample mean for the multivariate distribution. In order to get R -/=- 0 we shall suppose that � lln a l � co and so � lln a l � co(8o) for 8o E F. If 'P8 o is regular for any 8o E F, 'PO will be said to be "regular". We prove now that and rp, it is clear that if � rp, then is more powerful than rp -in the classical sense- because Ee ? Eerp for any e.)

Proposition 10.5.1 For a � 1 and � lln a l � co , if 'P O is a regular level-a non­ standard test, then .(e)) , so infeE8o J( [JR, >. (e)) J( [JR, >. ( 8o)) verifies J( fJR, >. (8o)) :-: _ I�a . Put c := o ( l !r�a l ) . The region where 'PO -/=- 0 is included in the external subset R'PO R(a, 8o) . We prove that R'P O c { � E A, J(�. Ao)� c} 8 oE.F {. -1 (�o) . If �0 E hal(Ao), then eo E So since A is a standard diffeomorphism from 8° onto A0 . L Suppose Po := d(�o , 8R(a, 8o)) > vn and so B(�o, Po) n A c R( a , 8o). As A contains the support of Pe0 , the nonstandard law of large numbers yields PlJo (X E R( a, So)) = 1 +0 which is impossible because PlJo ( X E R( a, So)) � a. Suppose then Po � Jn · Let 6 E fJR( a , Go) n hal(�o); the n-regularity at �1 implies that 3 6 E hal(�I ) , ::Jp E ] Jn 0] such that B(6 , p) C R( a, Go) . As d(�o , 6) :-: 0, 6 E >.(So) and e2 := A-1 (6) E So. As previously, Pl); ( X E R( a, So)) = 1 + 0 which is impossible because PlJ; ( X E R( a, So)) � a. If �o t/. hal(Ao), we can write J (�o, >.( 0So)) :-: J(0�o , >.(0So)) J(�o , >.(So)) < J(0�o , >. (So)) :-: J( .;o, >. (So)) < c. �

-

=:

:=

n

=

,

•

A

�

-

,

•

�

10.5. Comparison with nonstandard tests based on X

167

But J( 0 �o , >. ( 0S o)) and J( 0 �o , >.(So)) -/=- 0 are both standard numbers; as >. ( So) c ( >. ( 0S o ) ) 0 we have J( 0.;o , >.( 0S o)) < J ( 0 �o , >.(So)) and so J (�o , >. ( Go)) ;:, c. 1 c: this contradicts On the other hand J(R(a, G 0 ) , >. (Go)) : ::o

�o E R(a, Go) .

10.5.2

Case when

80

a

-

::o

D

is convex

For any 8o E F and any e 1 E 8 \ hal ( 8o) , there is a level-a test \[!de o which is the most powerful at e 1 (for any level-a test cp testing e E 8o against e E 8 1 , Ee 1 cp ::; Ee 1 W�0 ) . This test is defined in the following way (see [13, p. 49]) let �

is the least favorable distribution. Then \[!�0 = 1 if f(X) < C and W�0 = 0 if f (X) > C where C is a constant depending on a, 8o , e1 . So, the rejection set (relatively to X) is { � E A : f ( �) < C}.

where

T

Definition

10.5.1

For e 1

E8

\ hal( So ) , let W� 1

:=

inf { W�0

: 8o E

F} .

We might think that W� 1 is "the most powerful at e1 nonstandard level-a test of Ho"; but in fact, in some cases,
Proposition

10.5.2 Jf Go is

convex, then for any e1

Proof. Let e1

E 8 \ hal( So) .

We prove that the rejection sets

A := { � E A :

!(�)

E 8 \hal (So) ,
W�0 .

< C}

are n-regular and then the proposition 10.5. 1 yields the result. The set So is convex and d ( e1 , Go) rj: 0 and so exists a standard cone � C �k and a standard positive number > 0 such that if u E � and ll u l l = 1 then ( eo e1 I u) < for any eo E So . Therefore, for any 8o E F, for any e E 8o and for any u E � \ {0} , (eo - e1 I u) < 0. Choose 8o E F. If u E � \ { 0} , f(x + u)

c

-c

-

le

:= .

o

ex

p ( n ( eo e l I u)) ·

. exp

-

(

n

( ( eo

-

e1 I x ) ?jJ ( eo) + 1jJ (e1 )) -

)

T

( deo)

< f(x).

168

10. Nonstandard likelihood ratio test in exponential families =

Let now a E 8A, then f(a) C. For any u E � \ {0}, f(a + u) < C, so a + (� \ {0}) C A . It is now possible to choose a point b E a + � such that d(a, b) = � ( for example) and such that B(b, nf12 ) C a + � - If a t/. hal ( 8A ) , this proves that A is n-regular in a. If E hal ( 8A) , we have to use Cauchy's principle like at the end of the proof of proposition 10.3. 1 D a,

References

[1] R. R. BAHADUR, An optimal property of likelihood ratio statistic, in Pmc. Fifth Berkeley Symp. Math. Statist. Prnb. , Vol. 1 , 1965. [2] R. R. BAHADUR Some limit theorems in statistics, SIAM, Philadelphia, 1971. [3] R.R. BAHADUR and S . L . ZABELL, "Large deviations of the sample mean in general vector spaces", Annals of Probability, 31 (1979) 587-62 1. [4] I . VAN DEN BERG, An external probability order theorem with applica­ tions, in F. and M. Diener editors, Non Standard Analysis in Practice, Springer-Verlag, Universitext, 1995. [5] J . B oSGIRAUD, "Exemple de test statistique non standard, risque ex­ terne", Pub. Inst. Stat. Paris, 41 (1997) 85-95. [6] .J . BoSGIRAUD, "Tests statistiques non standard sur des proportions", Pub. Inst. Stat. Paris, 44 (2000) 3-12. [7] .J . BoSGIRAUD , "Tests statistiques non standard pour les modeles a rap­ port de vraisemblance monotone ( premiere partie ) ", Pub. Inst. Stat . Paris, 44 (2000) 87-101 [8] J . BoSGIRAUD, "Tests statistiques non standard pour les modeles a rap­ port de vraisemblance monotone ( seconde partie ) ", Pub. Inst. Stat. Paris, 45 (200 1) 61-78. [9] .J . BoSGIRAUD , "Nonstandard chi-squared test", Journal of Information & Optimization Sciences, 26 (2005) 443-470 . [10] L . D . B ROWN, Fundamentals of Statistical Exponential Families, Institute of Mathematical Statistics, Hayward, California, 1986. [11] F. DIENER and G. REEB, Analyse non standard, Hermann, Paris, 1989. [12] P. G ROENEBOOM and .J . OoSTERHOFF, "Bahadur efficiency and proba­ bilities of large deviations", Statistica N eerlandica, 3 1 ( 1977) 1-24.

References

169

[13] W . C . M . KALLENBERG , Asymptotic Optimality of likehood ratio tests in exponential families, Mathematisch Centrum, Amsterdam, 1978. [14]

F. KouDJETI and I. VAN DEN B ERG, N eutrices, external numbers and external calculus, in F. et M. Diener editors: Non Standard Analysis in Practice, Springer-Verlag, Universitext, 1995.

[15]

Radically Elementary Pr-obability Theory, Princeton Univer­ sity Press, 1987. E . NELSON ,

[16] S . R . S . VARADHAN , Large Deviations and Applications, SIAM, Philadel­ phia, 1984.

A finitary approach for the representation of the infinitesimal generator of a markovian sem1group Scherazade Benhabib *

Abstract Ill, where the central question was: under suitab le regularity condit-ions, what is t he form of the infinitesimal generator of a Markov semigroup?

This work is based on Nelson's paper

the elementary approach using 1ST [2] , the idea is to replace the ![{ with a finite state space X possibly containing an unlimited number of points. The topology on X arises

In

continuous state space, such as naturally

from

the

probability theory.

For :1: E X, let. 'I.r.

be

the set

of all

h E M vanishing at x where M is the multiplier algebra of the doma.iu 'D of the iufiui tcsimal generator. To desaihe the st.rneLm:e of the scmigroup

l::y EX\{x} a(a; , y) h(y) so that of the points far from x appears separately. A defini tion of the quantity O:ah ( x ) = l:yEF a.(;t, y) h(y) is given using the least upper bo und of the stun..<; on all intemal sets T-V included in the external set F. Tl:ris leads to the characterization of the

generator A, we want

to

split Ah(a;)

the contribution of the external set Fx

global part of

11.1

=

the infiuitesimal generator.

Introduction

Let X be a finite state space possibly containing an uulimit.ed unrnbcr N of points. For all x in X, let a(x, y ) be a real number for each y on X such that a(x, y) is posi tive if y -=/=- x and L y a(x, y) = 0. Then A = (a(x, y)) is a finite N x N matrix, with positive off- diagonal elements and row sums 0. L n t'��" exists and is a markovian semigroup with infinitesimal Thus p t generator A. =

•EIGSI, 17041 La Rochelle, France. scherazade .benhabib@eigs i . fr

11.1. Introduction

171

Then we can define externally the domain of definition of A and its multi­ plier algebra as follows. Definition 1 1 . 1 . 1

A is the external set

The domain D of definition of the infinitesimal generator

D = { f E � X / f limited and Af limited}

(11.1.1)

And its multiplier algebra is the external set M = { f E D/ Vg E D fg E D}

(11. 1.2)

The topology on X arises naturally from the probability theory and a proximity relation is defined on X, more particulary from the way the functions in the multiplier algebra M of the domain D of the infinitesimal generator act on X. Definition 1 1 . 1 .2

and only if

Let u and v be in X, we say that u is close to v (u Vh E M h(u)

�

�

v) if

h(v)

Thus, the elements of M have a macroscopic property of continuity for­ mally analogous to the S-continuity. With this relation, we define the external sets :

�

Cx

=

X}

(11. 1.3)

Fx

= {y E X y rf. X}

( 11.1.4)

{y E X y

of points close to x and :

of those far from x. For each x in X, (a(x, y)) are positive real numbers if x -/=- y and can take unlimited values. Let Ix be the set of all h in M vanishing at x, and for h in Ix let us define ( 11.1.5 ) Ah(x) = L a(x, y) h (y) yE X \{x}

To describe the structure of the semigroup generator, we want to split Ah(x) so that the contribution of the points far from x in X. appears separately. One is tempted to define directly the quantity

aa h (x) = L a(x, y)h (y) yEFx

( 11.1.6 )

11. The infinitesimal generator of a markovian semigroup

172

which alas has no meaning because of the external character of the set of indices F:r Nevertheless, we will show in Section 1 1.2 how we can attach a definite meaning to it. That leads in Section 1 1 .3 to a definition for an ax-integrable function. Theorem 11.3.1 establishes the representation of Ah(x) for functions in M which have a zero at x of the third order. This is the global or integral part of A. Theorem 11.3.2 and its corollary extend the notion of ax-integrable to functions which have a zero of the first or the second order. 11.2

Construction of the least upper bound of sums in 1ST

Let E be an external subset of X and f a positive function defined on X. If there exists a standard natural number no such that for all internal sets W contained in E we have

Lemma 1 1 . 2 . 1

L f(y) :S: no

(11.2.1)

yEW

then there exists a unique up to an infinitesimal least number a( ! ) such that

L f(y) � a( ! )

(11.2.2)

yEW

for all internal sets W contained in E. Proof. Let no be a standard natural number such that for all internal sets W contained in E we have ( 11.2.1). By external induction, there is a least no such that (1 1.2.1) holds. Now for each standard natural k there is a largest natural number j with 0 :S: j :S: 2 k such that for all such W (11.2.3) L f(y) :S: no - k yEW

1

Let a(k) no - ik . Then for each standard k we have a standard a ( k ) , so there is a standard sequence k -+ a( k) . This sequence is decreasing, therefore it has a standard limit a (f) . Moreover, for all standard k and all such W , D ( a (f) - Ly Ew f(y)) :S: 21k · This proof uses very elementary tools. It can easily be formulated in the minimal non-standard analysis introduced by Nelson in [3] in which "standard" =

1 1.2. Construction of the least upper bound of sums in 1ST

1 73

a(O)

Step 0:

no - 1 Step 1:

no - 1

/,� i)\ : : J)\ n

(2

Step 2:

no - 1

no a(1) no a(2) no

Figure 11.1 applies only to natural numbers. On the contrary, if all the force of 1ST is used, one could check that s a (f) sup x E �+ ; 3 intw C E L f(y) :S x

=

{

yEW

}·

Figure 11.1 exhibits the way the sequence is built. Definition 1 1 . 2 . 1 1.

If f is a positive function defined on X, and if the conditions required for Lemma 1 1 . 2. 1 are verified the quantity (11.2.4) af = L f(y) yEE

is defined to be the standard limit a (f) . 2. If the conditions for Lemma 1 1 . 2. 1 are not verified, let af be the formal symbol x and say that (1 1 . 2. 2) is not defined. 3. In the general case, let f f + - f - be the decomposition of f into the difference of two positive functions. Say that af is defined in case both aJ + and a1- are defined, and let a f be their difference:

=

Notation 1 1 . 2 . 1

of af .

For each x in X, if f depends on x, write aJ (x) instead

11. The infinitesimal generator of a markovian semigroup

174

11.3

The global part of the infinitesimal generator

For x E X and h E M , let f = a(x, ·)h, then h is said to be ax-integrable if and only if aJ (x) is defined.

Definition 1 1 . 3 . 1

Notation 1 1 . 3 . 1

In that case write aa h (x) instead of aJ (x) .

Definition 1 1 .3.2 1.

h is a function in Ix if and only if h E M and h(x)

2. h is a function in I� if and only if h E M and h limited and eb gk ·in Ix .

= 0.

=

I:f=l e k gk with p

3. h is a function in I� if and only if h E M and is of the form h Ll e k fk gk with e k , fk , gk E Ix and p limited. 4. We say that h has a zero at x of the first order when h E Ix, of the second order when h E I�, of the third order when h E I� . If h has a zero at x of the third order, then h is ax-integrable and Ah ( x) is infinitely close to aah ( x) .

Theorem 1 1 . 3 . 1

Proof.

The proof goes through the following three steps:

1 . Let h be in I� , therefore it is of the form h and fk , gk in Ix .

=

I:i f� gk with q limited

,

2. I f f g E I.:r then P g is ax-integrable and so is h. These steps are proved as follows:

1. Let h E I�, as 1 1 1 efg = 4 (e + f) 2 g - 2 e 2 g - 2 f 2g, therefore h is of the mentioned form.

I L a(x, y)f2 (y)g+ (y) I w

:::;

l l g i i A(f2 ) (x)

1 1.3. The global part of the infinitesimal generator

175

As l l g l l and A(f 2 ) are limited, they are both infinitely close to a standard number. Taking the integer part of o (l lgi iA(f2 ) (x)) plus 1, we get the existence of the natural number no required by Lemma 11.2.1. The same argument applies to f 2g - . Thus aa( J2 g + ) (x ) and aa(j 2 g-) (x ) exist and as O!a( J 2 g) (x ) = aa( J 2 g + ) (x ) - aa(j 2g-) (x ) , then f2 g is ax­ integrable, and so is h. 3. Let (3 » 0. Since g E Ix there is an internal set W of Fx such that g + (y) :::; (3 for y E W 0 . We have

I A(f2 g + ) (x) - aa( J 2g+) I

=

IL

yE X \{x}

a (x , y) f2 (y)g + (y) - aa( J 2g + )

I

L a (x , y)f2 (y)g+ (y) - L a (x , y)f2 (y)g+ (y) L L a(x, y)f2 (y) :::; (3 L a (x , y) f 2 (y)

<

yEW

yE X \{x}

yEWr"\{x}

:::; {3

=

yEW0\{x} yE X \{x}

(3A(f2 ) (x)

Since A(f2 ) is limited and (3 » 0 is arbitrary, A(f 2 g + ) aa( J 2 g + ) (x ) . The same applies to f 2 g - . Then A(f 2 g - ) (x) ::: aa( J2 g-) (x) and aa( J2 g) = aa( J2 g + ) (x) + aa( J2 g-) (x) . D Thus A(f2 g) (x) ::: aa( J 2g) (x ) . Therefore A(h) (x) ::: a(ah) (x ) . :::

If h zs positive and has a zero at x of the first order, then h is ax-integrable and ah is less or infinitely close to Ah(x) . Theorem 1 1 .3.2 Proof.

Let W be any internal subset of Fx and h a positive function in I� , so

0 :::;

L a (x, y)h (y) :::; A(h (x w

)

)

As A (h) (x) is limited, we argue as in Theorem 11.3.1 and apply Lemma 1 1.2.1. One concludes that h is ax-integrable. As the inequality above holds for all internal subset W of Fx , we get that aah (x ) is less or infinitely close to A(h) (x) . D

176

11. The infinitesimal generator of a markovian semigroup

Corollary 1 1 . 3 . 1

If h has a zero at

integrable.

x

of the second order, then h is ax­

Let h be a function in I� , as fg = -41 (e + g) 2 - -41 (e - g) 2 h is also of the form h = Lk = l Pk ff with Pk = ±1, q limited and f'f in I"} . Consequently we can apply Theorem 1 1 .3.2 and h is ax-integrable. D

Proof.

1 1 .4

Remarks

The next goal will be to find an 1ST construction of the pure diffusion part of the generator of the semi-group and to complete a work already in progress concerning the rescaling of the time parameter for the markovian semigroup pt .

I would like to thank Professor E. Nelson for the initial suggestion of this problem and numerous enlightening discussions. I would also like to thank Professor G. Wallet of the University of la Rochelle for his support.

Acknowledgment

References

[1]

E. NELSON, " Representation of a Markovian Semi-group and Its Infinitesi­ mal Generator", Journal of Mathematics and Mechanics, 7 ( 1958) 997-988.

[2]

E. NELSON, "Internal Set Theory: a new approach to nonstandard analy­ sis", Bulletin Amer. Math. Soc., 83 ( 1977) 1165-1 198.

[3]

E. NELSON,

[4]

Press, 1987 E.

B.

Radically Elementary Probability Theory, Princeton University

DYNKIN,

Markov Processes, Springer-Verlag, 2 vols .. 1965.

[5] W. FELLER, An Introduction to Pmbability Theory and Its Applications, John Wiley & Sons. [6] I . VAN DEN BERG, Non Standard Asymptotic Analysis, Lectures Notes in Mathematics, 1249 , Springer-Verlag, 1987.

On two recent applications of nonstandard analysis to the theory of financial markets * Frederik S. Herzberg

Abstract Suitable notions of "unfairness" that measure how far an empirical dis­ counted asset price process is fi·om being a martingale arc introduced for complete and incomplete-market settings. Several limit processes are i nvolved each tirue, prompting a nonfltandard approach to the this concept. This leads to

( rather

traded

than a

011

martingale

several stock

a

analysi

s of

existence result for a "fairest price measure"

measure ) for an

asset

exchanges. This approach

describing the impact of

12 . 1

an

that is simultaneously also proves useful when

currency tr ansaction tax.

Introduction

Nonstandard analysis is most successful when it concerns itself with

ematical concepts that. i nvolve several limit

math­ p ro ecss cs in a non-trivial way. One

such exampl e is the quantification of a stochastic process's distance from being a martingale.

between the existence of a martingale mea..sure for a discounted asset price process and the non-existence of atbitrage opportunities, there is a natural economic interpretation to the compru·ison of two positive stochastic processes in regard to "how fru·" they actually ru·e from being a martingale. Vole will l'encler two economic questions related to such­ conceived "unfairness" mathematical : (1) Is there a "fairest price measure" (rather than a m�u·tingale measure ) for an asset that is sinmltaneously traded at mul t iple stock exchanges? (2) Does a currency transaction tax imposed on a currency that is sub j ect to herd behaviour among traders have a fairnessGiven the well-known correspondence

· Mathematical Institute, University of Oxford, Oxford OXl :3LB, United Kingdom. herzberg@maths . ox . ac . uk / herzberg@wiener . iam . uni -bonn . de

12.

178

Applications to the theory of financial markets

enhancing impact on the currency exchange rates preceding a financial crash? Both questions will be shown to allow for an affirmative answer. Techniques from nonstandard analysis have already been successfully ap­ plied to mathematical finance in the work of Cutland, Kopp and Willinger [5] . This contribution is based on two more recent papers by the author [8, 9] . 12.2

A fair price for a multiply traded asset

In this Section, we will en pass ant motivate notions of "unfairness" for both complete-market as well as incomplete-market settings. Theorem 12.2.1 Let n E N, p > 0 and c, N > 0 . Consider an adapted probability space (r, Q, IP') and B[O, 1 ] ® Y1-measurable geometric Brownian mo­ tions with constant multiplicative drift g� : r x [0, 1 ] �d·n , i E { 1 , . . . , n}. Then there exist processes gi , i E { 1, . . . , n} on an adapted probability space (D, F, CQ) , equ.ivalent to the processes g� on r (in the sense of adapted equiv­ alence [7}), such that there is a probability measure Mm on n in the class of measur-e s Q probability measure, VA E F1 iJ · CQ(A) :S: Q(A) :S: N CQ(A) , > vz .../.r. J. E { 1 , . . . , n } 0 I( ) I s c minimising ____,

1 1 CovQ((g,)s, (gJ)Jd .t.Q 9t9J s

}

·

w·

_

�

in the class of Loeb extensions of finitely additive measures from (*C) (D, G) (G being an arbitrary lifting of g). Analogously, there is a probability measure Mn minimising

(where nn ( Q, g) is defined to be +oo if the derivative in 0 in this definition does not a.s. exist as a continuous function in t) in the class of Loeb extensions of finitely additive measures from (*C) (D, G) . Moreover, inf, mn (·, g) :S: inf mr ( , g) C(D g )

as well as

C(r,g)

·

inf nr (·, g ) . inf nn (-, g) :S: C(r,g)

C(D,g)

12.2. A fair price for a multiply traded asset

1 79

The original paper [8} does not state the Theorem and the subs equent Lemmas as precisely as it is done here, although the exact result becomes clear from the proofs in that paper.

Remark 12.2. 1

The proof for this Theorem can be split into the following Lemmas which might also be interesting in their own right.

Using the notation of the previous Theorem hyperfinite adapted space n [12},

Lemma 12.2.1

12. 2. 1,

for any

inf mn (· , g) :::; inf mr (·. g) . C (r,g)

C(D.g)

Under the asswnptions of Theorem 12.2. 1 and choosing n to be any hyperfinite adapted space, the infimum of mn (- , g) in the class of Loeb extensions of finitely additive measures from (*C) (D, G) is attained by some measure Mm E C(D. g) . Proofs for both of these Lemmas can be found in a recent paper by the author [8] . Although they look very similar, it is technically slightly more demanding to prove the following two Lemmas ( which in turn obviously entail the second half of the Theorem, i.e. the assertions concerned with the map n) . Lemma 12.2.2

Using the notation of the previous Theorem hyperfinite adapted space n [12},

Lemma 12.2.3

12. 2. 1 ,

for any

nr (· , g ) . inf nn (· , g) :::; C inf (r,g)

C(D.g)

Under the assumptions of Theorem 12.2. 1 and choosing n to be any hyperfinite adapted space, the infimum of nn (·, g) on C(D, g) is attained by some measure Mn E C(D, g) .

Lemma 12.2.4

Easy results are Lemma 12.2.5 A semimartingale X is a ?-martingale on n if and only if 1 mn (P, x ) = 0 . The function mn (P, · ) v on the space of measurable processes of n satisfies the triangle inequality and is 1-homogeneous. For p = 2, it defines

an inner product on the space [.

:=

{x : n

x

[0, 1]

--+

�d

:

x

measurable, rn(P, x )

which becomes a HilbeTt space by this constr-uction.

<

+oo }

12. Applications to the theory of financial markets

180

A semimartingale X is a ?-martingale on n if and only if The function nn (P, ·) on the space of measurable processes of n remains unchanged when multiplying the argument by constants (scaling invariance) .

Lemma 12.2.6 nn (P, x) = 0.

We can generalise this to the following Definition 12.2.1

Let D be an adapted probability space. Define

£ (Sl, �d ) :

:=

{ X : Sl

X

[0, 1] --+ �d :

X measurable } .

A function Y £ ( n, �d ) --+ [0, +oo] is an incomplete-market notion of un­ fairness if and only if it satisfies the triangle inequality, is 1-homogeneous, and assigns 0 to a semimartingale y if and only if y is a martingale. Y is said to be a complete-market notion of unfairness if and only if it remains unchanged under multiplication by constants and Y vanishes exactly for those semimartin­ gales that are in fact martingales. 1

By Lemmas 12.2.5 and 12.2.6 respectively, the function mn (Q, - ) P (for a fixed probability measure Q) is an example for an incomplete-market notion of unfairness, whereas similarly nn (Q: - ) , again for a fixed probability measure Q, is an example for a complete-market notion of unfairness.

The distinction between notions of unfairness for complete and incomplete mar-kets can be justified by the following reasoning: If it is, under assumption of completeness, possible to buy as much of an asset as one intends to, multiplication of the discounted price process by a constant does not enhance the arising arbitrage opportunities at all; therefore, for complete markets, a suitable notion of unfairness should be scaling invariant in that it does not change under multiplication of the argument - which is conceived as being a discounted price process - by constants.

Remark 12.2.2

12.3

Fairness-enhancing effects o f a currency transaction tax

In this Section, we shall analyse the empirical price of an asset that is subject to herd-behaviour preceding a financial crash and how the resulting distortion can be mitigated through imposing a transaction tax. This is of particular relevance if the crashing asset in question is a currency, given the massive (usually destabilising and therefore adverse) macroeconomic conse­ quences of such financial crises.

12.4. How to minimize "unfairness"

181

We shall restrict our attention to a model of currency prices where extrap­ olation from the observed behaviour of other traders, up to white noise and constant inflation, completely accounts for the evolution of the currency price. This is to say that, in analogy to a Nash equilibrium, we assume that every agent is acting in such a manner that he gains most if all other agents follow his pattern. In our model this pattern will consist in using some average value of past currency prices as a "sunspot", that is a proxy for a general perception that depreciation or appreciation of the particular currency in question is due. In order to introduce the model for the pre-crash discounted currency price, consider a � ----+ � ' a piecewise constant function, such that a = a(1) > 0 on (O, + x) a = a(- 1) < 0 on (-oo, O). We assume that the logarithmic discounted price process x ( v ) is governed, given some initial condition that without loss of generality can be taken to be x ( v ) 0, by the stochastic differential equation :

=

(

x �v ) = a x �v ) - L Pu . x ((�2 u ) vo uEI

d

J2

dt + J . dbt - 2 '

)

X{

(v ) _ '>'

I xt

. (v)

dt I } (12.3. 1)

L.. n E I Pn x ( t- n)VO ;:o:v

where r > 0 is the logarithmic discount rate (assumed to be constant) , b is the one-dimensional Wiener process, (Pu)uEI is a convex combination - i.e. I C (0, + oo) is finite, Vu E I Pu > 0 and L uEI Pu = 1. The parameter v depends very much on the tax rate we assume. If p is the logarithmic tax rate and T the expected time during which one will hold the asset (that is the expected duration of the upward or downward tendency of the stock price) , one can compute v as follows: v = T · p. The equation (12.3.1), is of course first of all only a formal equation that thanks to the boundedness of a can be made rigorous using the theory of stochastic differential equations developed by Hoover and Perkins [11] or Albeverio et al. [2] for instance. We will introduce the following abbreviation: n;,(v) - X{ 1 · 1 2: v } a . 'f/ -

·-

.

12.4

,

How to minimize "unfairness"

Intuitively, the map nn (P, · ) (for a fixed probability measure P on a proba­ bility space D) when applied to a discounted asset price process measures how

12. Applications to the theory of financial markets

182

often ( in terms of time and probability ) and how much it will be the case that one may expect to obtain a multiple ( or a fraction) of one's portfolio simply by selling or buying the stock under consideration. For the following, we will drop the first component of n ( indicating the probability measure ) if no ambiguity can arise; for internal hyperfinite adapted spaces and Loeb hyperfinite adapted spaces, we will assume that the canon­ ical measure ( the internal uniform counting measure and its Loeb measure, respectively) on the space is referred to. First of all we derive a formula that will make explicitly computing n easier in our specific setting: Lemma 12.4.1 If x ( v ) satisfies (12. 3. 1) for some v > 0 on an adapted prob­ ability space (r, g , CQ), then the discounted price process exp x� v) : t ::0: 0)

( ( )

is of finite unfairness. More specifically,

( ( ))

nr exp xl"l

[ IE [ 1 ,;,1" (x;"1 fr p;x:;1 i)vo) I] dt [ IE [ I d: l .,�o IE [ xi�¥,] + �' I ] dt I

�

-

The proof is more or less a formal calculation, provided one is aware of the path-continuity of our process and the fact that the filtrations generated by b and x(v ) are identical. For this implies that, given t > 0, the value Pro of.

'1/J ( v )

(

)

( ) ( v)u - ""' xt+ 6 Pi X ( t+ u -� ) vo ( w ) i EI v

does not change within sufficiently small times u - almost surely for all those paths w where x�v ) (w) - L i E I p� x/��� ) vo(w) tf. { ±v }, this condition itself being satisfied with probability 1 . Now, using this result and the martingale property of the quotient of the exponential Brownian motion and its exponential bracket, we can deduce that for all t > 0 almost surely:

How to minimize "unfairness"

12.4.

183

Analogously, one may prove the second equation in the Lemma: for, one readily has almost surely

In order to proceed from these pointwise almost sure equations to the assertion of the Theorem, one will apply Lebesgue's Dominated Convergence Theorem, yielding

For finite hyperfinite adapted probability spaces, an elementary proof for the main Theorem of this Section can be contrived: Lemma 12.4.2 For any hyperfinite number H we will let for all v > 0 denote the solution to the hyperfinite inztial value problem

x6v) { x ( v ) - x (v ) - � } (x(v) - "" . x(v) ) X{

\it E

=

0, 0, . . . , 1 t

t+ J, -

x (v)

a

!

-

t

+ J . 1ft+ J,

6 Pu

uEI .

( t-u) VO

2 1 J 1 (2H ! ) 1/'2 2 H !

I

(v )

xt

-

(u)

L Pu ·X (t- u )V O

uEI

I

?: v

}

.� H!

(1 2.4. 1)

12.

184

Applications to the theory of financial markets

(where 1'1£jH ! : n { ± 1 } H ' --+ { ± 1 } is for all hyperfinite e <::.:: H! the projection to the f!-th coordinate) which is just the hyperfinite analogue to (1 2. 3. 1) . Using this notation, and considering a hyperfinite adapted probability space of mesh size H!, one has for all k < H!, =

""' IE Xk(v)/H ! - IE [X (v ) I FkjH !] - 2J2 1 H! -"-±f

<::; 6

If

k
for all 2: v . As a consequence, m (xO! n) is monotonely decreasing for all hyperfinite adapted probability spaces Sl = { ±1 } H ! . Pro of of Lemma 12.4.2. It suffices to prove the result for finite ( rather than merely hypcrfinitc) adapted spaces. By transfer to the nonstandard universe, we will obtain the same result for infinite hyperfinite H as well. The proof of this Lemma for finite H relics on exploiting the assumption that a is piecewise constant, since a = a ( 1 )x(O,+=) + a ( - 1 )x(-=.O) + a (O)x{o} yields Vk < H! Vv > 0 T

[ (v) I J v) (

J2 1

IE X kJ} Fk/ H ! - X J'r, + 2 H!

v

(

- �a x( l - 6 ""' pu kj H ! H!

_

1

_

- H!

(

·

u EI

a(1)x

{

(v

X ( �,l -u)v o

(c, · J

c'·l

) {l

xk/H ' - L Pu·X .J£. uEI HI

+ a( - 1 ) x

{

C uJ

xk/H' - L Pu ·X uEl

u

x

1t

) VO

(C kJ - u ) JTf

(v) xk/H'

;:::v }

VO

::; -v

" p " L-

uEl

)} '

.x (v )

k - u) V O (w

l>-v}

which immediately follows from the construction of Anderson's random walk [3] B. = v�Jr. H, and the recursive difference equation defining the process x(v) . However,2 this last equation implies Vk < H! Vv > 0

I

[ (v) I ] (v)

J2 1

IE IE X k}i,1 Fk/ H ! - X J'r, + 2 H! 1

= -

H!

a ( 1)X

IE

{

(v )

I

((v)

:.:C: v } s-v } )

xk/ lf ' - L Pu ·X k m u) V O uEl

+ a(-1)x

{

X

��� , - uLE I Pu X ((vm� - u

VO

12.4. How to minimize "unfairness"

185

} "'"' IP' "'"' { 6 XkjH! - 6 Pu · X �, -u)vo ? v

Now all that remains to be shown is that (v)

k< H! and

(v)

(

uEI

"'"' IP' { XkjH! - "'"' 6 Pu . X );, -u)vo 6 (v)

k< H!

(v)

(

uEI

<::;

-v

}

are monotonely decreasing in v. The former statement is a consequence of the following assertion:

Vv' ::; v \If! < H!

IP' { xt�! - L Pu . xt�, -u)vo v } L k�R uEI "'"' "'"' Pu . X --", -u)vo 6 1P' { Xk/ H ' - 6 '2

H

<::::

(v' )

k�R

(v')

(H

uEI

2::

v

1

}

(12.4. 2)

One can prove this estimate by considering the minimal f! such that the point­ wise inequality

L k
(v)

(v)

xk/H' - L Pu ·X ( _k_ ;:o: v uEI HI u V O

)

} ::; L k
(v')

(v')

::?: v xk f H' - L Pu · X ( _b_ HI u VO uE I

)

fails to hold. Then one has an w E S1 such that Vk < f! (w ) (vJ CvJ

X{

xk/H ' - L Pu · X ( k uEI HT - u

= X{

1 0

(v')

)

vo

::?:v

}

(v' )

xk/H' - L Pu · X ( k ::?: w u VO v uEI

X{ = X{

(v )

(v)

(v' )

(v')

)

' } (w) ,

)

::?:v xf! H ' - L Pu · X ( £ uEI HI u VO

)

'}

::?: v XR f H' - L Pu ·X ( £ uEI HI u VO

(12.4.4 )

} (w) ,

( 12.4.5 ) ( 12.4.6)

) ' } (w .

But equation ( 12.4.4 ) implies, via the difference equation for inductively in k the relation Vk < f! Xf(w ) = X;;:' (w ) .

(12.4.3)

x ()

(12.4. 1),

12.

186

Applications to the theory of financial markets

If one combines this with

(which is equation (12.4.5)) and v ::; v' , one can derive - again via the recur­ sive difference equation (12.4. 1) - that x , :::: x ! applied in t = �! as well as the estimates

i;2

X { XR/H' - L (v ' )

uEI

Pu · X

(v ' ) k

( H!

-u

) VO

2:v ' } (w)

> >

X{ X{ X{

i;1

(v ' )

xf/H' - L Pu · X uEI

(v )

xf/ H ' - L Pu X uEI

(v )

·

xt/ H ' - L Pu · X uE l

1.

This contradicts equation (12.4.6) . Hence, the estimate tablished for all k < H!, leading to (12.4.2). S imilarly, one can prove

(v )

(t

w -u

(v )

(f

w-u

(v )

w_ - u

(t

)VO ) VO ) VO

(12.4.3)

2:v ' } (w) 2:v' } (w) 2:v } (w)

has been es­

(v) - L Pu · X (v ), - O ::; -v } L IP' { Xk/H' ( H u)V k�R uEI "" IP' { Xk/H' "" Pu . X((v')l,) -u)vo ::; ' } (v' ) - L._., ::; L._., k�R uEI which entails that Lk < H! IP' { Xk/�! - 'I:, Pu · X (v1 _ u)vo ::; - v } must be monouEI ( H ! Vv' ::; v W <

H!

·

k

-v

tonely decreasing in v.

D

Using nonstandard analysis and the model theory of stochastic processes as developed by Keisler and others [10, 12, 7] , we can prove the following result:

Suppose (y(v) : v > 0 ) is a family of stochastic processes on an adapted probability space r such that y(v) solves the stochastic differential equation (12. 3. 1) formulated above for all v > 0. Then the function J f---+ nr (y(" l ) attains its minim·um on [0, S] in S.

Theorem 12.4.1

Pro of. B y the previous Lemmas 12.4.2 and the formula for n from Lemma 12.4. 1, the assertion of the Theorem holds true internally for hyperfi­ nite adapted spaces r, if we replace y(o-) by a lifting y(o-) and if we let nn when applied to internal processes denote the hyperfinite analogue of the standard nn . Now, according to results by Hoover and Perkins [11] as well as Albeverio

187

References

et al. [2] , the solution x(v ) of the hyperfinite initial value problem (12.4.1) is a lifting for the solution x(v ) of ( 12.3.1) on a hyperfinite adapted space for any v 2: 0. Now, y c---+ nn ( y ) is the expectation of a conditional process in the sense of Fajardo and Keisler [7] . Therefore, due to the Adapted Lifting Theorem [7] , we must have Vv 2: 0 °nn x(v ) = nn ( x(v ) )

( )

(where we identify n with its internal analogue when applied to internal pro­ cesses) . Since the internal equivalent of the Theorem's assertion holds for internal hyperfinite adapted space, the previous equation implies that it is also true for Loeb hyperfinite adapted spaces. Now let (y (v ) : v > 0) be a family of processes on some (not necessarily hy­ perfinite) adapted probability space r with the properties as in the Theorem . Because o f the universality o f hyperfinite adapted spaces [7, 12] , we will find a process x (v ) on any hyperfinite adapted space n such that x and y are automor­ phic to each other. This implies [2] that x ( v ) satisfies ( 12.3.1) as well. Further­ more, as one can easily see using Lebesgue's Dominated Convergence Theorem, Vv

>

0 nn ( x ( v) )

=

nr ( y ( v ) ) .

Due to our previous remarks on the solutions of (12.3.1) on hyperfinite adapted spaces, this suffices to prove the Theorem. D Acknowledgements. The author gratefully acknowledges funding from the German National Academic Foundation ( Studienstiftung des deutschen Volkes ) and a pre-doctoral research grant of the German Academic Exchange Service ( Doktorandenstipendi1Lm des Dwtschen Akademischen A 1Lstauschdienstes ) . He is also indebted to his advisor, Professor Sergio Albeverio, as well as an anony­ mous referee for helpful comments on the first version of this paper.

References

[1]

S . ALBEVERIO, Some personal remarks on nonstandard analysis in prob­ ability theory and mathematical physics, in Prnceedings of the V!Ilth In­ ternational Congress on Mathematical Physics - Marseille 1986, eds. M . Mebkhout , R. Seneor, World Scientific, Singapore, 1987.

[2]

S . ALBEVERIO , J. FENSTAD , R. H0EGH-KROHN and T . LINDSTR0M ,

Nonstandard methods in stochastic analysis and mathematical physics, 1986.

Academic Press, Orlando,

[3]

R. M . ANDERSON, "A nonstandard representation for Brownian motion and Ito integration", Israel .Journal of Mathematics. 25 (1976) 15-46.

188

12. Applications to the theory of financial markets

[4] A. CoRCOS ET AL . , "Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos", Quantitative Finance, 2 (2002) 264-281. [5] N. CUTLAND, E . KoPP and W . WILLINGER, "A nonstandard treatment of options driven by Poisson processes", Stochastics and Stochastics Re­ ports, 42 ( 1 993) 1 1 5- 133. [6] l\1 . D REHMANN, .J . 0ECHSSLER and A . RoiDER, Herding with and with­ out payoff externalities - an internet experiment , mimoe 2004. [7] S . FAJARDO and H. J. KEISLER, Model theory of stochastic Lecture Notes in Logic 14, A. K Peters, Natick (Mass . ) , 2002.

processes,

[8] F . S . HERZBERG, "The fairest price of an asset in an environment of tem­ porary arbitrage", International Journal of Pure and Applied Mathemat­ ics, 18 (2005) 121 - 131. [9] F . S . HERZBERG, On measures of unfairness and an optimal currency transaction tax, arXiv Preprint math. PR/0410543. http : //www . arxiv . org/pdf/math . PR/04 10543 [10] D . N . HOOVER and H . .J . KEISLER, "Adapted probability distributions", Transactions of the American Mathematical Society, 286 ( 1 984) 1 59-201. [ 1 1] D . N . HooVER and E . PERKINS, " Nonstandard construction of the stochastic integral and applications to stochastic differential equations I, II", Transactions of the American Mathematical Society, 275 ( 1 983) 1-58. [12] H. J. KEISLER, lnfinitesimals in probability theory, in Nonstandard anal­ ysis and its applications, ed. N. Cutland, London 1Iathematical Society Student Texts 10, Cambridge University Press, Cambridge, 1988. [13] P . A . LOEB , " Conversion from nonstandard to standard measure and ap­ plications in probablility theory", Transactions of the American Mathe­ matical Society, 2 1 1 ( 1 974) 1 13-122 . [14] E . NELSON, Radically elementary probability theory, Annals of Mathemat­ ics Studies 1 17. Princeton University Press, Princeton (NJ) , 1987. [15] E. PERKINS, Stochastic processes and nonstandard analysis , in Nonstan­ dard analysis-recent developments, ed. A.E. Hurd, Lecture Notes in Math­ ematics 983, Springer-Verlag, Berlin, 1983. [16] K . D . STROYAN and .J . M . B AYOD, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics 1 1 9, North­ Holland, Amsterdam, 1986.

Quantun1 Bernoulli exp eriments and quantum stochastic processes Manfred

\iVolff*

Based on a

W ...-algebraic

Abstract

approach to quantmn probability theory we

construct basic discret e in terna l quantum stochastic pro cesses with in­

d ep eiHlent increments. We ob tain Bernoulli

exp eriments

a one-parameter family of (classical)

as linear combinations of these basi c processes .

Then we use the nonstandard hull of the internal GN S-Hilbert space corresponding to the chosen state

T

( the

?-{,.

un derlying quantum probabil­

i ty measure) in o rder to derive nonstandard hulls of our internal pro­ cesses.

Finally continuity requirements lead to the specification of a

certain flUhf;pace £ of

nal pro cesses to

the

can

Hr

to which the nonstandard hulls of our inter­

be restricted and

wh ch

i

turns out to be isomorphic

Loeb-Guicltardet space intro d uced by Lci tz-Martini

space of £ the.n ifl shown to he isom orphic to

F+

(L2 ([0, 1] , >.))

1 101.

A sub­ space

flymmetric Fock

and om· basic processes agree with the processes of

Hudson and Pa.rtltasarathy on this

13. 1

t.he

subspace.

Introduction

In the early 1980's Hudson and Parthasarathy [5], Hudson and Lindsay [4] . and Hudson and Streater [6] started the theory of quantum stochastic pro­ cesses, in pcu·ticular of quantum Brownian motion. In [12[ Parthao:;arathy showed one way how to make the passage from quantum random walk to diffu­ sion. All theS(' topics are extensively treated in P.A. Meyer's compendium [ 1 1 [ . For another general introduction to this field see fl3] where an extensive mo­ tivation ti·om qua.ntum physics may be found. The Hudson-Part.hasarathy ap­ proach is based on t.he classical symmetric Fock space over L2 (R+ , >.) where ·Mathematisches Institut, Eberhard-Karls-Univ. Tiibingen, manfred . wolff�uni-tuebingen . de

190

13.

Quantum Bernoulli Experiments

A denotes the Lebesgue measure (see section 13.6 ) . Guichardet [3] gave an interesting representation of this Fock space as an L 2 -direct sum of spaces L� = L 2 (Xn) where Xn is the space of all subsets of the unit interval [0, 1] having exactly n elements. The measure on Xn is defined via the conesponding Lebesgue measure on �f- , This representation was extensively used by Maassen developing his kernel approach to general quantum stochastic processes. Journe [7] invented the so called toy Fock spaces (bebe Fock in French) as discrete approximations of the symmetric Fock space. Let Do = { 0, 1} be equipped with the uniform distribution f-LO · Then the toy Fock space of order n is the space L 2 (D0, p, � n) . A rigorous discrete approximation of the Guichardet space and the basic quantum stochastic processes defined on it which uses the N space L2 (D � , p,� ) as a toy Fock space is given by Attal [2] . There is another approach based also on the Guichardet space to prove such approximation theorems using nonstandard analysis which was developed by Leitz-Martini [10] . He discretized the Guichardet space in the following manner: let T = { � : k E *Z , 0 :::; k :::; N } be the hyperfinite time line where N E *N is infinitely large. Let r n be the set of all internal subsets of T of internal cardinality n and set r = u�=O rk . Now let ]1.1 be an internal subset of r and set where IAI denotes the internal cardinality of A. Then m is an internal measure on r with m(f) � e. The space L 2 (r, Lm) , where Lm denotes the Loeb measure associated to m, contains the Guichardet space as a Banach sublattice. In fact L 2 (r, Lm) is the nonstandard version of the Guichardct space, so to speak the Loeb- Guichardet space over the time interval [0, 1] , Among other things Leitz-Martini constructed all the relevant discrete quantum stochastic processes and showed that their nonstandard hulls exist in a precise manner and form the basic quantum stochastic processes: the time process, the creation and annihilation processes, and the number process. All approaches mentioned so far start with some kind of discrete approxi­ mation of the symmetric Fock space or the isomorphic Guichardet space over �+ , [0 , 1] respectively. In contrast to these approaches we present here a new one based on the more natural W* -algebraic foundation of quantum stochastic as described e.g. in [8] , In [13] the author sketched a different though similar approach in the finite-dimensional setting but he was not able to make the transition from the finite-dimensional to the continuous infinite-dimensional case. Our access is exactly the noncommutative version of Anderson's way [1] to Brownian mo­ tion. In particular we arc able to ju.s tify the usc of the symmetric Fock space (or equivalently the Guichardet space) in the context of quantum stochastic

191

13.2. Abstract quantum probability spaces

processes. We believe that by our approach the applications of nonstandard analysis as given by Leitz-Martini [ 10] are also better understandable. We begin with the discrete quantum Bernoulli experiment modelled in the von Neumann algebra of all 2 n x 2 n-matrices over CC. Then we prove a l\loivre-Laplace type theorem for quantum Bernoulli experiments, i.e. we show that these discrete stochastic processes converge to quantum Brownian motion in the same sense in which the classical approximation of the usual Brownian motion is treated by Anderson [ 1] . Furthermore we construct the basic quantum stochastic processes out of the corresponding discrete versions in the case of a vacuum state. Finally we show how the symmetric Fock space approach fits into our setting. Applications to the theory of stochastic processes are under preparation .

13 . 2

Abstract quantum probability spaces

Recall that almost all information about a given probability space (D, I:, JL ) is to be found in the pair (A, JL) with A = L= (n, I:, JL). Let us denote by [! ] the equivalence class modulo negligible functions in which the essentially f( w ) dJL( w ) = : bounded measurable function f is contained. Then [f] ----+ JL(f) defines a positive, linear, order continuous normalized functional on A. Analogously a quantum probability space is a pair (A, T) where A denotes a W*-algebra and T is a positive, linear order continuous normalized functional on A, or a normal state for short. The reader not familiar with the abstract theory of W*-algebras may look at them as subalgebras of the algebra of all bounded operators on an appropriate Hilbert space which arc closed under involution * : a ----+ a* and which are also closed with respect to the strong operator topology. In this context order continuity means the following: whenever a downward directed net (aa) of nonnegative selfadjoint operators from A converges to the operator 0 with respect to the strong operator topology then lima T(aa) = 0 holds. In particular one may interpret L= (n, l::JL ) as the W*-algebra of all multiplication operators MJ : g ----+ MJ (g) = fg (! E L= (n, I:, JL)) on the Hilbert space L 2 (D, I:, JL ) . Let (A, T) be a quantum probability space. Let A1 , A2 be W*-subalgebras generating the W*-subalgebra A3; moreover let Tk be the restriction of T to Ak (k = 1 , 2, 3). Then A1 and A2 are called stochastically independent if (A3 , T3) is isomorphic to (A1 ®A2 , Tl ®T2) where ® denotes the W*-tensor product .

fn

Remark: Let us point out that this notion of independence agrees with the usual one in the classical (commutative) case A = L= (n, I:, JL) but that

13. Quantum Bernoulli Experiments

192

there are other notions of independence in the quantum stochastic setting also generalizing the classical one (e.g. free independence) . Two probability spaces (A. T) and (A', T1) are called equivalent if there exists an order continuous algebraic isomorphism rp : A --+ A' with T1 = T o rp . Now let us construct the L 2 -space corresponding to the probability space (A, T). It is nothing else than the so called GNS-space (after Gelfand, Naimark, and Segal) . To this end we introduce the C-valued mapping (al b) = T(a*b) on A x A. It is scsquilincar, that means, it is linear in the second argument, antilinear in the first argument, and it satisfies (a l a) 2 0 for all a E A. The space LT = {a : (a I a) = 0} is a left ideal, and on A/ LT there is well-defined a scalar product by (a l b) = ( a l b) where a = a + LT denotes the equivalence class in which a is contained. 7-{T is the completion of A/ LT with respect to the associated scalar product norm 11 a11 = � = JT(a* a) . The representation JrT : A --+ J:(HT ) is given by JrT (a)(b + LT) = ab + LT which is well-defined since LT is a left ideal. The representation is injective whenever A is simple (i.e. has no nontrivial closed ideals) , e.g. A = Mn (C), the algebra of all n x n-matrices. In case A = L = (n, p,) the space 7-{T turns out indeed to be the space 2 L (D, p,) and in this way L = (n, p,) is represented as the algebra of operators of multiplication by L =-functions (see above) . 13 . 3

Quantum Bernoulli experiments

Let us begin with the easiest example of a quantum probability space: Let

A = M2 (C) be the W*-algcbra of all complex 2 x 2 matrices a = (a;k ) and choose A E] 1/2, 1] . Then T.A (a) = >-au + ( 1 - >.)a22 defines a normal state on A. As in the case of classical probability we will often write IE ,\ (a) in place of a.

T.A (a) , and we call it the expectation of Now consider the following model of tossing a fair coin: choose Ao as the commutative W*-algebra C { O . l } and let p,o : Ao --+ C be given by p,o(f) = � (! ( 0) + f ( 1)). The underlying classical probability space is to be rediscovered in Ao as the set of two idempotents 1 { o } and 1 { 1 } where 1 A denotes the indicator function of a subset A of a given set. Ao is spanned also by the elements 1r

We now construct all injective *-algebra homomorphisms : Ao --+ A satisfy­ ing T,\ o = f-LO · In the sense of section 13.2 this means that we will find all abstract probability spaces (B, T,\ 1 13 ) which arc equivalent to (Ao, p,o). An easy calculation shows that they are given by := with 1r

1r

1rz

-

z

1

) = : Qz

13.3.

Quantum Bernoulli experiments

and linear extension, where Pz - Q z =

( � 0 ) = : Xz .

z E lC

193

is a parameter with l z l

= 1. n2 (X) =

In summary we have seen that the simplest quantum probability space (A. T.\) contains a one-parameter family (Az , T.\ IAJ of models of classical coin tossing. For w = eis, z = e it ( -n < s, t :<:: n) the commutator [Xw , X2] . = XwXz ­ X2Xw is easily determined: [Xw , Xz] = 2 i

sin(s - t)

( � -� ) .

So Xw and X2 commute if w = z or w = -z. We obtain JE.\ ( [Xw , X2] ) = 2 i (2>. - 1) sin(s - t). The absolute value of it attains its maximum at - t = ±n /2. The pair (Xw , X2) with z = - iw is called a quantum coin tossing. It is unique up to automorphisms. More precisely this means the following: choose w = e it (t E] - n, n] ) , and consider the automorphism 1.{) on A given by I.{)( a) = u*au where 0 e- it/2 u. 0 e it/2 Then i.p(Xl ) = Xw ' i.p(X- i ) = x- iw and T.\ 1.{) = T.\ . So in the following we need only to consider the quantum coin tossing (X1 , X_ i ) . Notice that X1 = ax , X_i = ay and [Xw , X_,w]x = 2ia2 , where ax , a y, az denote the Pauli matrices. The following important matrices can be constructed by (ax , ay) : 1 0 1 . (13.3. 1 ) (ax + wy ) , b2 0 0 1 0 0 . (13.3. 2) b+ 2 (ax - wy ) , 1 0 1 0 0 ( 13.3.3) 1 - [ax , ay] · bo 0 1 2i Obviously the following formulae hold: s

_

(

( ( (

0

)

) ) )

ax = b+ + b - , ay = i · (b+ - b - ) , [b - , b+] = a2 = 1 - 2b0 , Xw = wb+ + wb - .

These formulae show that the random variables b+ and b- are in a certain sense more basic than Xw though they are not selfadjoint. As in the classical case we describe the experiment of n ( E N) independent coin tossings by the n-fold tensor product (A®n , T�n) . The algebra correspond­ ing to the k-th trial is described by the elements 1 129 · · • 129 1 ®x 129 1 129 · · · 129 1 = : X k '-v-" (k- 1 ) times ( n -k) times

'-v-"

194

13.

Quantum Bernoulli Experiments

where x E A is arbitrary. Describing k times repetition of our quantum coin tossing we obtain Sw. k := L;= l Xw1 . We have k [Sw ,k, s- iw, k] = 2i k·

La

J =l

z,

The pair (Sw, k l S- iw, k ) is called a quantum Bernoulli experiment. Setting Ak := A0 k ® 1 ® · · · ® 1 = {a ® 1 ® · · · ® 1 : a E A0 k } '-v-" '-v-" n - k times n - k times we obtain a natural filtering lC · 1 C A1 C · · · C An = A0n . Moreover there also exists a conditional expectation !Ek from An onto Ak which is given by IEk (a ® b) = T; ( n- k ) (b) · a , where b E A0 ( n- k ) , and linear extension.

13.4

The internal quantum processes

Now we consider a polysaturated model *V(JR:.) of the full structure V(JR:.) over JR:.. As usual we replace the standard integer above by an infinitely large integer N. Then all considerations in the previous section remain true for internal objects. We use the discrete time interval T = { � : 0 <:.:: k < N } denoting its elements by §., t, JJ . . . . Moreover we rescale also the family (Ak) by setting At- := ANt · Set p � JN · Then we introduce n

a±t_ a'l_ •

at

pbf , b'j , p

2

(0 �). 1

( 13.4. 1 ) ( 13.4.2) (1 3.4.3)

These elements are the internal increments by which we construct the basic internal stochastic processes A£ = L: o :S;�
( A i ) -t is called the internal creation process, (Ai )t the annihilation pro­ ( A! ) t the time process and finally (Ai)t th� �umber process. The internal Brownian motion corresponding to the parameter w E lC then is given by Bw,t wAt + wA_t ( l w l = 1 ) . It satisfies the internal difference equation dBw,t = w at+ + w at .

ce ss ,

13.4. The internal quantum processes

195

Like in the previous section we consider the state Tf: N obtain

T.\ ,N and we

JE.\,N ( Bw,t_ ) : = T,\,N ( Bw,t_ ) = 0, JE.\ ,N(B�,t. ) = t_.

(13.4.4)

The second equation follows from the facts that

((wa; wa-:;- ) 2 )

for §. # 11:, but T>-,N - + The commutator is

= T>-,N (1/N) =

1/N.

The so-called internal Ito-table for the increments is easily computed:

a• a+o a§. a;,

pa-2:a: I a0-; I a0-� I p2a-a:;; p2 a; 0 0 p2 a� 0 a+§. a§.o 0 0 a• a;, 0

p-tv =

A-� + lvi B-"-,t. + l v l 2 A-; lvl

s

s

§.

Let 0 # v E *IC be arbitrary. Then the process (P.�_) given by is called the Poisson Process with parameter v . Now we compute the characteristic operator functions of these pro­ cesses. Whenever V = (ifi)t_ is a stochastic process its characteristic oper­ ator function is defined by fv (u, t) = exp( i uvt), where u E *IF!:.. Obviously A ( t ) l u= O = iifi and fv ( u, t) is unitary iff Vi is selfadjoint . The processes V = (Vi) we considered so far are all of the form u,

Vi =

I: d§.,

O:S;§.
where

d8 = 1 Q9 · · • Q9 1 Q9 d Q9 1 Q9 · · · Q9 1 '-v--"' '-v--"' ' N13- 1 times N-N13 times and d E A is appropriately chosen. Since d'IJ,, dy_ commute for 11: # 1!. we obtain

fv (u, t)

=

@ (exp( i ud) ) 13• O:S:i3<.1.

13. Quantum Bernoulli Experiments

196

The most important formulas are the following ones:

@ (cos( pu)! + i sin(u)Xw).e_ O <;.s
13.5

From the internal to the standard world

The operator norm topology on A is too fine in order to obtain reason­ able nonstandard hulls. Moreover the processes discussed so far are internally bounded but not necessarily S-bounded. For example Bw, kjN is selfadjoint, hence its operator norm is equal to its spectral radius. This number is easily computed as )N, or in other words I I Bw.t. ll = VN · t In the following we only consider T;.. for >. = 1 . The other case will be treated elsewhere. We set T1 = : T and we construct the Hilbert space 7-{T corresponding to (A, T) (A the algebra of 2 2-matrices, see section 13.2) . To this end we denote the columns of the 2 2-matrix a by a = (ai , a�) . Then T(a* b) = (ai I bi ) where the latter is the canonical scalar product on IC 2 . Hence LT = {a : a � = 0} and nT (a) (b + LT) = abi + LT. In particular 7-{T S=' IC2 , where the isometric isomorphism is given by a + LT ---+ ai. We have previously introduced the random variable X1 ax . Its equiva­ lence class in 7-{T is denoted by xl = x l + LT We set eo = b�1 = ( 6 ) = xp and e 1 = b11 = ( ? ) = X1 . Now we consider TN = T® N Obviously 7-{TN S=' (*IC2)® N holds. Let w E {0, 1 }T be an arbitrary internal function. Then we define ew by e w = @�=[} e w ( k/N ) · The set {e w : w E {0, 1 V } forms an orthonormal basis in 7-{TN · 7-{TN is an internal Hilbert space, so its nonstandard hull if.TN is a Hilbert space. The formulae (13.3.1) up to ( 13.4.3) above lead to the following equations, which result from the fact that 7-{TN is the GNS-Hilbert space of (A N , TN ) · In order to make the formulae as simple as possible we denote the addition mod 2 by EB, i.e. we consider D = {0, 1 V as the cartesian product of the group (Z2 , EB). Then we set w + = 1r EB w and w - = w for w E {0, 1 }T . Moreover we write 1.t. in place of 1 {.1.} . x

x

=

13.5. From the internal to the standard world

197

With all theses conventions we obtain

w± (.t.) e

(13.5. 1)

w+ (.t.) ew - a - a + ew,

(13.5. 2)

VFJ wEB1t '

t t ----_;::;w (.t.) ew = N · ata-; ew. -

13. 5 . 1

(13.5.3)

Brownian motion

First of all we show that our internal Brownian motion ( Bw,t) leads to the standard Brownian motion on a subspace of HTN . To this end we recall that the standard Brownian motion (f3t) on L 2 ( D , L'") (with D = {0, 1}r, f-L f-LN f-L� N , L the corresponding Loeb measure) can be viewed as a family of multiplication operators f {3tf on L2 ( D L'") which are selfadjoint but unbounded (cf. the end of section 13.2) . From equation (13.4.4) it follows that Bw,t is S-bounded in 7-{N . We claim that classical Brownian motion (or better to say: Anderson ' s Brownian motion) is equivalent (in the sense of section 13.2) to our CBw , ot) where (Bw.ot) denotes the family of selfadjoint operators on a subspace of HTN coming from the family (Bw,t_). To this end let l w l = 1 and =

=

'"

f---+

,

Po = � 2

( -w

-w 1

l

).

For w E n we set Pw = @o-::;. k < N pw ( kjN) · It follows TN (Pw) = 2 - N = !-L({w}) . We denote the internal W*-algebra generated by {Pw : w E {0, 1}T} by B ( w ) . The formula p bw ,t .· - - - (P1 - - p1 ) w at+ + w at t - 2 t , shows that ( Bw t) L o < s < t bw.t is contained in B ( w). The internal L= -sp;-e -L= (n, f-L) is nothing else than L=({0, 1}, f-Lo) 0 N . Thus it is not very hard to see that 11N := 11� N (see section 13.2) maps L= (n, f-L) onto B(w) with TN 11N IB (w ) = f-L. Now let D = {f E L= (n, f-L) : f is S-bounded} . Then -

=

+

c

o

11N ( D ) C {a E AN : llall is S-bounded} C { x E 7-{TN : llxll 2 = VT ( x* x ) is S-bounded}.

Moreover for j, g E D we have

13.

198

Quantum Bernoulli Experiments

because of 1rN (Jg) = 1rN (j)*nN (g) . Since L 2 (D, L'") is the completion of ef : f E D} the mapping 0/ ---+ ;;;(}) can be extended to a linear isometry U from L2 (D, L'") onto a subspace M , say, of 7-{TN · For t E [0, 1] set t min{§. E T : §. > t}. We show that U(f3t ) Bw, t · Defi ne =

Yt

(w)

=

{ -11

=

w(.t.) w(.t.)

=

1 0 =

and l'i = }N L O :S.£oc f3t li\1n = f3t holds in L2 (D, L'") . Therefore for every standard E > 0 there exists a standard no with llf3t(1D - 1 Mn ) ll 2 < E for n 2 no standard. But this in turn implies 11Yt(1D - 1 MJ ll 2 < E for n standard, n 2 no . Since 7rN is an isometry also for the internal L 2 -norm on 7-{TN we obtain IIBw, t7rN(1 D - 1 Mn ) ll 2 < E for these n 2 no. I I Bw. t 7rN (1 Mn ) l l 2 = I I Yt1 Mn ll 2 � n implies that Bw, t 7rN ( 1 M, ) E 7-{TN ,fin and U(f3t 1 MJ Bw ,t--;;;{i AJJ . But this in turn gives U(f3t) = Bw, t · That the operator of multiplication by f3t is mapped onto the operator of multiplication by Bw ,t follows from an easy additional argument. =

13.5 . 2

The nonstandard hulls of the basic internal processes

The deeper problem is to find a joint subspace JC of HTN such that all pro­ cesses have standard parts as closable operators densely defined in JC. We begin the construction with some more notations: For w E { 0 , 1}T =: D we set Mw = {.t. : w(.t.) = 1 }, and lwl := I Mw l , that is the (internal) car­ dinality of lvlw . Using these notations we obtain the following formulae by means of ( 1 3.4.4)-(13.5.2) . Let y LwED Yw ew be arbitrary. Then =

A_t y Aiy Aty Az y

L �L O:S,s.
) ( L( L ) ) L (� L ) L (� L w

+

( •)

(13.5.4)

Yw'w

wED

wED


tpED

O:S,s.
w (')

O:S,s.
(13.5.5)

Yw'w

11,an,'P- ( •)

1!,an,'P

+

(,)

'"

(1 3 . 5 . 6 )

'"

(13.5. 7)

13.5. From the internal to the standard world

199

Consider now the projection IE.t. from 7-{ onto 1-l.t. which is the Hilbert space coming from A..t. (see section 13.3 ) . IE..t. is given by lEt- ( ew ) Then we obtain IE_.t.A�

=

{ ew 0

w (�) = 0 for all � 2> t

else

A�IE..t. for each rt E family (A�h< 1 is ada�ted t� (H..t.h < 1 ·

{ • , o, + , - }

=

or in other words the

We prove now some continuity properties. To this end we introduce 7-{ ( n)

:=

{ L Yw ew : Yw E *lC} l w l= n

and we set 7-{ : = 7-{TN for short. Then 7-{ = 1_;{= 0 7-{ (n) where _i denotes the orthogonal direct sum. More­ over A; (H( n) ) C 7-{ ( n) Ai (H( n) ) C 7-{ ( n) Ai (H( n ) ) C 7-{n + 1 , and finally Ai- (H( ;;:) ) C 7-{ ( n-1 ) where 7-f ( -1 ) := {0 } = : 7-{ ( N+ 1 ) . For y = L l w l =n Yw ew and 0 :S §. < t we obtain ,

,

L I� L

II A� y - A� y ll 2

lw l =n

§."5cy
w+ ( s ) I 2 1Yw l 2

1 1 2 1 Yw l 2 L L I� "5cy <.t. w

<

=

( t - �) 2 II Y II 2 . (13.5. 8)

l l=n §. It follows that II A; - A: ll :S (t - §.) , in particular the mapping t S-continuous from -T to L(H) with respect to the operator norm. Let Y = L l w l = n Yw E 7-{ ( n) , and 0 :S � < t

II Ai y - A� y ll 2

=

L(L

l w l=n §."5cY.<.t.

f---+

A� is

w( �) r1Yw l 2 ·

It follows that II A� IJt(n) I :S but the mapping t A�y is not S-continuous in general. So finally we have to choose an appropriate subspace in order to get S-continuity of t --+ A�y. Next we show the following inequality: n,

Proposition 1

holds.

For- 0 :S � < t

f---+

(13.5.9 )

13. Quantum Bernoulli Experiments

200 Proof.

Let y = L l w l = n Ywew E Ji( n ) be arbitrary. Then

holds. If ip(u)

=

0 then

l iP

EB

t,,,l

=

<

n + 1 hence Y'Pffil:Yc

=

0. It follows

n+1 N

Let w E D be arbitrary with lwl = n . Then there exist at most N(t. - ..s.) - lw · 1 [� ,![ 1 different ip with ip EB 11' = w and .§. :S 1J < t_. This gives I (Ai - A! ) Y II2 :S D N (t;�) ( n + 1 ) II Y II2, and the asserted inequality follows. Corollary 1

( 13.5. 1 0)

Ai is the adjoint of Ai. By co�struction we have 'H( n) l_Ji( m ) for n # So the subspace JC= of H , generated by all the spaces i{( n ) ( n E N) is nothing else than the orthogonal direct sum JC= = l_ n ENo i{(n ) . Recall the definition t = min{JJ E T : > t.}. Proof.

m.

Then from the inequality (13.5.8) it follows that by

1J

A;,n Y (A[,ny ) /\ there is uniquely defined an operator on i{( n) bounded by t. So we obtain the operator A� = l_ n ENo At . n on JCoc which is selfadjoint and bounded by t. Moreover I I A� - A: II :S t - holds. D In order to define the nonstandard hull of one of the other operators A-� (� E { +, - , 0}) we choose the following joint domain of definition: JC� : = { Y E }(00 : Y = l_n ENo Yn, Yn E i{(n ) , � nll :iJn ll § < 00} =

s

which is dense in JC=. From the inequalities (13.5.9) to (13.5.10) we obtain that by

13.5.

201

From the internal to the standard world

:

the operators A� K0 --+ K 00 are uniquely defined. Moreover the following proposition holds: Proposition 2

A�

is selfadjoint and Ai and Ai are adjoint to each other.

Proof. Set At n = A� I joint, whereas Atn and is obvious.

H17• These operators are bounded and A�, n is selfad­ A�( n+ l ) are adjoint to each other. Now the assertion D

The final problem we have to solve is to find an appropriate filtration with the necessary conditions of continuity. To this end we calculate the difference IE.t - IE.e_ for :<:: §. < t For y E 7-{ ( n) we obtain

0

_e.

(IE_t - IE Let t. =

)y =

k/N, = l/N. It follows §.

l

1 Yw 2

l w l = n , w9 [o: ..t[

l

max 1 Yw 2

<

lwl = n

(k n- l)

17 max 1 Yw 2 Nn (t_ - §.)

l

lwl =n

•

n 1 1 (1 - k --l ) · · · ( 1 - k -- l ) . I

So the mapping y --+ IE.tY will be S-continuous if max lwl = n Yw 1 2 N17 will be finite. This consideration leads to the internal measure m on D = given by m ( w } ) = NL . Then (D, m) is the direct sum of (D17, mn)n :5o N with Dn = = and mn = m bn · Notice that m(Dn) = (�) · Jn , so that N m(D) = L�= O ( � ) · = ( + it ) :::::J e. Moreover L�= L m(D k ) :::::J for all L :::::J oo. Let (D, Lm) be the Loeb measure space associated to (D, m) . Then D ) = e and D is an Lm - null set. Finally L 2 (D, Lm) � _l n N L (D17 , LmJ where Lmn is the Loeb measure associated to m17 • Consider the external subspace of all S-bounded internal *complex val­ ued functions on D . Then for E the complex-valued function : --+ is in L 2 (D. Lm) and the space E is dense in L 2 (D, Lm) . Mor eover = 0fk in the L2 -sense, where = Now we map onto a subspace Y of 7-{ by the mapping V defined as follows: ( ) - I wl / 2 ew , V( ) =

{ {w E D : l w l

n}

Jk

Lm (Uk EN k E 2 0( j (w ))

0f

1

{0 . 1 }T 0

\ (U kE N Dk)

X f X

LkENo

X

ef : f X} fk f Ink .

f Lfw N wED

°f

w

13.

202

Quantum Bernoulli Experiments

and set

W(j) = 0]). Then Jr2 l f l 2 dm II V ( f ) ll 2 . Let .C b e the closure o f {W(j) : f E X} = : .[( b) in JC=. Then the last equation shows L2( D, Lm ) � .C. Moreover Xk = {f E X : f ( De ) = 0 for £ # k} is mapped onto a subspace of Ji ( k) ( k standard) . Let .C k be the closure in JC= of {W(f) : f E Xk } =: .C�b) . Then .C = ..lkENLk and .Ck � L 2( D b Lm k ) . The following facts arc easily seen to hold:

Theorem 1

_c k(b) .Ai• _c k(b) Aio _c k(b) .A �+ _c (b) k .A - _c k(b) �

c JC'ff for k E No c .C k c .C k c _c (b) k+l

c .c�bl_l .

Moreover .C n JC0 is dense in .C and it is the joint domain of definition of the operators A� , � E { +, - } which are closable and essentially selfadjoint in case � = and adjoint to each other in case � = +, - . •, o,

•, o,

Remark: We map an internal set !vi of T onto its indicator function This mapping 7/J is bijective and therefore we obtain an isometric isomorphism U of m ) onto the internal Guichardet space of Leitz-Martini. This shows that our considerations concerning the continuity of the filtering leads in a very natural manner to the Loeb-Guichardet space for the basic quantum stochastic processes cf. section 13. 1).

1M E {0, 1 }T =

D.

L2( D,

(

13.6

The symmetric Fock space and its embedding into .C

Let 'H be a Hilbert space and let 'Hr:om be the nth tensor product of 'H . We denote the full symmetric group of n elements by Then by

§n .

there is uniquely defined an orthogonal projection, the range of which is called the nth symmetric tensor product The symmetric Fock space

JiO n .

13.6.

The symmetric Fock space and its embedding into .C

203

+ ( H ) over H is the orthogonal sum :F+ ( H ) = l_':= o H O n where as usual H O 0 lC. For the special case H = L 2 ( [0, 1], >.) (>.: Lebesgue measure) the tensor n

:F

=

product H® is nothing else than L2 ( [0, 1] 71 , .\71 ) where A 71 denotes the dimensional Lebesgue measure. and the projection Pn is given by

n­

Now we give the more or less classical definition of the basis quantum processes. To this end we set

h o · · · fn : = Pn ( h Q9 · · · � fn ) · 0

For

0 :S t

<

1

the definition of the creation operator is given by

Ai (( h · · · fn ) : = 1 [o, t [ h · · · fn 0

0

0

0

0

(13.6. 1)

and linear extension t o the largest possible domain. Similarly the annihilation operator is given by

n Ai ( h o . . . o fn ) = L (1 [o, t [ l fJ ) h o . . . o jj o . . . o fn , J= l

(13.6. 2)

where the expression jj indicates that this factor is omitted. The number operator is given by

A� (fl o · · · o fn ) = 1 [o, t [ · fl o · · · o fn + · · · + fl o f2 o · · · o fn - l o 1 [o, t[ · fn , (13.6. 3) whereas the time-operator is

A� (h o · · · o fn ) = t fl o · · · o fn ·

(13.6.4)

Remark: These operators are those ones introduced by Hudson and Parthasarathy [5] as the basic quantum processes. Consider now the set Xn = {(t 1 , t 2 , . . . , t n ) : 0 :S t 1 < · · · < t71 :S 1}. It is a measurable subset of [0, 1] 71 • Let f be an element of L 2 (X71, .\ 71 ) . Setting

f-( t 1 , · · · , tn ) =

{ OJ (t 1 , . . . , tn)

(t 1 , . . . , t n ) E Xn

otherwise

13 . Quantum Bernoulli Experiments

204

we obtain an element j of L 2 ( [0 , l]n, An) . Then Pn (]) E L 2 ( [0, l] , A) O n and II Pn (]) ll 2 = fxn l l 2 An , i.e. llfll = II Pn (]) ll holds. On the other hand let E L2 ( [0, 1] , A )0 n be arbitrary. Then = Pn (jl xn ) (equality in the L2sense) . It follows that the mapping ---+ Pn (]) is a linear isometry from L2 ( Xn , An) onto L 2 ( [0, 1] , A) O n. Its inverse is given by 91 xn =: Un (g ) . Now we embed L2 (Xn , An) isometrically into the space Ln constructed at the end of the previous section. To this end let w E Dn be arbitrary. Set (w) = min(� : w(�) = 1) and by induction �k+ 1 (w) = min(� > h : w(�) = 1). Then I(w) := (h (w) , . . . , �n (w)) E *Xn n Tn . Moreover if f := (kl /N. . . . , kn/N) E *Xn n Tn then f = f( w) for w = 1 M where M = {h (w) , . . . , �n (w) }. We consider the dense subspace Cn of L2 (Xn , An) consisting of the re­ strictions to Xn of continuous functions on the closure Xn . For E Cn we set S(f) (w) = * (f(w)) . This is an element of L2 (D n , mn ) satisfying l l f l l 2 :::::J II S(f) ll 2 · It follows that Cn is linearly and isometrically embedded into L2(D n , LmJ , and this embedding can obviously be extended to the whole of L2 (Xn , An) . Since L2 (D n , Lmn ) is linearly and isometrically embedded in Ln , we obtain that the symmetric tensor product L2 ( [0, l] , A) O n can be em­ bedded in Ln . An obvious extension then yields an embedding denoted by U of the symmetric Fock space F+ ( L2 ( [0 1 ] , A) into £. Call y the image of this embedding and set Yo = G n K0 .

fd

f

f

f n!

g n! --->

t1

f

f

,

2 Yo is dense in y. Moreover the restrictions of A� to Yo yield closed densely defined operators which are selfadjoint for � E { •} and adjoint to each other in the other cases and which moreover satisfy

Theorem

on u- 1 (Qo) where to (13. 6.!J

o,

A�

aTe the operators given by equations

(13. 6. 1)

'UP

The proof is obvious.

References [ 1 ] R. M . ANDERSON, "A nonstandard representation for Brownian motion and Ito integral", Israel .J. Mathern. , 25 15-46. [2] S . ATTAL, Approximating the Fock space with the toy Fockspace, in .J. Azema, I\1 . Emery, M. Ledoux, I\L Yor (eds.), XXXVI, Lecture Notes in Mathematics 180L Springer-Verlag, Berlin New York, 2003.

Seminaire de Probabilites

References

[3] A.

GUICHARDET,

205

Symmetric Hilbert spaces and related topics, Lecture

Notes in Mathematics 26L Springer-Verlag, Berlin, 1972.

[4] R.

L . HUDSON and J . M. LINDSAY, "A noncommutative martingale rep­ resentation theorem for non-Fock- Brownian motion", .T . Funct. Anal., 6 1

(1985) 202-221 .

[5] R.

R . PARTHASARATHY , "Quantum Ito ' s formula and stochastic evolutions", Comm. Math. Physics, 93 (1 984) 311-323. L. HUDSON and K .

[6] R.

L. HUDSON and R. F. STREATER, "Ito ' s formula is the chain rule with Wick ordering", Phys. Letters, 86 ( 1982) 277-279.

[7]

"Structure des cocycles markoviens sur respace de Fock", Prob. Theory Re. Fields, 75 (1987) 291 -316.

.T . L . .T OURNE.

[8] B .

KDMMERER and H. MAASSEN , "Elements of quantum probability", Quantum Probab. Communications, QP-PQ, 10 (1998) 73-100.

[9]

H . MAASSEN, Quantum Markov processes on Fock space described by integral kernels, in L. Accardi, W. von Waldenfels, Lecture Notes in Math­ ematics 1136, Springer-Verlag, Berlin. New York, 1985.

Quantum probability and applications II, Proceedings Heidelberg 1984, [10] M. LEITZ-MARTINI. Quantum stochastic calculus using infinitesimals, Ph.-D. Thesis, University of Tubingen, 2001 . [11] P . A . MEYER. Quantum probability for probabilists, Lecture Notes Mathematics 1 538, Springer-Verlag, Berlin, New York, 1993.

[12]

m

K. R. PARTHASARATHY, The passage from random walk to diffusion in quantum probability, (1988) , Ap­ plied Probability Trust, Sheffield, 231 - 245.

Celebration Volume in Appl. Probability [13] K. R. PARTHASARATHY, An introduction to quantum stochastic calculus, Birkhauser. Basel, 1992.

Applications of rich measure spaces formed from nonstandard models Peter Loeb*

Abstract \i'Ve review some recent work by Yeneng Sun and the author. Sun'i:l work shows that there

are

results, some used for decades without a rigourous

that arc only true for spaces with tho rich structmo of Loeb measure spacel:l. His j oint work with the author uses that structure to extend an important result on the purification of measure valued maps. foundat.ion,

14. 1 In

Introduction 1975 [ 1 1] ,

the author constructed a class of standard measure spaces

fanned on nonstandard models.

These spaces, nmv called "Loeb spaces" in

the literature, arc very close to the underlying internal spaces, and ru·c rich in

s t ruc t ure.

(Sec

1 121 by Yeneng Sun

the author's throe clmptcrs and Osswald's two chapters in

for background. ) 'vVe will briefly review here some recent work

that uses these measure spaces. Sun has shown that there are results, some

used in applications for decades without a rigorous foundation, that are only true for spaces with the rich structure of Loeb measure spaces. We will fol l ow the description of Sun'::; work with a detailed proof of a special case of t.hc recent

result in

[H] by Yeneng Sun and the author on A counterexample shows that the

valued maps.

the purification of measure­

result we give is false if the

Loeb space we use is replaced by the unit interval with Lebesgue measure. We

note in passing hero that the result in

[13]

by Osswald, Sun, Zhang and the

author presents yet another example needing rich measure ::;paces.

*

Department

of

Mathematics, 'University

loeb@math. uiuc . edu

of

Illinois, Urbana, IL 61801.

14.2. Recent work of Yeneng Sun

14.2

207

Recent work of Yeneng Sun

We begin with some work in the late 1990's of Yeneng Sun ( [16] , [17] , [18] , and Chapter 7 of [12] ) . That work, Sun's alone, is published elsewhere, so what is said here is an invitation to read, and not a substitute for, the original articles. To set the background we consider the following question: Does it make sense to speak of an infinite number of independent individuals or random variables indexed by points in a uniform probability space? Can one, for example, reasonably consider an infinite number of independent tosses of a fair coin with the tosses indexed by a uniform probability space, and if so, does it make sense to say that half the tosses should be heads? Of course, there is no problem in speaking of independent random variables indexed by the first integers with each integer having probability 1/ An infinite index set with a uniform probability measure, however, must be uncountable; there is no uniform probability measure on the full set of natural numbers. The problem thus evolves to finding the possible meaning of an uncountable family of independent random variables. Whatever way one approaches this problem, the usual measure-theoretic tools fail. A natural attempt to generalize coin tossing replaces the natural numbers with points in the interval [0, 1], and thus replaces sequences of 1 ' s and - 1 ' s with functions from [0, 1] to the two-point set { -1, 1 } . This is a reasonable model for an uncountable family of independent coin tosses. In 1937, Doob [2] exhibited a problem with this and similar spaces of functions when standard techniques arc applied. To sec the problem, let denote the set of { - 1, 1 }­ valued functions on [0, 1] , and let be the product measure on constructed from the measure taking the values 1 / 2 at 1 and 1/2 at - 1 . In the usual construction of an appropriate a-algebra on measurable sets are formed from the algebra of cylinder sets, with functions in each cylinder set restricted at only a finite number of elements of [0, 1] to take either the value 1 or - 1 . It follows that each measurable set in is determined in a way described below by a countable subset of [0, 1 ] , and so the following result holds.

n

n.

P

D

D

D,

D Proposition 14.2.1 Fix any h E D. Set Mh := {w E D : w(t) = h(t) except for countably many t E [0, 1 ] } . Then Mh has P-outer measure 1 . Proof. For each measurable B � D, there is a countable set C [0, 1] such that for all o:, (3 in D, if o:(t) = (3 (t) for all t E C, then E B if and only if (3 E B. Suppose Mh � B . Given w E D, Let w' agree with w on C and agree with h on [0, 1] \ C. Then w' E Mh, so w' E B. It then follows that w E B . Thus B = D, so the outer measure P*( Mh ) = 1 . a

C

D

14. Applications of rich measure spaces

208

Remark 14.2.2 Note that if w E lvh, then since Lebesgue measure A of a countable set is 0, w = A-a.e. on [0, 1 ] . This is true if is nonmeasurable, or if 1, or if - 1 . Now outer measure when applied to the intersection of measurable sets with Mh is finitely additive, hence countably additive. Since the outer measure of Mh is 1 , one can trivially extend P to a measure P with P(Mh) = 1 . Thus, no matter what might be, one can claim that P almost every function is equal to at A-almost very point of [0, 1]. This, and other highly questionable arguments have been used for decades to work around the measure theoretic problem indicated by Doob's example.

h

=

h

=

h

h

h

h

Another approach to representing a continuum of independent random vari­ ables is to consider a function w , called a process, where is an index from an uncountable probability space called the parameter space (the probability measure need not be uniform) and w is taken from a second probability space called the sample space. The question then is whether it makes sense to work with the usual product of these two probability spaces. Sun has shown in Proposition 7.33 of [12] that no matter what kind of measure spaces, even Loeb measure spaces, one might take as the parameter space and sample space of a process, independence and joint measurability with respect to the classical measure-theoretic product, i.e. , formed using measurable rectangles as in [15] , are never compatible with each other except for a trivial case. Here is the exact statement of that proposition.

f(i, )

i

Let (I, I, ) and (X, X, v) be any two probability spaces. Form the classical, complete product probability space (I x X, I 129 X, 129 v ) . Let f be a function from I x X to a separable metric space. If f is jointly measurable on the product probability space, and for 11-almost all (i 1 , i 2 ) E !xi, h and fi2 are independent (call this almost sure pairwise in­ dependence), then, for tt-almost all i E I, f( i. ) is a constant f1m ction on X. Proposition 14.2.3 (Sun)

f1

f1

f1 129

·

Sun notes that Proposition 14.2.3 is still valid when f1 has an atom A . The almost sure pairwise independence condition implies the essential constancy of the random variables fi for almost all i E A. In the articles cited at the beginning of this section, Sun has shown that a construction overcoming these measure-theoretic problems is obtained by forming the internal product of internal factors, and then taking not just the Loeb space of each factor, but also the Loeb space of the internal product. This yields a rich extension of the usual product a-algebra formed from the Loeb factors, one that still has the Fubini Property equating the integral over the product space to the iterated integrals over the factor spaces forming that product. Here in more detail is that construction.

14.2. Recent work of Yeneng Sun

209

Sun's starts with internal spaces ( T, T, A) and ( D, A, P) . The space T may be a hyperfinite set with A given by uniform weights. He then forms the Loeb spaces (T, L>. (T) ) ) and (D, L!L (A) , P ) . He lets A Q9 P denote the internal product measure, while T Q9 A denotes the internal product a-algebra, and L>. (T) Q9 Lp (A) denotes the classical product a-algebra formed from L>. (T) and Lp (A) as in [15] . On the other hand, forming the Loeb space from the internal product T Q9 A produces a larger a-algebra L>.0P (T Q9 A) on T n . ( Sec the following propositions. ) Here arc some properties of what we shall call the big product space (T x D, L>.0p (T Q9 A) , �) . X

The big product space depends only on the Loeb factor spaces (T, L>. (T) , >:) and (D, Lp (A) , P) . Proposition 14.2.5 (Anderson [1] ) If E E L>. (T) Q9 Lp (A) , then E E L>.0p (T � A) and � (E) = ); ® P(E) . Proposition 14.2.6 (Hoover (first example) , Sun [ 1 7] ) The inclusion of L>. (T) Q9 Lp (A) 'in L>.0P (T Q9 A) is strict if and only if both ); and P have non-atomic parts. Proposition 14.2. 7 (Keisler [6] ) A Fubini Theorem holds for the big prod­ uct space, Proposition 14.2.4 (Keisler-Sun [7] )

In his articles, cited above, Sun shows that to work with independence in a continuum setting, as has been attempted in informal mathematical applica­ tions without rigor, and for decades, one needs a rich product a-algebra such as the a-algebra of a big product space. Here is that general result as stated in Proposition 7.4.1 of [12] and proved in Theorem 6.2 of [17] .

Let X be a complete, separable and metrizable topological space. Let M (X) be the space of Borel probability measures on X, wher·e M(X) is endowed with the topology of weak convergence of measures, Let be any Borel probability measure on the space M(X) . If both ); and P are atomless, then there is a process f from (T n , L>.0P (T Q9A), GP) to X such that the random variables ft = f ( t, ) are almost surely pairwise independent (i,e,, for �-almost all (t1 , t2 ) E T x T, fh and ft2 are independent ), and the probability measure on M(X) induced by the function Pft- 1 from T to M (X) is the given measure Remark 14.2.9 Here, for any given t E T, P ft- 1 is the probability measure on X induced by the random variable ft D ---+ X. It is the measurable mapping from T to M (X) taking the value P ft- 1 at each t E T that induces Proposition 14.2.8 (Sun) f-L

·

X

f-L .

:

the measure f-L on M (X).

14. Applications of rich measure spaces

210 Remark 14.2.10

(

A consequence of the proposition is that when the space

T. L>.. (T), :\) is a hyperfinite Loeb counting probability space and 'ji is a non­

atomic Loeb probability measure, it makes sense to use the big product space as the underlying product space for an infinite number of equally weighted, independent random variables or agents. As part of his work with large product spaces, Sun has extended the law of large numbers. The usual strong law states that if random variables Xi , E N, are independent with the same distribution and finite mean m, then � L:i= l Xi tends almost surely to the constant random variable m. That is, for almost all samples the value of the sequence at tends to the constant m.

i

w, w Theorem 14.2. 1 1 (Sun) Let f be a real-valued integrable process on the big product space. If the random variables ft := f(t, ·) are almost surely pair­ wise independent, then for almost all samples w E D, the mean of the sample function fw := f(·,w) on the parameter space T is the mean off viewed as a random variable on the big product space. There is no requirement of identi­ cal distributions. Another facet of this same work deals with independence. It is well known that for a finite collection of random variables, pairwise independence is strictly weaker than mutual independence. Sun has shown that for processes on big product spaces, pairwise independent and mutual independent coalesce. and they coalesce with other notions of independence that are distinct for a finite number of random variables. This implies asymptotic results for finite families of random variables where the families are ordered by containment and have increasing cardinality.

14.3

Purification of measure-valued maps

We now turn to a special case of the joint work of Yeneng Sun with the author in [14] . That special case generalizes the following celebrated theorem of Dvoretzky, Wald and Wolfowitz (see [3] , [4] , [5] ) .

Let A be a finite set and M(A) the space of probability measures on A. Let (T, T) be a measurable space, and I-Lk ! k = 1 , · · · , finite, atomless signed measures on (T, T) . Given f : T ---+ M (A) so that for each a E A, f( ·) ( {a}) is T-measurable, there is aT-measurable map g : T A so that for each a E A, and each k l f(t)({a})df-Lk (t) = f-Lk ({t E T : g(t) = a}). Theorem 14.3.1

m,

---+

:<:: m,

14.3. Purification of measure-valued maps

211

This theorem justifies the elimination, i.e., purification, of randomness in some settings. For example, in games, T represents information available to the players, and A represents the actions players may choose, given E T. Every player�s objective is to maximize her own expected payoff, but that payoff depends on the actions chosen by all the players. For each player, a mapping from T to A is called a pure strategy. A mapping from T to M (A) is called a mixed strategy; in this case, each player chooses a "lottery on A". A Nash equilibrium is achieved if every player is satisfied with her own choice of strategy given the choices of the other players. In quite general settings, such an equilibrium can be achieved using mixed strategies. When results such as Theorem 14.3.1 apply, those mixed strategies can be purified yielding an equilibrium with the same expected payoffs. We re­ fer the reader to [14] for more information on the game-theoretic consequences of Theorem 14.3.1 and its extension. We will concentrate here on the proof of that extension, a proof simpler than that of the more general result in [14] . We extend the DWW Theorem from a finite set A to a complete, sepa­ rable metric space. There always exists a Borel bijection from such a space to a compact metric space. That is, if A is uncountable, then it follows from Kuratowski's theorem (see [15] , p. 406) that there is a Borel bijection from A to 1]. On the other hand, for a countable set A, one can use a bijection from A to {0, 1 , 1/2, . . , 1 / , . . . } . Therefore, it suffices to let A be a compact metric space. For simplicity we will work with a finite set of measures, but may extend to a countable collection.

t

[0,

. n

Let K be a finite set, and let A be a compact metric space. For each k E K, let be a nonatomic, finite, signed Loeb measure on a Loeb measurable space (T, T) . If f is a T-measurable mapping from T to M (A), then there is a T -measurable mapping g from T to A such that for each k E K and for all Borel sets B in A, f f (t ) ( B) ( dt) = (g- 1 [Bl) . This is equivalent to the condition that for each k E K and for any bounded Borel measurable function e on A, l l B( a)f(t)(da )f-Lk( dt) l B(g( t )) !-Lk (dt). Theorem 14.3.2

f-Lk

r

f-Lk

f-Lk

=

Example 14.3.3 Sun and the author show by a counter example in [14] that the Loeb measures of the theorem cannot be replaced with measures absolutely continuous with respect to Lebesgue measure on [0, 1 ] . In their example, the compact set A is the interval [ -1, 1], and the measure-valued map is given by E [0, 1 ] . The example uses just two measures, : = (5t + L t )/2 for each /-L l = >. on [0, 1 ] , and f-L 2 = 2 >. on 1]. If there is a function that works, then it is not hard to see using the functions on A given by and

f(t)

t t [0 ,

g B(a) l a l B(a) a2 =

=

14. Applications of rich measure spaces

212

(t l g(t)1 ) 2 >.. ( dt) g

g(t )

that J01 must take the value t or - t >..- a.e. = 0. That is, on [0, 1 ] . It is also not hard to see that this is impossible. The moral is that a Lebesgue measurable can't switch values on [0, 1] fast enough to work.

fik

To set up the proof of the theorem. we let be the signed internal measure generating for each k E K. Also, we set Po = where is the internal total variation of and c = Then Po is an internal probability measure on and for each k E K, < < Po . Let The probability measu � P is the Loeb measure generated P=

f-Lk

fik

T,

st(c) L.kE K l f-Lk 1.

f-Lk f3k

�L.k EK 1/ikl , L.kEK 1/ik I (T). fik

f3k f-Lk

1/ikl

by P0 , and for each k E K, < < P. Let be the internal Radon-Nikodym derivative of with respect to Po . Since < < P, is S-integrable with � respect to Po and := st is the Radon-Nikodym derivative of with respect to P. We let B denote the collection of Borel subsets of The following lemma is the only part of the proof that needs nonstandard analysis.

fik

f3k

{ik

A.

f-Lk

Let { ¢, : i E N} be a countable, dense (with respect to the sup­ norm topology) subcollection of the continuous real-val1Led functions on A. As­ sume that there is a sequence ofT-measurable mappings {gn , n E N} from T to A such that for each i E N and k E K, the sequence c/J2 (gn(t)) f3k (t)P(dt) converges; let c,,k E rn:. denote the limit. Then, there is aTfr-measurable mapping g from T to A such that for each i E N and k E K, l c/J, (g(t)) f3k (t ) P (dt) ci k Proof. For each n E N, let hn : T ---+ *A be a To-measurable lifting of gn with respect to the internal measure Po . Then for each and i E N and each k E K, Lemma 14.3.4

=

,

·

n

and so

�i�

n --t CXJ

( l l *¢i(hn(t)){ik (t)Po(dt) - i,k l ) st

c

hn

= 0.

Using N 1 -saturation, we may extend the sequence to an internal sequence and choose an unlimited integer H E *N so that for every E N and each k E K,

The desired function

t E T.

i

g is obtained by setting g(t) := st (hH ( )) E A at each t

D

14.3. Purification of measure-valued maps

213

( )

Now for the proof of Theorem 14.3.2, we note first that since M A is a compact metric space under the Prohorov metric p which induces the topol­ ogy of weak convergence of measures , there is a sequence of simple functions Un } �= l from T, T to M A such that

(

)

( )

(

)

Vt E T, nlim p (jn (t) , j(t)) = 0. ---t oo Assume we know that for each E N, there is a T-measurable mapping n

9n from T to A such that for each k E K, and any bounded Borel measurable function e on A ,

fr l e(a) fn(t) (da)p,k (dt) = fr e(gn(t))!Lk (dt) .

Let g = { cPi : i E N} be a countable dense subset of the continuous real-valued functions on A with the sup-norm topology. For a given cPi E g and each t E T, A ¢� (a)fn (t) (da) ----+ A cjJ, (a)j (t) (da) . Moreover, each integral is bounded by the maximum value of l¢i l on A , so by the bounded convergence theorem, for each k E K,

f

f

L cPi (gn (t) )fJk (t)P(dt) L cP2 (gn (t))tLk (dt) = L l cPi (a)fn (t) (da)p,k (dt) ----+ L l cPi (a)j (t) (da)p,k (dt) =

By the lemma, there is a T-measurable mapping g from T to A such that for each k E K,

L cPi (g(t) )tLk (dt) = L cPi (g(t) )f3k (t) P(dt) = L l cPi (a) j(t) (da)tLk (dt) .

Since this is true for each cPi in Q, it is true with cPi replaced by an arbi­ trary bounded Borel measurable function e on A. Therefore, without loss of generality, we may assume that f : T ----+ M A is simple. Next, we fix a sequence of Borel measurable, finite partitions pm { A , . . . , A } of A such that the diameter of each set in Pm is at most 1/2m , and Pm + l is a refinement of Pm . Also for each l, 1 ::; l ::; lm , we pick a point a[ E A . Since f is a simple function from T to M A , there is a T-measurable partition { Sj }f= l of T such that f = "'/j E M A on Sj . It follows that for

( )

1

=

Z:

[

( ) ( )

14. Applications of rich measure spaces

214

each B E B, and each k E K, fr f(t ) ( B ) p,k (dt) = Lf= l "(j ( B ) p,k(Sj ) . By Lyapunov's Theorem, each 51 can be decomposed by a finite, T-measurable partition {Tf' m , . . . , Tz:":'} so that for every l :::.; lm and k E K, f1k (Tf 'm ) = ,.YJ (A[ )p,k (Sj ) , whence

l j(t) (A[ ) p,k (dt) f; "(j (A[n ) !1k ( Sj ) =

N

N

=

( )

L /1k Tf 'm . j= l

(As an alternative to the above use of the Lyapunov theorem, one can with small modifications of the proof here use the author's Lyapunov theorem [10] , but then all of the simple functions must be modified on a P-null set T0 so that for each of them, the corresponding partition sets Sj of T are internal. ) Now for each m 2: 1 , we define a T-measurable mapping 9m : T ----> A so that for each l :::.; lm and each j :::.; N, 9m ( t) = a[ on T/'m . For each continuous, real-valued 7/J on A , for each m _2: 1 and each k E K,

lm N

= L L ( 7/J (a [ )rj (A[ ) ) f1 k (Sj ) l=l J = l

lm N

( )

= L L 7/J (a[ ) !1 k T/'m l=l j = l

approximates as m ----> oo the integral

£ (l >/� (a)f(t) (da) ) Mk (dt)

�

·

·

t, £ (1r t, � 1, (1r

)

>,b (a) j(t) (da) �k (dt)

)

>,b(a) o, (da) Mk (dt ;

215

References so

1/J (a[ ) !Lk (Tzj,m ) ----. L (1 1/J (a)j(t)(da) ) fLk (dt) It follows from the lemma that there is a T-measurable mapping g from T to lm N =LL l=l j = l

A such that for each k E K and for each continuous real-valued function e on A from a countable dense set of such functions, and therefore for each bounded Borel measurable function e on A,

L B(g(t))tLk (dt) L l B(a)j(t)(da) fLk (dt). =

D

References [1] R. M . ANDERSON, "A nonstandard representation of Brownian motion and Ito integration", Israel J . Math . , 25 (1976) 15-46. [2] .T . L . D ooB, " Stochastic processes depending on a continuous parameter", Trans. Amer. Math. Soc., 42 (1937) 107- 140. [3] A. DVORETSKY, A. WALD and .J . WoLFOWITZ, "Elimination of random­ ization in certain problems of statistics and of the theory of games", Proc. Nat . Acad. Sci. USA, 36 (1950) 256-260.

[4] A. DVORETSKY, A. WALD and .J . WoLFOWITZ, "Relations among certain ranges of vector measures", Pac . .J. Math. , 1 (1951) 59- 74. [5] A . DVORETSKY, A . WALD and .J . WoLFOWITZ, ""Elimination of ran­ domization in certain statistical decision problems in certain statistical decision procedures and zero-sum two-person games", Ann. Math. Stat . , 22 (1951) 1 -21.

[6] H . .J . KEISLER,

An infinitesimal approach to stochastic analysis, Memoirs

Amer. Math. Soc.

48, 1984.

[7] H .. T . KEISLER and Y . N . SUN, "A metric on probabilities, and products of Loeb spaces", .Jour. London Math. Soc . , 69 (2004) 258-272.

14. Applications of rich measure spaces

216

[8] 1\ I . A . K HAN , K . P . RATH and Y. N. SuN. "The Dvoretzky-Wald­ Wolfowitz theorem and purification in atomless finite-action games", In­ ternational Journal of Game Theory, 34 (2006) 91 - 104.

[9] M . A . KHAN and Y. N . Su N , "Non-cooperative games on hyperfinite Loeb spaces", J. Math. Econ. , 31 (1999) 455-492. [10] P . A . LOEB, "A combinatorial analog of Lyapunov's Theorem for infinites­ imally generated atomic vector measures". Proc. Amer. Math. Soc . , 39

(1973) 585-586. [ 1 1] P . A . L OEB, " Conversion from nonstandard to standard measure spaces and applications in probability theory", Trans. Amcr. Math. Soc., 2 1 1

(1975) 1 1 3- 122.

Nonstandard Analysis for the Working Mathematician. Kluwer Academic Publishers, Amsterdam, 2000.

[12] P . A . LOEB and M. WoLFF, cds.,

[13] P . A . L OEB, H . OsswALD, Y. SuN and Z . ZHANG, "Uncorrclatcdncss and orthogonality for vector-valued processes", Tran. Amer. Math. Soc . , 356

(2004) 3209 -3225. [14] P . A . LOEB and Y. Su N , "Purification of measure-valued maps", Doob Memorial Volume of the Illinois Journal of Mathematics, 50 (2006) 747- 762. [15] H . L . RoYDEN,

Real Analyszs, third edition. Macmillan, New York, 1988.

[16] Y. SuN. "Hyperfinite law of large numbers", Bull. Symbolic Logic, 2 (1996) 189-198. [17] Y. Su N , "A theory of hyperfinite processes: the complete removal of in­ dividual uncertainty via exact LLN". J. Math. Econ. , 29 (1998) 419-503. [18] Y. SU N , "The almost equivalence of pairwise and mutual independence and the duality with exchangeability", Probab. Theory and Relat. Fields, 1 1 2 (1998) 425-456.

More on S-measures David A. Ross

15.1

*

Introduction

In their important. (but. often overlooked ) paper [1] . C. Ward Henson and Frank \iVattenberg introduced the notion of S-measura.biz.ity, and showed that S-measurable functions are "approximately standarcP' ( in a sense made precise in the next section) . In a recent paper ([3]), the author used this machinery to transform a well­ known nonstandard plausibility argument for the Radon-Nikodym theorem into a correct and complete nonstandard proof of the theorem. The problem of finding an "essentially nonstandard" proof for Radon-Nikodym had been one of long standing. A ltho ugh Luxemburg gave a nonstandard proof as long ago as 1972 ([2] ) , he obtained the result as a consequence of another equally­ deep theorem in analysis clue to Riesz. ( Beate Zimmer [6] has recently proved vector-valued extensions of Radon-Nikoclym starting from the same plausibility argument, though using different standardizing machinery than that in [3] or the present paper. ) In this paper I use an S-measure argument very like the one in [3J to give an intuitive nonstandard proof of the Riesz result used by Luxemburg. Along the way [ gi ve a new proof for the main technical result from [1] on S-measures, a uew noustaudard proof for Egoroff's Theorem, and a nonst.
15.2

Loeb measures and S-measures

I will assume that we work in a nonstandard model in the sense of Robin­ son , and that this model is as saturated as it needs to be to carry out all constructions; in particular, it is au enlargement. *Department of Mathematics, University of Hawaii, Honolulu, HI 96822. ross@math . hawai i . edu

218

15. More on S-measures

Suppose X a set and A is an algebra on X. There are two natural algebras on *X : Ao = {*A : A E A} (the algebra of subsets of *X) , and *A. Note that except in the simplest cases, (i) neither Ao nor *A are a-algebras; (ii) Ao is external; and (iii) *A is vastly larger than Ao . These leads to two distinct a-algebras:

standard

1 . As

=

the smallest a-algebra containing Ao

2. AL

=

the smallest a-algebra containing *A

Recall that if IL is a (finitely- or countably-additive) finite measure on (X. A) then *IL maps *A to * [0 , oo ) , and in particular is not normally a measure, un­ less the range of IL is finite. However, 0*1L takes its values in [0 , oo ) , and so (*X, *A, 0*1L ) is an external, standard, finitely-additive finite measure space. The following was first noticed by Loeb. It is an immediate consequence of N rsaturation and the Caratheodory Extension Theorem, or can be proved directly; sec [4] for details. Theorem 1 °*1L VE E AL

extends to a a-additive measure ILL on (*X , AL ) . Moreover, ILL(E)

=

=

inf{ILL (A) : E c:;; A E *A} sup{ILL(A) : A c:;; E, A E *A}

( 1 5 . 2. 1 ) ( 1 5 . 2 . 2)

Of course, the measure ILL remains a-additive when restricted to any a­ algebra contained in AL, in particular As; this corresponds to replacing *A by Ao as the generating algebra. However, it is not so obvious that the approx­ imation properties ( 1 5 . 2 . 1 ) and ( 15 . 2 . 2) hold with this replacement as well. The next lemma, which asserts that they do, guarantees that a set E c:;; *X has 0" in the sense of Henson and Wattenberg [1] precisely when E c:;; D for some D E As with ILL(D) = 0. For this lemma, and the duration of the paper, it will be convenient to assume that A is a a-algebra.

"S-measure

Lemma 1 VE E As, �LL (E)

=

=

inf{ �L(A) : E c:;; *A, A E A} sup{IL(A) : *A c:;; E, A E A}

Proof. Let

'B

{ E E As : Vc > 0 ::JA, B E A such that *A c:;; E c:;; *B and IL(B \ A) < c} { E E As : Vc > 0 ::JA, B E A such that *A c:;; E c:;; *B and ILL (*B) < ILL(E) + c and ILL (*A) > ILL (E) - c } .

(1 5.2.3) ( 1 5 . 2 . 4)

219

15.2. Loeb measures and S-measures

E

E

Evidently *A 'B for every A A. It suffices to show that 'B is a a-algebra, so that As � 'B. If A � E � B then B e � Ec � A c and f-LL(B c \ A c ) = f-LL (A \ B) , so 'B is closed under complements. It remains to show that 'B is closed under countable unions. Let E = U n En where En 'B ( n N) increases to E. and let c > 0. For some N, ILL( EN ) > f-LL(E) - c/2, and for some A A, *A � EN and f-LL (*A) > f-LL (EN ) - c/2; it follows that *A � E and f-LL(*A) > f-LL(E) - c. For the exterior approximation, there exists Bn A with En � *Bn and f-LL (*Bn \ En ) < c2 - (n + 1 ) . Put B = U n Bn , then E = U n En � U n *Bn � *B ' and

E

E

E

E

f-L(B)

=

f-LL ( *B)

Call a function

(

)

((

) )

/-L L *B \ U *Bn + ILL U *Bn \ E + f-LL(E) n n n 1 < 0 + L c T ( + ) + f-LL(E) S f-LL(E) + c. n S

f : *X ---+ rn:.

D

approximately standard provided:

1. f is As-measurable;

f

2.

the restriction to JR:. ; and

g = f ix

3.

f :::::J *g almost

everywhere (with respect to

of

to X is an A-measurable function from X

f-LL) .

For example, suppose h : X ---+ rn:. i s a bounded A-measurable function. For r rn:. , 0*h - 1 ( - oo, = n nE N * h - 1 ( - oo , + 1/n) , so 0 *h is As-measurable. For X, = = 0*h(x) , so h = (0*h) lx. It follows that 0*h is approximately standard. The main result of this section is the theorem of Henson and Wattenberg, that As-measurable functions are approximately standard. The proof here differs from theirs. and makes it possible to prove some new results about integrability. Denote by the set of all approximately standard functions. The following enumerates some useful propert ies of

r] x EE h(x) *h(x)

r

all

APS

APS.

Lemma 2

APS is closed under finite linear combmations. 2. If f, g E APS then min{!, g} E APS and max{!, g} E APS. 3. If fn E APS, n E N, and fn mono tonically increases pointwise to f, then f E APS . 1.

220

4.

15. More on S-measures

Let h : X rn:. be an A-measurable function, and put E = {x E *X : l *h(x) l < oo}. Then (i) E E As, (ii) f-LL(*X \ E) = 0, and (iii) 0(*hxE ) E APS (where XE is the characteristic function of E). ---+

Proof.

f, g *X rn:. and (af (3f) l x = (af l x f3f l x ) . 2. As in ( 1 ) , note that max{f, g} l x = max{ f l x , g l x } and min{!, g} l x = min{ f l x , g l x }. 3. Let 9n = fn l x , An = { x E *X : *gn (x) ';f:; fn(x)}, and A = U n EN A n. Since f-LL( An) = 0 for each f-LL( A) = 0. Let g = f i x and note g = upn 9n · Fix a standard c 0, and for E N let En = {x E X : 9n (x) g(x) - c }. Since X = Un EN En, f-LL(*En) = f-L(En) increases to f-L(X) = f-LL(*X), so f-LL(A U (*X \ U nE N *En )) = 0 . Suppose x E ( *X \ A) UnEN *En . For some m E N, x E *E and fm(x) f(x) - c. It follows that *g (x) < *g (x) + c ::::::; fn (x) + c f(x) + c < fm(x) + 2c ::::::; *gm (x) + 2c *g(x) + 2r::n. Since c was arbitrary, *g (x) ::::::; f(x). 4. Put En = {x E X : l h (x) l < } , so E = Un EN *En E As . X = U n EN En, so f-LL(*En) f-L(En) increases to f-L(X) f-LL(*X), and f-L L(*X \ E) 0. 1 . Follows immediately from the observation that if E rn:. then + +

a , (3

---+

n,

s

>

n

>

n

n,

:S

:S

n

=

n

>

=

=

The proof of (iii) is now like the example preceding the statement of this lemma. D

Every As-measurable function is approximately standard. Proof. Note that if A E A then X*A = *xA = 0*XA is in APS by Lemma 2 ( 4) .

Theorem 2

By ( 1 ) and (3) of Lemma 2 and a suitable version of the Monotone Class Theorem (for example, Theorem 3 . 1 4 of [5] ) , every bounded As-measurable function is in APS. If 2 is As-measurable then = supn max{f, n so is in APS by (2) and (3) of Lemma 2. Any general As-measurable can be written = max{!, - max{ so is in APS by (1) and (2) of Lemma 2. D

f 0

f

f

f

},

0} f, 0} For example, suppose E E As ; denote by S(E) the set X n E of standard elements of E . If f is the characteristic function of E then g = f i x is the characteristic function of S(E) . By the theorem, S(E) is A-measurable and f-L(S(E)) = f-LL(*S(E)) = f *g df-LL = f f df-LL = IL L( E) . Corollary 1 Let f : *X be approximately standard, and p 0. Put g = f i x . Then f E £P(ttL) if and only if g E £P(tt), in which case J fPdf-LL = J gPdf-L and l f l v = l g l v · ---+ JR:.

>

221

15.3. Egoroff's Theorem

Proof. Without loss of generality, f 2: 0. Let s 71 : *X ---+ rn:., n E N, be simple functions increasing pointwise to fP. Put t71 = s 71 l x, then t71 is a sequence of simple functions on X increasing pointwise to gP. Moreover, if s71 = L7:=1 CYk XEk then t n = L 7:= 1 CYk XS ( Ek ) , and j s n df-LL = L7:=1 CYk f-LL(Ek ) = '5:."7:= 1 CYk f-L(S(Ek )) = J t 71 df-L. Let n ---+ oo, and it follows that J fPdf-L L and J gPdf-L arc either both infinite or arc both finite and equal. D The next result lets us extend the notion of approximate standardness to completion-measurable functions.

Let f : *X ---+ JR:.; the following are equivalent: (i) f is measurable with respect to the ILL-completion of As; (ii) f i x is measurable with respect to the f-L-completion of A, and f :::::J * (f i x) off a f-LL-nullset in As .

Lemma 3

Proof. Recall (without proof) the standard fact that a function is completion measurable if and only if it agrees with a measurable function off a nullset . (i =? ii) There is an As-measurable f' and a f-LL-nullset A E As such that f = f' off A. Let g = f i x and g' = f' l x , then g' is A-measurable, *g' :::::J f' off a f-LL-nullset B E As, and {x E X : g(x) # g'(x)} � S(A) . Since f-LL(A) = 0, f-L(S ( A)) = 0 by the example above, and so g' is measurable with respect to the f-L-completion of A. Note also that *f-L(S(A)) = 0, so f-LL (A U B U *S(A)) = 0. For x E *X , x rJ. A U B U *S(A) , f(x) = f'(x) :::::J g'(x) = g(x) , proving (ii) . (ii =? i) Let A E As be a f-LL-nullset with f :::::J * (fix) of!:" A. Let g = fix, g' A-measurable such that g = g' off a f-L-nullset B E A, E = {x E *X : l *g'(x) l < oo }, and f' = 0(*g' XE ) · By Lemma 2, f' is in APS. For x rJ. E U *B U A, f'(x) :::::J *g ' (x) = *g(x) :::::J f(x) , so in fact f'(x) = f(x) . Since f' is As­ measurable, f is completion measurable. D

15.3

Egoroff's Theorem

The most striking application [1] made of the S-measure construction was a proof of Egoroff's Theorem. Their proof relied on a nonstandard condition shown by Robinson to be equivalent to approximate uniform convergence. Here I use the machinery above to give an alternate proof of the theorem not depending on Robinson"s condition. :

Let (X, A, f-L) be a finite meas1Lre space, j, fn X ---+ rn:. measur­ able, fn ---+ f a. e. Then Vc > 0 3A E A f-L(A) > f-L( X ) - c & fn ---+ f uniformly on A.

Theorem 3

222

15. More on S-measures

Proof. For n E N put 9n = inf m> n fm and h n = supm > n fm · Note that for almost all x, hn (x) - gn (x) is nonnegative and nondecreasing in n with limn_, = hn(x) - 9n(x) = 0. Put ?Jn = 0 *gn , h n = 0 *h n . Let E = {x E *X : limn _,= hn(x) - ?Jn (x) c/c 0}. Note E E As, so fLL(E) = p,(S(E)) = p,(0) = 0. It follows that for some A E A, *A c;;;; E C and p,(A) > p,(X) - c . It remains to show that fn converges to f uniformly on A; equivalently, that hn - 9n converges to 0 uniformly on A. Fix 6 > 0. For x E *A there is a k E N such that hk(x) - gk(x) < 6. Note *h k( x ) - *g k (x) < 6 as well. Let ¢(x) be the least k such that *h k (x) - *g k (x) < 6. The function ¢ : *A ---+ N is internal and finite-valued, so has a standard upper bound N E N. Then *h k - *g k < 6 on *A for every k 2 N, so h k - 9k < 6 on A; this completes the proof. D

15.4

A Theorem of Riesz

Let T be a continuous linear real functional on .C 2 (X, A, fL ); then there is a g such that for every f E .C 2 (X, A, p,), T(f) = J fg dp,.

Theorem 4

g is actually in .C 2 (X, A, p,) . By saturation let A be a *-finite algebra with A0 c;;;; A c;;;; *A . There

After the proof we show that the function

Proof. is an internal *-partition II of *X which corresponds to A in the sense that the latter is the internal closure of the former under hyperfinite unions. Now, define

i (x)

:=

{

*T( xv ) *p, (p )

0,

'

x E p E II and *p, (p )

>

otherwise.

Let a : II ---+ *X be an internal choice function, that is, For any bounded, measurable f : X ---+ rn:.,

T (f) = L * T (*f xv ) p p p

0;

ap E p for p E II. ( 1 5.4. 1 ) ( 1 5.4.2) ( 1 5.4.3) (1 5.4.4)

223

1 5 .4. A Theorem of Riesz

1p � L 1 *j(x yY (x) *df1 p p = J *f :Y *d/1 =

L *f(avYf (x) *d/1

(1 5.4. 5)

p

( 1 5.4. 6) ( 1 5.4. 7)

The steps from ( 1 5.4. 1 ) to ( 15.4.2) and (15.4.5) to ( 1 5.4.6) follow from boundedness of f and the definition of A. For (15.4.2) to ( 1 5 . 4.3) note that by continuity of T, if p E II and *!1(P ) = 0 then *T( xv ) = 0. Moreover, if *f is replaced in the above by any internal function h : *X ---+ *JR:. with the property that h is constant on each p E II, then all but the first equality in the above still holds, and we obtain *T(h) = J hi d*f1 . In particular, if h = XA for some A E A then *T( x A ) = fA :Y d*f1. The proof now proceeds as follows : (i) Show :Y is finite almost every­ where (and therefore 1 =0 :Y exists almost everywhere) . (ii) Show that :Y is S-integrable (and therefore 1 =0 :Y is integrable) . (iii) Put G = IE[riAs] (the conditional expectation of 1 ) . (iv) Let g be the restriction of G to X; by Theorem 2 g is A-measurable. (v) Show that g works. For (i) , write [:Y < n] = {x E *X : :Y(x) < n } ( n E *N) , and [i < oo] = U n EN [:Y < n] . Suppose (for a contradiction) that f1L ( [:Y < oo] ) < 1 - r for some standard r > 0. Then !1 ( [:Y < n ] ) < 1 - r for each standard n E N, so !1 ( [:Y < H] ) < 1 - r for some infinite H. But then oo > T(1) � *J1:Yd*f1 (by the note above) 2 *J [:Y ? H] :Yd*/1 2 H*!1 ( [:Y 2 H] ) > rH, which is infinite, a contradiction. For (ii) , the reader is referred to [4] for a discussion of S-integrability. In particular, to show that :Y is S-integrable, it suffices to show that VH in­ finite, *J :Y > H] :Yd*/1 � 0. (Indeed, this can be adopted as the definition of I S-integrability. ) So, fix such an H. Note [:Y > H] E A; it follows that * J :Y > H] :Yd*/1 = I *T(X[:Y > HJ ) · Suppose (for a contradiction) that *T(X[:Y > HJ ) > r > 0, r stan­ dard. By (i) , *!1 ( [:Y > H] ) � 0. It follows from transfer that for every stan­ dard n E N there is a Bn E A with T(XBn ) > r and f1(Bn) < 1 / 2 -n . Put B n := Um > n Bm . Then 11(B n ) < L m > n 2 - m = 2 - n + l , but T(XBn ) > r. As n ---+ oo, X;;n ---+ 0 in £ 2 but T(XBn ) > -:;., contradicting continuity of T. It follows from S-integrability of i that 1 =0 :Y is integrable, so its con­ ditional expectation with respect to As, IE[riAs] , exists; put G = IE[riAs] . (Conditional expectation is defined in the next section, where a simple non­ standard proof for its existence is given. The reader is referred to chapter 9 of [5] for properties of the conditional expectation.)

224

1 5 . More on S-measures

Finally, let g G l x , which ( as noted above ) is A-measurable by Theorem 2. It remains to show that g satisfies the conclusion of Theorem 4. Let f E J:} (X, A, p,) , and suppose first that f is bounded. This ensures that (*!)1 is S-integrable, and (0*j)r E J:} (*X, AL, p,L) . Then =

J *FI d *p, � J (0*!) { dp,L = j IE W *f)r i A s ] dp,L j C *f) IE [riAs] dp,L = J (0*j)G dp,L = J fg dp,

T(j) �

=

( 1 5.4.8) ( 1 5.4.9) ( 15.4. 1 0) ( 15.4. 1 1 ) (15.4. 12) (15.4. 1 3)

The step from ( 15.4.8 ) to ( 15.4.9 ) is by S-integrability, ( 1 5.4. 10 ) to ( 15.4. 1 1 ) is a standard property of conditional expectation ( using the fact that 0*j is As-measurable ) , and (15.4.12 ) to ( 1 5.4. 13 ) is Corollary 1 with p = 1 . For unbounded f E J:} (X, A, p, ) and n E N, let fn = max { -n, min { !, n } } . By the result above , T(jn ) = J fn g dp,. By continuity of T and Lebesgue's Dominated Convergence Theorem ([ 5 ] , Theorem 5.9 ) , T(j) = J fg dp,. D The Riesz Theorem is usually stated in the following nominally stronger form.

Let T be a continuous linear real functional on .C 2 (X, A, p,) ; then there is a g E .C 2 (X, A, p,) such that for every f E .C 2 (X, A, p,), T(j) = J fg dp,.

Corollary 2

Proof. It suffices to show that any g satisfying the conclusion of Theorem 4 is already in .C 2 (X, A, p,). Since T is continuous, it is bounded, so there is a constant C such that for any f E .C 2 ( X, A, p,) , T(j) � CIIJII 2 · Put 9n = min { g, n } . Then ll9n ll� = Jg� dp, S fgn g dp, = T(gn ) S Cll9n l l 2 · Divide both sides of this inequality by ll9n ll 2 and square, obtain J g� dp, ll9nll � S C 2 . The result now follows by Fatou's Lemma ([5] , Lemma 5.4) . D

15.4. A Theorem of Riesz 1 5 .4 . 1

225

Conditional expectation

Theorem 5 Suppose (X, A, f-L) zs a probabdzty measure, that 'B C:: A is another a-algebra, and that j E ,.C} (X, A, tt) . There is a 'B -measurable function g : X ___, ffi. such that for any B E 'B, f df-L 9 df-L.

JB

=

JB

The function 9 is called the conditwnal expectation of f on 'B, and denoted by IE [f i'B] . To avoid circularity, it is necessary to show that IE [f i 'B] exists without use of the Riesz Theorem (or related results, such as the Radon-Nikodym Theorem) . Such a proof appears in [3] . This section presents a modification of that proof which is even more elementary, in that it does not require the Hahn decomposition. Without loss of generality f 2 0. Put 9 {9 E ..CJ (X, 'B, f-L) : VB E 'B , 9 df-L <:.:: f df-L }. Note if 9 1 , 92 E 9 then max{9 1 , 92 , 0} E 9 . Let r = sup{fx gdf-L : g E 9}. and for n E N let 9n E 9 with fx gn df-L > r - l/2 n ; we may assume that 0 <:.:: 9 1 <:.:: 92 <:.:: · · · . Put 9 supn 9n · Note that if B E 'B , 9 df-L = supn 9n df-L <:.:: f df-L, so 9 E 9. Moreover, fx 9 df-L = r . It remains to show that for any B E 'B. f dtt = 9 df-L. Suppose not; then there is a B E 'B and c > 0 with f df-L - 9 df-L = c. For some 6 > 0, c 1 = JB (j - g)df-L > 0 where g = 9 + 6 · g almost witnesses a contradiction; the perturbation by 6 needs to be localized to a slightly smaller set. As in the proof of Theorem 4 let 23 be a * -finite algebra with {*B : B E 'B } C:: 23 C:: *'B, and let II be the internal hyperfinite *-partition of *X corresponding to 23 . Let B + = U{b E II : * (! - g )d *f-L > 0}. Put s = o * (! - g )d *f-L . Note that for every C E 'B , if C C:: B then Jc(f - g) df-L = * (! - g )d*f-L <:.:: s : in particular. c1 <:.:: s. For n E N consider the statement,

JB

JB

JB

=

JB

JB

JB

=

JB

JB

JB+ f.c

Jb

As this holds in the nonstandard model (with B + transfer in the standard model. For m E N put Bm B00 n m B m . Note B= E 'B and B = C:: B, so g)df-L > s - 2 -n , the other hand. since always Ln >m 2 - n = s - 2 -m , therefore . (j - g)df-L 2 s . Put 9 1 = 9 + If C E 'B, =

6XB=· _

for Bn) , it holds by U n >m Bn , and put (j - g)df-L <:.:: s. On

=

JB=

JBJJ JB

JB m (j - g)df-L > s ­

CX)

Jc\ Bx u

r u 9') df-L = r

Jc

JB XB

_

9) df-L +

.fCnB=u - g) df-L .

15. More on S-measures

226

The first term in this sum is nonnegative since g E 9 . The second is nonnegative since otherwise g')df-L 2 0, so g' E fj )df-L > s. It follows that 9 . Since s > 0, !-L(BcxJ) > 0, so g' df-L g df-L + 6f-L(B00) > r. a contradiction.

JBx\C ( j -

J

=

J

Jc ( J -

References [1] C . WARD HENSON and FRANK WATTENBERG , "Egoroff ' s theorem and the distribution of standard points in a nonstandard model", Proc. Amer. Math. Soc., 81 ( 1981) 455-461 . [2] W . A . J . LUXEMBURG, O n some concurrent binary relations occurring in analysis, in Contributions to non-standard analysis (Sympos., Oberwol­ fach, 1970), Studies in Logic and Found. Math. , Vol. 69, North-Holland, Amsterdam. 1972. [3] DAVID A. Ross, Nonstandard measure constructions - solutions and problems, in Nonstandard methods and applications in mathematics, Lec­ ture Notes in Logic, 25, A.K. Peters, 2006. [4] DAVID A . Ross. Loeb measure and probability, in Nonstandard analysis (Edmburgh, 1996), NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., vol. 493, Kluwer Acad. Publ., Dordrecht, 1997. [5] DAVID WILLIAMS, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. [6] B EATE ZIMMER. "A unifying Radon-Nikodym theorem through nonstan­ dard hulls", Illinois .Journal of Mathematics, 49 (2005) 873-883.

A Radon-Nikodym theorem for a vector-valued reference measure G. Beate Zimmer*

The conclusion of be represented

a

Abstract

Radon-Nikodym theorem iH that

as �m

that for all measurable sets Lebesgue)

a

measure f-1. can

integral with respect to a reference measure

A, p(A)

such

JA .f1,. (x) d).. with a (Bochner or

=

integrable derivative or density .fw The mcasme ).. is usually a

countably additive <J-finit.e measure on the given measure space and the

mcasmc JL is absolutely continuous with respect to ).. . Different theorems

have different range spaces for p., which could be the real numbers, or Banach s paces with or without the Radou-Nikodym property.

Iu this

paper we ge1ieralize to derivatives of vector valued measures with re­ spect a vector-valued reference measure. theorem for vector

measures

Vvc present a R.adon-Nikodym

of bounded variation that arc absolutely

continuous with respect to another vector measme of bounded variation. W'hile it is easy in settings sud1 on the interval

[0, 1]

as

Jl < <

,\ , where ,\ is Lebesgue measure nonstan­

and Jl· is vector-valued to write down a.

clard Radou-Nikodym deri vative of

Lhe form t.p :

*[0, 1]

-->

fiu ( *E) by

t.p1, (:1:) 2:!1 :���:\ lA, (.1; ) , a vector valued reference measure does uot. =

allow this approach.

as

the quotient of

two vectors

in different B anach

spaces is undefined. Furthermore, generalizing to a vector valued control mcmmre uecessita(;es the usc of a generalization of the Bartle

bilinem· vector integral.

16 . 1

integral , a

Introduction and notation

For noustandard notions and notatious not defined here we refer for exam­

ple to the book by Albeverio, Fenstad, 1-Ioegh-Krohn and Lindstrom

t.he survey on nonstandard hulls by Henson and 1\tloore *Department of Tvfathematics and Statistics, Texas Corpus

Christi, TX 781112.

[1]

and

[6] or the introduction

A&M Universit.y -

Corpus Christi,

16. A Radon-Nikodym theorem

228

to nonstandard analysis in [7] . Plenty of information about vector measures can be found in [5] . Throughout, D is a set and I: is a a-algebra of subsets of D . E, F and G denote Banach spaces. f-L : I: ---+ E and v : I: ---+ F are countably additive vector measures of bounded variation. The variation of v is defined as I vI (A) = sup L 1rEII llv(A") II where A E I: and II is a finite partition of A into sets in I: . B y Proposition I . l . 9 i n [5] the variation lvl of a countably additive measure v is also a countably additive measure on I:. We think of the nonstandard model in terms of superstructures V(X) and V(*X) connected by the monomorphism * : V(X) ---+ V(*X) and call an ele­ ment b E V(*X) internal, if it is an element of a standard entity, i.e. if there is an a E V (X) with b E *a. We assume that the nonstandard model is at least N-saturated, where N is an uncountable cardinal number such that the cardinality of the a-algebra I: of I vi-measurable subsets of D is less than N. Let (*D, Llvi (*I: ) , lvl ) denote the Loeb space constructed from the nonstan­ dard extension of the measure space (D, I:, I v i ). This measure space is obtained by extending the measure 0*1vl from *I: to the a-algebra generated by *I:. _I he completion of this a-algebra is denoted by Llvi (*I: ) . The Loeb measure l v l is a standard countably additive measure. The functions we work with take their values in a Banach space E or its nonstandard hull E. The nonstandard hull of a Banach space E is defined as E = fin(* E)/ the quotient of the clements of bounded norm by clements of infinitesimal norm. The nonstandard hull is a standard Banach space and contains E as a subspace. We denote the quotient map from fin(*E) onto E by or 'if£ . Whenever we encounter products of Banach spaces, we equip them with the floo norm: l i e J l l = max(ll e l l , I I J I I ) . B y a result of Loeb in [8] , there exists an internal *-finite partition of *D consisting of sets A1 , . . . , AH E *I: (H E *N ) such that the partition is finer than the image under * of any finite partition of D into sets in I:. The proof uses a concurrent relation argument. From this fine partition we discard all partition sets of *!v i -measure zero. The remaining sets still form an internal collection, which we also denote by A1 , . . . , AH . ::::J ,

1r

16.2

X

The exist ing literature

Ross gives a nice detailed nonstandard proof of the Radon-Nikodym the­ orem for two real-valued a-finite measures f-L < < v on a measurable space in [9] . Previously we have studied nonstandard Radon-Nikodym derivatives of vector measures tt : I: ---+ E with respect to Lebesgue measure on [0, 1 ] . We have shown that through nonstandard analysis a unifying approach to

16.2. The existing literature

229 or

Radon-Nikodym derivatives independent of the Radon-Nikodym property lack thereof of E can be found. The generalized derivatives we constructed are not necessarily essentially separably valued and therefore not always Bochner integrable. However, a generalization of the Bochner integral in [10] for func­ tions with values in the nonstandard hull of a Banach space allows one to integrate the generalized derivatives. In [ 1 1] with methods similar to those used by Ross [9] or Bliedtner and Loeb in [3] , and with the use of local reflex­ ivity we "standardize" the generalized derivatives from maps *D ---+ E to maps D ---+ E" with values in the second dual of the original Banach space E. The vector measures book [5] by Diestel and Uhl is still considered as the authoritative work on Radon-Nikodym theorems. The standard literature yields very little on vector-vector derivatives; only Bogdan [4] together with his student Kritt has made an initial foray into this field. In their article they assume that they have two vector measures f-L : I: ---+ E and v : I: ---+ F with f-L < < I v 1 . and they assume that the range space of f-L has the Radon-Nikodym property and the range space of v is uniformly convex. They then define a derivative and integral in the form f-L(A) = fA ��I (w) t l (w) dv, where tl is a function from D into F', the dual space of the range of v . The integrand is then of the form �� ( w) = e . !' ' a function from n into E X F'. A trilinear product on E x F x F' with values in E can be defined as e · J' · f f' (f) · e. This is needed to integrate simple functions of the form L�=1 e, . !I · 1 A , : n ---+ E X F' with respect to the F-valued measure v as follows: ·

=

L t e; · JI · 1A, dv = t e; · JI · v(A; ) = t e, · J; (v(A; )) E E . ,=1

,=1

,=1

The main difficulty is the definition of tl , the derivative of a real-valued measure with respect to a vector-valued measure, the opposite setting from the ordinary Radon-Nikodym theorems. Bogdan uses his assumption that F is uniformly convex to find this derivative. Since v < < l v l and since uniform convex spaces are reflexive and hence have the Radon-Nikodym property, there is a Bochner integrable function d��l : D ---+ F such that for all measurable sets A, the vector measure can be written as a Bochner integral v(A) = fA d�� l dl v l . A standard theorem ( e.g. Theorem 11.2.4 in [5]) about vector measures states that the norm of the Radon-Nikodym derivative is a Radon-Nikodym derivative for the total variation of the vector measure, i.e. for all A E I: l v i (A)

=

i l d�� � � d l v l .

This implies that I vi-almost everywhere on D, II d��1 1 1 1 . For uniformly convex spaces F there is a continuous map g : Sp ---+ SF' , where Sx denotes =

230

16 .

A Radon-Nikodym theorem

the unit sphere of X, such that ( !, g(f)) = 1 for all f E Sp. In this case, the composition g o d�� l is a map from D into the unit sphere of F'. The derivative of f-L with respect to v is defined as �� � · (g o d�� l ) : D ----+ E F', or, if one regards the product e f' as a rank one continuous linear operator from F to E, the derivative can be considered a map: �� : D ----+ L(F, E) . Since d��l and d�� l arc both Bochner integrable and g is continuous and g o d�� l is almost everywhere of norm one, it is easy to show that the result is integrable with respect to the vector measure v with an integral that uses approximation by simple functions. =

x

16.3

x

The nonstandard approach

With nonstandard analysis we can significantly weaken the assumptions on the Banach spaces E and F. We use Bogdan's idea of writing the derivative dlvl -_ djV[ dp · ----;IV dfl . (g o djV[ dv ) . as djV[ The derivatives ��I and d��l are derivatives of a vector measure with respect to a real valued measure. They can be found and integrated with the methods described in [10] , [11] or [12] . In [10] we defined a Banach space M(lvi, E) of e�ended integrable func­ tions as the set of equivalence classes under equality lvl-almost everywhere of functions f : *D ----+ E for which there� an internal, *-simple, S-integrable f l vl-almost everywhere on *D. Such a cp : *D ----+ *E such that 71£ o cp cp is called a lifting of f. On M ( lvl, E) we defined an integral by setting JA f d!vl 71 ( fA cp d * lvl ) , for all A E *I:, where the integral of the internal simple function cp is defined in the obvious way. This integral can be extended to sets in the Loeb a-algebra and generalizes the Bochner integral in the sense that M ( lvl, E) contains L1 ( lvl, E) and the integrals agree on that subspace. However, .iH(Ivi, E) also contains functions which fail to be essentially separa­ bly valued and hence fail to be measurable. Lemma 1 in [11] translates to this situation as: =

=

Let E be any Banach space and let f-L : I: ----+ E be a ! v i ­ absolutely continuous countably additive vector measure of bounded variation. Then the internal *-simple function cp 11 : *D ----+ *E defined by

Lemma 16.3 . 1

is S -integrable, where A1 , . . . , AH is the fine partition of *D introduced above.

16.3. The nonstandard approach

231

S-integrability implies that the function is l v l -almost everywhere fin(*E)­ valucd. This allows us compose an S-intcgrable internal function with the quotient map 71E : fin(*E) ----+ E to make it a nonstandard hull valued function defined on the Loeb space ( *D , L i v i (*L: ) , !vi) . Define f11 E M(lvi, E) by

f!l = 71£ o CfJw

Theorem 4 in [11] asserts that if the vector measure f-L has a Bochner integrable Radon-Nikodym derivative, then the generalized derivative "is" the Radon­ Nikodym derivative in the following sense:

Let f-L : I: ----+ E be a countably additive vector measure with a Bochner integrable Radon-Nikodym derivative f : D ----+ E. Define the gener­ alized Radon-Nikodym derivative f11 : *D ----+ E as above. Then 71£ o *f = f11 on each set Ai in the fine partition of *D.

Theorem 16.3.2

Of course, the same arguments also work for finding a F-valued generalized derivative fv E lVI(Ivi, F ) of the vector measure v : I: ----+ F with respect to its total variation I v i . Differentiating v with respect to its total variation gives one extra feature of the derivative:

Let F be a Banach space and let v : I: ----+ F be a countably additive vector measure of bounded variation. Define

Lemma 16.3.3

Then fv = 71F o C{Jv is I vi -almost everywhere of norm one. Since v << lvl , the S-integrability of fv follows from Lemma 16.3.1 . The norm condition follows from an adaptation of a basic standard result (Theorem 11.2.4 in [5] ) : if f is Bochner integrable and a vector measure F is defined by F(A) = fA f dlvl, then I F I (A) = fA ll f l l dl v l . By construction of the generalized derivative, Proof.

for all A E *I:. In this situation it implies that * l v i (A) = * d* l vl and hence * II CfJv (w) ll

=

l I d��� �

1 for lvl-almost every w E *D.

D

232

16.4

16. A Radon-Nikodym theorem

A nonstandard vector-vector integral

The next theorem allows us to generalize the generalized Bochner integral to a vector-vector integral which for any simple function obviously agrees with the Bartle integral developed in [2] . For the definition of the Bartle integral, E, F and G are Banach spaces equipped with a continuous bilinear multipli­ cation E x F ----+ G and v : I: ----+ F is a countably additive vector measure of bounded variation. Such a product can arise by taking E F G to be a C*-algebra, or by taking E = F' and G = rn:., or by taking E = L ( F, G ) . A function f : D ----+ E is Bartle integrable if there is a sequence of simple functions sn that converges to f almost everywhere and for which the se­ quence An ( A ) fA S n dv converges in the norm of G for all A E I:. Then fA f dv = limn_, = >-n (A ) exists in norm uniformly for all A E I:. Essentially bounded measurable functions are Bartle-integrable. The Bartle integral is a countably additive set function. =

=

=

Let E, F and G be Banach spaces equipped with a continu­ ous bilinear multiplication E x F ----+ G. Assume that v : I: ----+ F is a countably additive vector measure of bmmded variation and f E Af (!vi, E) . The integrand f has an internal lifting i.p J and there is an internal *-simple lifting i.pv of the generalized derivative of v with respect to I v i . Then for all A E *I:, the integral fA f dv defined by

Theorem 16.4 . 1

l f dv

= no

(l

i.pJ · i.pv

d* l v l

)

is a well-defined countably additive G -valued integral. The function f E 1\!I (!vi, E) has a internal, S-integrable and *-simple lifting i.pJ : *D ----+ fin ( *E ) . By Lemma 16.3.3 i.pv : *D ----+ fin ( *F ) is internal, *-simple, S-integrable and of norm one almost everywhere. Continuity of the multiplication E x F ----+ G implies that the product i.p j ' i.pv is an S-integrable and internal *-simple function from *D ----+ fin ( *G) . This means that it is a lifting of an extended Bochner integrable fun�on in M(!vi, G ) and hence the G-valued generalized integral with respect to l v l applies and gives a well-defined vector­ vector integral. Since the extended Bochner integral is countably additive, so is this integral. D Proof.

The use of nonstandard analysis simplifies standard arguments here, since in the standard setting this argument would only work if the Banach space F had the Radon-Nikodym property.

16.5. Uniform convexity

16.5

233

Uniform convexity

Bogdan needed the assumption that F is uniformly convex to convert the F-valued derivative d�� l into the F'-valued derivative tl . He composed the derivative d�� l with a continuous map g from the unit sphere of F to the unit sphere of its dual space F' such that for all f E Sp, (!, g(j)) 1 . The resulting map g o d�� l : D ----+ F' is lvl measurable and inherits the integrability of d�� � · If the Banach space F happens to be uniformly convex, we compose the nonstandard extension of g with VJv. In this case, (!, g(j) ) 1 for all . f E SF t ransl ates . t o /\ **l vv i((AA,, )) , *g *l*vv (i (AA,,)) - 1 for . - 1 , . . . , H 1.e. =

(

m

))

can

-

z -

=

Let E be any Banach space and let F be a uniformly convex vector space. If the vector measure f-L : I: ----+ E is absolutely continuous with respect to the total variation of the vector measure v : I: ----+ F, then there is an extended Bochner integrable function f : *D ----+ ExF' S1Lch that for all A E I:

Theorem 16. 5 . 1

f-L(A) =

J*A f dv.

Define VJ J Yp · (*g o VJ v ) , where Yp is a lifting of the generalized derivative d��l , tpv is a lifting of the generalized derivative d�� l and g is the continuous map from Sp to Sp . This lifting is internal, S-integrable, *-simple and takes its values in fin(*E) *Sp . Define f 71ExF' o :PJ · Then

Proof.

=

fn f d v

x

=

71£ 7IE rrc

(!n tp (w)

=

) (�� *lvi*f-L(Ai)(Ai) . *g ( *lv*v(Ai)i (Ai) ) *v(Ai) ) (f ,;��(�;) 'l vi (A;)) p

· *g ( VJv ( w ))

d *v

·

71£ ( *f-L(*D))

f-L(D) ,

and similarly for f-L(A) , where (Ai) is replaced by (A; n *A) throughout. Since the fine partition refines the standard partition of D into A and its complement, Ai n *A is either the empty set or A,. D

16. A Radon-Nikodym theorem

234

If E has the Radon-Nikodym property, then this lifting coincides with the image of the derivative found by Bogdan in the nonstandard hull of the space of vector-vector integrable functions on D.

16.6

Vector-vector derivatives without uniform convexity

We can weaken the assumption on F: it follows from the Hahn-Banach theorem that for each nonzero f E F there is an f' E F' with 1 1 .!' 1 1 1 and f'(f) = llfll · Without uniform convexity, the choice of f' is not necessarily unique, and the map f f----+ f' need not be continuous on the unit sphere of F. We can forego the use of a function defined on all of Sp. All we need is a derivative on each set A, in the fine partition of *D. =

g

Let F be any Banach space and let v : I: ----+ F be a countably additive vector measure of bounded variation. Then there is a function 9i vl E M(lvi, P) such that Theorem 16.6. 1

1A 9ivl dv = o(*lv i ( *A)) = lvi (A)

for all A E I: .

Proof. As the collection of sets A, is internal, we can use the axiom of choice to get an internal collection of norm one functionals E *Sp with ( *v(A; )) = * lv i (A; ) for i = 1 , . . . , H.

g;

g;

Define an internal *-simple S-integrable function .

1/Ji vl

=

H

L i=l

:

1/J i vl *D ----+ *Sp by

g; . A

1 ,·

This function is a lifting of an extended integrable function 9i vl E M(lvi, P) . We can define a continuous bilinear multiplication F' x F ----+ JR; by f' · f = f' (f) and then use the integral from theorem 16.4. 1 . In this case the quotient map no becomes the standard part map o : fin (*JR;) ----+ R By the choice of the fine partition. A; n *A is either the empty set or equals Ai. d*v dv = o

lA 9ivl

(lA 1/l i vl ) 0(1A t, g; ·1A, d*v)

16.6. Vector-vector derivatives without uniform convexity

(� o(� o

gi · *v(Ai n *A) *lvi (Ai n *A)

o ( * l v i ( *A ))

)

235

) =

for any set A E I:. This finishes the proof, as for all A E I:, lvi (A)

0 ( * 1 v i ( *A )) . D

The last theorem constructed a derivative for any countably additive vector measure of bounded variation with respect to its total variation as an element of M ( lvl, P) . This was the main difficulty in differentiation with respect to a vector measure. Now we can differentiate one vector measure with respect to another vector measure. :

Theorem 16.6.2 Let E and F be any Banach spaces, let f-L I: ----+ E and v I: ----+ F be a countably additive vector measures of bounded variation such that f-L < < lvl . Then there is a function h E M(lvi, ExF' ) such that for all A E I: f-L ( A ) h dv. :

=

j

*A

x

x

The continuous trilinear multiplication E F' F ----+ E defined by f' (f) · e extends to a continuous bilinear multiplication ExF' F ----+ E defined by � f') n(f) 7rE(J'(f) · e). The extended integrable function h *D ----+ E F' is defined through its lifting cp cp11 · 1P1v1 , the product of the lifting cp11 of d��l E M(lvi, E) and the lifting 1/J i v l of t l E M(lvi, P) found in theorem 16.6. 1 . Setting Proof. x

x

e f' f

= :

x

x

=

x

=

*

x

we obtain a simple internal function with values in fin ( *E ) *SF' . The S-integrability of cp follows from the S-integrability of cp11 and Lemma 16.3.3 which asserts that *111/l i v l ll :::::J 1 almost everywhere. Then for all A E I: 1rE ( *f-L ( *A )) f-L (A )

"" (f (�� 1rE

M ( A; n 'A )

'

)

*fJ (Ai n *A ) * l v i ( A n *A) * l vi (Ai n *A ) ·

2

)

16. A Radon-Nikodym theorem

236

:

and h *D . df.lv t 1ve d .

16.7

----+

ExF' is an extended integrable generalized vector-vector derivaD

Remarks

All the constructions above depend on the choice of a fine partition A 1 , . . . , AH of *D that refines all finite standard partitions of D into measurable sets. A different choice of the partition would yield different derivatives, especially in the case of non-uniformly convex Banach space F. However, for the standard analyst there is no noticeable difference between different choices of fine partitions, as the integrals over any standard set will be the same, no matter which fine partition is used. Representing a vector measure v I: F as an integral of its generalized derivative d1� l simplifies some of the results on Loeb completions of internal measures [13] by Zivaljevic. a

:

----+

References [1] S. ALBEVERIO, J . E . FENSTAD, R. H 0EGH-KROHN and T. LINDSTR0M,

Nonstandard methods in stochastic analysis and mathematical physics,

Academic Press, Orlando, F L, 1986.

[2] R. G . BARTLE, "A general bilinear vector integral", Studia Math. , 15 ( 1956) 337-352. [3]

.J . BLIEDTNER

and P . A . LOEB, "The Optimal Differentiation Basis and Liftings of L00", Trans. Amer. Math. Soc., 352 (2000) 4693-4710.

[4] W . M . BOGDANOVICZ and B. KRITT, "Radon-Nikodym Differentiation of one Vector-Valued Volume with Respect to Another", Bulletin De L' Academic Polonaise Des Sciences, Scric des sciences math. astr. ct phys. , XV ( 1967) 479-486.

References

237

[5] J. DIESTEL and J . J . U HL , Vector Measures, Mathematical Surveys 1 5, American Mathematical Society, Providence, RI, 1977. [6] C .W . HENSON and L. C . MOORE, Nonstandard Analysis and the the­ ory of Banach spaces, in A.E. Hurd (ed.) Nonstandard Analysis-Recent Developments, Springer Lecture Notes in Mathematics 983, 1983. [7] A . E . HURD and P .A. LOEB, An Introduction to Nonstandard Real Anal­ ysis, Academic Press, Orlando, FL, 1985. [8] P . A . LOEB , "Conversion from nonstandard to standard measure spaces and applications to probability theory", Trans. Amer. Math. Soc., 2 1 1 (1 975) 1 13- 122. [9] D. A. Ross, Nonstandard Measure Constructions - Solutions and Prob­ lems submitted to NS2002, Nonstandard Methods and Applications in Mathematics, June 1 0-16 2002, Pisa, Italy. [10] G . B . ZI M MER , "An extension of the Bochner integral generalizing the Loeb-Osswald integral", Proc. Cambr. Phil. Soc., 123 ( 1998) 1 1 9- 131. [ 1 1] G . B. Zrl\IMER, "A unifying Radon-Nikodym Theorem through Nonstan­ dard Hulls", Illinois Journal of Mathematics, 49, (2005) 873-883. [12] G . B. Zil\IMER, Nonstandard Vector Integrals and Vector Measures, Ph.D . Thesis, University of Illinois at Urbana-Champaign, October 1994. [13]

R.

T. Z IVALJEVIC, "Loeb completion of internal vector-valued measures", Math. Scand. , 56 (1985) 276-286.

Differentiability of Loeb measures Eva Aigner*

Abstract We introduce a general definition of S-diffcrcntiability of m1 internal measure and compare different special cases. It will be shown how S-differentiability of an illtcrnal measure yielc Is differeutiability of the associated Loeb measure. vVc give some exmnplcs.

17. 1

Introduction

In this pa per we present some new results about differential properties of Loeb measures. The theory of differentiable measures was sugges ted by Fomin [9] as aJl infinite dimensional substitute for the Sobolev-Schwartz theory of distributions and has extended rapidly to a strong field of research . In partkular during the last ten years it has become the foundation for many applications in different fields such as quantum field t heory ( sec e.g. [10] or [14]) or stochastic analysis ( see e.g. [4] , [5] , [6] and [151 ) . There are many different. notions of measure differentiability. The following two are most, common (see e.g. [3[ , [6[ , [9[ , [13[ and [151). Let v be a Borel measure on a locally convex space E and y an element of E. l) The measure 11 is called Fom·in-d-ifferentiable along y if for all Borel sub­ sets B C E the limit lim

r�O, rElR

11(B + ·ry) - z; ( B) T

ex i sts . v is called Skorohod-differentiable with respect to y, if there exists another Borel measure v1 on E, such that for all continuous real-

2) The measure •

Ludwig-Maximilians- Universitiit , Miincheu , Germany.

17.1. Introduction

239

valued bounded functions g on E limr

r ---+ 0.

J g(x - ry) dv( x) - J g(x ) dv( x) ER r

=

J g(x ) dv'(x ) .

The relationships between these two approaches were studied by Averbukh, Smolyanov and Fomin [3] and Bogachev [6] . A very general definition of differentiability of a curve of measures is given by Weizsacker in [16] . In this paper we define measure differentiability as follows.

Let (D, :F, v) be a measure space, (vr) - c
=

=

J

Note that Definition 1 7. 1 . 1 covers the cases mentioned above, since for a locally convex space D and a fixed vector y E D a curve (vr ) - c
=

:

1 7. Differentiability of Loeb measures

240

17.2

S-different iability of internal measures

Standing Assumption

Throughout this paper let D be an internal set, A an internal *t'Y-field on D and f-L :2: 0 an internal *t'Y-additive S-bounded measure on A. Let ( !-Lt)tEJ be an internal curve of nonnegative *t'Y-additive S-bounded measures on A. Since we want to obtain an external curve of Loeb measures, we assume that for some c E rn:. + the internal parameter set J is either an interval of *JR containing the standard interval I =] - c; c [ or J is a discrete interval { Jl , -r;-l , . . . , ki/ , fl } with H E * N \ N, k E { 1 , . . . , H } and c :'S: .fl . Moreover, the internal curve ( !-Lt)tEJ shall be S-continuous in the following sense. If t, s E J with t :::::J s, then f-Lt( A) :::::J f-Ls (A) for each A E A. Finally we assume that f-L = f-LO · We now introduce S-differentiability for internal measures.

Suppose we have a (not necessarily internal) set C of in­ ternal *JR:.-valued functions on D, each being A-measurable and S-bounded. We say that the internal measure f-L is S -differentiable with respect to the set C if there exists an internal S- bounded signed measure f-L1 on A so that for all f E C and for all infinitesimals t E J, t # 0 J f(w) df-Lt (w) - J f(w) df-L(w) :::::J f(w) df-L (w ) .

Definition 17.2.1

We call f-L1 an (internal) derivative of f-L ·

J

'

Note that a derivative is not uniquely determined by the above definition. If f-L is S-differentiable with respect to a set C and if for some infinitesimal t E J the internal measure 11 ' ;11 has limited values, then 11 ' ;11 is a derivative of f-L· In this section we will regard and compare S-differentiability for differ­ ent sets C of functions. Following the standard literature we define S-Fomin­ differentiability.

The internal measure f-L is called S -Fomin-differentiable if the differentiability is with respect to C = {1 A A E A} where 1 A is the indicator functwn. Note that in the case of S-Fomin-differentiability each measure 11 ' ;11 with t :::::J 0, t E J \ {0}, is a derivative of f-L · The following proposition shows the

Definition 17.2.2

:

power of S-Fomin-differentiability.

If f-L is S-Fomin-differentiable and f-L1 is a derivative of f-L, then f-L is S-differentiable wzth respect to the set C of all S-bounded *JR:.­ valued A-measurable functions. The Fomin-derivative 1/ is also a derivative with respect to C . Proposition 17.2.1

17.2. S-differentiability of internal measures

241

Let f-L be S-Fomin-differentiable and let f-L1 be a derivative of f-L · Let t # 0 be an infinitesimal of J and set [L := 11 ' ;'" . Since f-L1 :::::J [L on A we obtain J f(w)df-L'(w) :::::J J f(w)d[L(w) for all S-bounded A-measurable functions f : D ----+ *R D Proof.

When considering measures with Lebesgue densities, then the following lemma is useful. =

Let D *JR:., A the internal field of Borel subsets, y a fixed element of *JR:., f-L an internal S -bounded measure on A. Take J = *JR:. and for all t E J define f-Lt (A) = f-L(A + ty) for all A E A. Assume that f-L has an internal Lebesgue density f satisfying the following three conditions: 1) f 2 0, f = 0 only on a set of internal Lebesgue measure 0. 2) f is * differentiable in the direction ofy (with derivative f� (x) := f'(x) · y) . 3) If t 0, then for all x E *JR:. with f (x) # 0 f� (x) � f(x + ty) _ 1 . t f(x) f(x)

Lemma 1 7. 2 . 1

:::::J

(

)

:::::J

Now if the internal f1mction (3� : *JR:. ----+ *JR:., defined by f� (x) . (3� (x) = f(x) if f(x) # 0 0 if f(x) 0,

{

=

is 511 -integrable, then f-L is S-Fomin-differentiable and if f-L1 is a derivative, then for all A E A f-L1 (A) (3� (x) df-L(x) . :::::J

l

Since (3� is 511 -integrable, A f---+ JA (3� (x)df-L(x) defines an S-bounded measure. Now let N = {x E *JR:. : f(x) = 0}. Then >-(N) = 0, where A is the internal Lebesgue measure. If t 0, t # 0, then

Proof.

:::::J

I

!-Lt (A) � f-L(A)

- l (3� (x)df-L(x) l l l f(x + ty) - f(x) - f3J,(x) . f(x) l d>-(x) r l � ( f(x + ty) _ 1 ) - f� (x) l df-L(x) f(x) f(x) JA \ N t �

= :::::J

0.

Here we have used condition 3 ) and the fact that f-L is S-bounded.

D

1 7. Differentiability of Loeb measures

242

Example 1 7. 2 . 1 Let D = *JR:., A be the internal field of Borel subsets and f-L defined by f-L(A) fA l;x2 d>.. ( x) . Fix an element y E *JR:. and define the curve by f-Lt (A) .JA + ty l ;x2 d>.. ( x) , t E *JR:.. If y is limited. then it ' s easy to see that the assumptions of Lemma 17.2.1 are satisfied. Hence the measure f-L is S-Fomin-differentiable and if f-L1 is a derivative, then f-L'(A) :::::: fA ���¥ df-L(x) for each A E A. If y is an unlimited element of *JR:., then f-L is not S-Fomin­ differentiable, because for t = � and for the internal interval A = * [ 0 , 1] C *JR:. the value flt ( A);p ( A) is unlimited. =

=

Note that Lemma 17.2.1 is also true for D *JR:.L , L E *N. In the author's thesis [1] this is used to show the S-Fomin-differentiability of a nonstandard representation of the Wiener measure introduced by Cutland and Ng in [8] . We now turn to other forms of S-differentiability. Again. following the standard literature, we define S-Skorohod-differentiability. =

Remark

Let D be a subset of *M where NJ is a metric space and let A be an internal *a-field on D. The measure f-L is called S -Skorohod­ differentiable if it is S -differentiable with respect to the set of all S -bmmded, A-measurable functions f : D *JR:. that are S-continuous. This means, f ( w) :::::: f ( w) for w , w E D, w :::::: w . Definition 1 7.2.3

----+

I f D is as described in Definition 17.2.3, then, as a consequence of Proposi­ tion 17.2. 1 , S-Fomin-differentiability implies S-Skorohod-differentiability. The following example shows that the converse is not true. =

Fix a natural number H E *N \ N and let D c *JR:., D * Z } . The measure f-L is the counting measure defined on the field of internal subsets of n, i.e.

Example 1 7.2.2

{ 1-£

·

z : z E

f-L(A)

=

lA

n

[o H - 1 ] I �H .

Here [0, Hif 1 ] {0, 1-£ , . . . , Hii l } and I A I is the internal number of elements. Now let J c D, J = [- Jr , Jr] with l E *N, Jr :::::: � and let (f-Lt)tE J be defined by f-Lt (A) f-L(A + t). Note that for any k E *N with fl E J we obtain: =

=

and

17.3. Differentiability of Loeb measures Now choose k E *N with fl

E

J and fl 0 and

::::::

243

0 and set A = [0, k}/ ] . Then f-L -k - f-L H-k H

(A)

1.

Hence f-L is not S-Fomin-differentiable. It is not hard to check that f-L is S­ Skorohod-differentiable with internal derivatives 1"1;'-1 for all infinitesimals t E J \ {0} . We will give a standard application of this example in Example 17.3.1 . Finally we consider S-differentiability with respect to *continuous func­ tions. We will see that this is equivalent to S-Fomin-differentiability. The following proposition can be easily shown by transfer of Urysohn ' s Lemma.

Let D be an internal *normal space, A the internal field of Borel subsets. If ( !-Lt ) t EJ is a curve of *regular measures, then f-L is S­ differentiable with respect to the set C of all internal * continuous S -bounded functions if and only if f-L is S- Fomin-differentiable. Proposition 1 7.2.2

17.3

Differentiability of Loeb measures

In this section we show how S-differentiability of an internal measure yields differentiability (in the sense of Definition 17.1 .1) of the corresponding Loeb measure. Recall the Standing Assumption on page 240. Now we can define a curve of Loeb measures in a unique way. Let c E rn:.+ as described in the Standing Assumption (page 240) , I =] - r:: , r:: [ . For each r E I choose t E J such that t :::::: r and set f-Lr := f-Lt· Let us denote the associated Loeb spaces by (D, L 11r (A) , (f-Lr )L) · Since the Loeb a-fields L 11r (A) are not necessarily identical we choose a joint a-field :F c n rE I Lllr (A) . We now define the curve ((f-LL)r)rE I of measures on :F by (f-LL)r := (f-Lr)L restricted to :F. Let us mention some obvious connections between S-differentiability and differentiability.

Let f-L be S -differentiable with respect to a set C of internal *JR:.­ valued, A-measurable and S -bounded functions on D and let 1-L' be an internal derivative of f-L · 1) Then for all f E C

Lemma 1 7. 3 . 1

rlim ---> 0

J_ ) d_ f-Lr_(w_)'--)--_ f (_ 0 (=J-_f_ -_ w_ ( w_)d_f-L_ ( w )) --0 -'(cc_ r

______

The convergence is uniform, if C is internal.

1 7. Differentiability of Loeb measures

244

2) Suppose C

:=

{g : D ----+ ffi. : there is an f E C with 0 ( f (w ) )

and

:F.,, :=

=

g ( w ) for all w E D}

(nr L.,r (A) ) n £.,, (A) . EI

Then the Loeb measure ILL , restricted to :F.,, , is differentiable with respect to the set C and the Loeb extension ( tl ) L of o ( tl ) , restricted to :F1L' , is a derivative (ILL ) ' of ILL . This means that for all g E C : J g (w)d(ILL)r (w) - J g(w )d�LL (w ) g (W )d( 1L' ) L (W ) . l. r1m --->0 r =

J

Note that, if IL' and 1 ' are two internal derivatives of IL, then J g(w)d(IL')L (w) = J g( w )d(r')L for all g E C, but the Loeb measures ( �L ')L and (r')L on A and hence also the a-fields £.,, (A) and £1, (A) may be different. In Example 17.2.2 we defined an internal counting measure IL· Since IL is S-Skorohod-differentiable with internal derivatives T, t E J \ { 0 } , t � 0, we can17apply Lemma 17 17.3 . 1 . But as we have seen in Example 17.2.2, for k E *N with E J and � 0 and A = [0, k}/]

Example 1 7. 3 . 1

( IL Hk 17- IL ) (A) L

=

IL =!5:. - IL 0 and ( HI'l ) (A) L

=

1.

Nevertheless, there exists a a-field, on which the Loeb measures ("1 �" ) L coin­ cide for all infinitesimals t E J \ { 0 } . Let B(ffi.) be the ( standard ) a-field of all Borel subsets of R For B E B(ffi.) let sC 1 [B] = {w E D : 0W E B} . According to the usual approach to standard Lebesgue measure by a nonstandard count­ ing measure ( see [ 7] or [ 12] ) , the external set st - 1 [B] is an element of L.,r (A) for all r E I and (ILr ) L ( s C 1 [B ] ) = vr ( B ) , where ur ( B ) = JB +r 1 [o . l j (x ) d>. (x ) (A) with standard Lebesgue measure >.. But st - 1 [B ] is also an element of L for all infinitesimals t E J \ { 0 } and PrP t

( ILt � IL )

L

( sC1 [B ] ) = 1B( 1) - 1 B (O) .

Hence the Loeb measures ( "1�" h , restricted to the a-field {sC1 [B ] : B E B(ffi.)}, coincide for all infinitesimals t E J \ { 0 } . If we define a measure v' on B(ffi.) by v'(B ) = 1 B ( 1) - 1B(O), then the S-differentiability of the internal counting measure tt yields the Skorohod-differentiability ( with respect to 1 E ffi.) of the standard measure v with derivative v'.

17.3. Differentiability of Loeb measures

245

We will now see that in the case of S-Fomin-differentiability the a-field :F doesn ' t depend on the chosen internal derivative and the derivative (ILL)' of ILL is uniquely determined. 1\Ioreover, the differentiability of ILL is true not only with respect to standard parts of internal functions, but also with respect to all :F-measurable S-bounded real-valued functions on D. We gather these facts together into the following, which is the main theorem of this paper.

Let 1L be S-Fomin-differentiable and :F = n rE I L11r (A). Then ILL is differentiable on :F with respect to the set C = { 1 B B E :F} and the differentiability is uniform on C. The derivative (ILL )' is uniquely de­ termined and is absolutely continuous with respect to ILL . If IL' is an internal derivative of IL, then the Loeb extension (IL')L is defined on :F and coincides with (ILL)' . In particular this is true for all internal measures � , where t E J \ { 0} is infinitesimal. Theorem 1 7. 3 . 1

:

Let IL' be a derivative of IL· We will show the following statements: (A) For all internal sets A E A the limit

Proof.

(ILL) -'--- ( A)--- -' ----�-'--LL (A) ---'-- - --'-ll. m --"-----r---'r--> 0 r exists and is equal to ( �L')L ( A) . The convergence is uniform on the inter­ nal field A. (B) If N E :F is a ILL-nullset, then (ILL) r (N) - �LL (N) lim = 0. r r--->0 The convergence is uniform for all ILL-nullsets of :F. Any ILL-nullset of :F is also a (IL')L-nullset of :F. (C) If B E :F and if A E A is ILL-equivalent to B, then (ILL) r (A) - ILL ( A) (ILL) r (B) - ILL ( B) = lim lim ' r--->0 r---> 0 r r in particular the left limit exists. The convergence is uniform on :F. (D) The Loeb extension (IL')L is defined on :F and for all B E :F

(A) follows from Lemma 17.3.1. To prove (B) let N b e a ILL-nullset of :F, i.e. there exists a sequence (N� )n EN C A so that for all n E N we have N C N1_ , Nn� c:;; N� and 1

1 7. Differentiability of Loeb measures

246

p,(Nl ) :<:: � · Let N all r nE I

:=

n�=l N� . Then N E :F, p,L (N)

=

0 and we obtain for

Since p,L (N) = 0 we get

Therefore it is sufficient to show that lim0 (!n ); (JV) r---+

=

0. Now since p,L (N)

=

0,

r

limn_, = ( !1L ) r ( N� ) - limn_, = /1L ( N� ) r

( !1L) r (N� ) - !1L (N� )

lim

" It follows from (A), that for each E N the limit limr---+ O '\ exists and is equal to (p,')L (N� ). Since (p,')L is defined on the smallest a-field containing A, the limit limn ---+ = (p,')L (Nl ) (p,')L (N) also exists. Because of the uniform convergence stated in (A) and since (N� )n EN C A we can exchange the limits as follows: (11L)r( N 1 )-pL( N 1 )

n

=

lim nlim --+CXJ r--+ 0

(p,L) r (Nl ) - p,L (N1_ ) n

r

lim lim r--+ 0 n--+CXJ

(p,L) r (N� ) - ttL (N� )

n

So the measure /1L is differentiable at N and the value of the derivative is (p,')L (N) . It remains to show that (p,')L (N) = 0. To this end we usc an argument due to Smolyanov and Weizsacker [15] . Let us consider the function

Then f is differentiable at t = 0 and f'(O) = (p,')L (N) . Since f is nonnegative and j(O) = 0, the first derivative of f in 0 must be 0. Hence (p,')L (N) = 0. The uniformity of the convergence can be seen by using again the uniform convergence on A. Of course (t/)L (N) = 0 since N C N. Hence the P,L­ nullsets of :F are also (p,')L-nullsets of :F.

17.3. Differentiability of Loeb measures

247

(C) Here we show the differentiability for an arbitrary element of :F. So let B E :F, A E A ILL-equivalent to B and r E I. Since ILL (B ) = ILL (A) , it is sufficient to show that (ILL )r ( B ) - ( ILL)r(A) O. rlim ---> 0 r Since ILL is nonnegative the following estimate is easy to verify. =

where A 6 B denotes the symmetric difference ( A \ B ) U (B \ A ) . Now (B) yields

I

(ILL )r(B) - (ILL)r( A ) rlim ---> 0 r

I

< lim - r---> 0

(ILL)r( A 6 B) I I T

=

O.

The uniform convergence follows from (A) and (B). Hence (C) is proved. (D) follows from (A), (B) and (C).

D

Let IL be a measure with internal Lebesgue density satisfying the assumptions of Lemma 17.2.1 for a fixed y E *R Recall that the internal curve is given by ILt( A ) IL(A + ty) for all Borel subsets A C *R Let c E rn:. + , I =] - c, r:: [ . Since IL is S-Fomiu-differentiable, we can apply Theorem 17.3.1. Hence ILL is differentiable on :F = n r EI L ,lr (A) with respect t o c = { 1B B E :F}. Moreover, for the uniquely determined derivative (ILL)' we obtain: Example 1 7.3.2

=

for all B E :F, where (3� is defined in Lemma 17.2.1 The power of Fomin-differentiability is also shown in the last result. Corollary 1 7. 3 . 1 If IL is S -Fomin-differentiable with an internal derivative IL' and :F is defined as in Theorem 1 7. 8. 1, then the Loeb measure ILL , restricted

to :F, is differentiable with respect to the set C of all :F -measurable real-valued bounded functions on D. The Loeb measure ( �L')L, restricted to :F, is the deriva­ tive of (IL )L .

Proof.

This is routine integration theory using the uniform differentiability on

:F and the approximation of measurable functions by simple functions. D Remark An application of Fomin-diffcrcntiability is given in [15] by Smolya­

nov and Weizsacker. There, a nonnegative measure v on a locally convex space

1 7. Differentiability of Loeb measures

248

E is considered which is Fomin-differentiable along all elements of a Hilbert subspace of E. Using this measure differentiability, Smolyanov and Weizsacker introduce an operator on a subspace of £ 2 (E, v) . In the Gaussian case this operator is the derivative operator of the Malliavin calculus (see [11]). The question of the assumptions that are needed for differential properties of Loeb measures to yield such a derivative operator is the subject of ongo­ ing investigation. References

[1]

A IGNER , Differentiability of Loeb measures and applications, PhD the­ sis, Universitiit Munchen, (in prep.) E.

[2] S. ALBEVERIO, .J . E . FENSTAD, R. H 0EGH- KROHN and T. LINDSTR0M, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986. [3] V.I.

AVERBUKH, O . G . S MOLYANOV and S .V. FOl\1IN, "Generalized func­ tions and differential equations in linear spaces, I. Differentiable mea­ sures", Trudi of Moscow Math. Soc., 24 (1971) 140-184.

[4] V . I .

" Differential properties of measures on infinite dimen­ sional spaces and the Malliavin calculus", Acta Univ. Carolinae, Math. Phys., 30 (1989) 9-30. BOGACHEV,

[5] V.I.

BOGACHEV, "Smooth Measures, the Malliavin Calculus and Approx­ imations in Infinite Dimensional Spaces", Acta Univ. Carolinae, Math. Phys., 3 1 (1990) 9-23.

[6] V . I .

BOGACHEV, "Differentiable Measures and the Malliavin Calculus", Journal of :tvlathematical Sciences, 87 (1997) 3577-3731 .

[7] N .

CuTLAN D,

Loeb Measures in Practice: Recent Advances, Lecture Notes

in Mathematics 1751 , Springer-Verlag, 2000.

[8] N .

CUTLAND and S .-A. N G, A nonstandard approach to the Malliavin calculus, in Applications of Nonstandard-Analysis to Analysis, Functional Analysis Propability Theory and Mathematical Physics (eds. S. Albeverio, W.A . .J. Luxemburg and M. Wolff) , D.Reidcl-Kluwer, Dordrecht. 1995.

[9] S. V. FoMIN, Differential measures in linear spaces, in Proc. Int. Congr. of Mathematicians. Sec. 5, Izd. Mosk. Univ. , Moscow, 1966.

249

References

[10]

A .V. K IRILLOV , "Infinite-dimensional analysis and quantum theory as semimartingale calculus", Russian Math. Surveys, No. 3 (1994) 41-95.

[11]

D. NuALART, The Malliavin Calculus and and its Applications, Springer-Verlag, 1995.

[12]

H. O sswALD, Malliavin calculus tion, Book manuscript, 2001.

[13]

A.V. S KOROHOD, Integration in Hilbert Spaces, Ergebnisse der Mathe­ matik, Springer-Verlag, Berlin, New-York, 197 4.

[14]

O . G. SMOLYANOV and H.V. WEIZSACKER, "Formulae with logarithmic derivatives of measures related to the quantization of infinite-dimensional Hamilton systems", Russian Math. Surveys, 551 (1996 ) 357-358.

[15]

O . G . S l\!OLYANOV and H . V . WEIZSACKER, " Smooth probability mea­ sures and associated differential operators", Inf. Dim. Analysis, Quantum Prob. and Rcl. Topics, 2 (1999) 51-78.

[16]

H .V . WEIZSACKER, Differenzierbare Maj3e und Stochastische Analysis, 6 Vortragc am Graduiertenkolleg 'Stochastische Prozesse und probabilistis­ che Analysis' . Januar/Februar, Berlin, 1998.

Related Topics,

Probability

in abstract Wiener spaces. An introduc­

The power of Gateaux differentiability

Vftor

N eves*

Abstract

The search for useful nou standard minimil:ation conditions on C1 func­ tiona]s defined on Banach spaces lead us to a very shows that

if a C1 function .f

:

E

--+

simple argument

which

F between Banach spaces is actu­

finite points along finite vectors, then it iS UJlllOrll!J.y Conti nUOUS On bounded SetS il' and ouJ.y if it is Jipschitziau

ally GatealL'C differentiable on

on bounded sets. The following is a development of these ideas starting from locally convex spaces.

18. 1

Preliminaries

This section consists of

an

informal description of Lools to fi·ame the dis­

course and therefore contains only a small number of theorems and few proofs. Ccu·eful foundational treat me nts may be found in [3] , [9[, [14[ or [ 1 [ ; we shall also use Nelson's quantifiers vst an d 3st as in [ 8] . 'vVe assume the existence of two set-theoretical structures A an d f3 and a function *(·) : A -+ B satisfying the properties we proceed to describe. Elements of eit:her of t he structures which are sets shall be called entities 1 •

(Pl)

A is a model of the Televant A nalysis in the sense that any object of Classical IVIathematical Analysis as well as the mathematical structures under s t udy have an interprctat.ion in A and theorems about them are true in A. In particul m the complete ordered field lR of real numbers or any other classical space cu·e elements of A.

•Departamento de Matemat;ica, Universidade de Aveiro. vneves®mat . ua . pt

Work for this article was partially supported by FCT via both the grant POCTI\ MAT\41683\01 and funds from the R&D unit CEOC. 1We shall also arlmit the existence of atoms i.e. objects wit.hout clements which arc not the empty set; for instance numbers are atoms.

18. The power of Gateaux differentiability

254

The image by *(-) of any given element a E A will be denoted *a and be called standard as well as a itself; elements of standard sets in B shall be called internal ; non internal sets in B are called external. A set of sets verifies the Finite Intersection Property, abbreviated f.i.p. , if all its finite subsets have non-empty intersection. Denote the cardinal (num­ ber of clements) of a set S by card(S) and its power set by P(S) . (P2) Polysaturation 2

Given E E A and C C *P(E), if C verifies the f. i.p. and card( C) card(A) then n C # 0

<

There is an encompassing formal first order language £ with equality = and E (to be read as the membership relation) , with the usual connectives ' for negation, 1\ for conjunction, V for disjunction, =? for implication, ¢? for equiva­ lence, universal V and existential ::J quantifiers and enough constants, predicate and function symbols to denote any elements, relations and functions under consideration either in A or in B. Say that a formula of £ is bounded if its quantified subformulae, hereby including the formula itself, are of the form

Vx [x E a =? ¢]

or

::Jx [x E a 1\ ¢]

for some constant a and formula ¢; a sentence is a formula without free variables. B contains a formal copy of A in the following sense. (P3) Transfer

If ¢( a1 , , an) is a bounded sentence with occurrences of constants ai ( 1 :<.:: i :<.:: n) and no more constants, then ¢( a 1 , , an) is true in A iff ¢(*a 1 , , *an) is true in B 3 . ·

·

·

·

·

·

·

·

·

It is always useful to keep in mind theorems 18. 1 . 1 and 18. 1.2 below. Say that a formula of £ is standard (resp. internal) if it is bounded and all its constants denote standard (resp. internal) elements of B. Theorem 18. 1 . 1 (Principles of Standard and of Internal Definition) A set B E B is standard (resp. internal) iff there exists a E A and a standard (resp. intemal) formula ¢ such that B = {x E *al ¢(x) } or, more informally,

a set in B is standard or internal iff it is definable by a respectively standard or internal formula. 2 Actually

formulate.

this is stronger than needed for our purposes, but it is powerful and easy to

38 is a kind of elementary extension of A with embedding *( · ) .

18.1.

255

Preliminaries

Call monad any intersection

n cEC *C with 0 # c E A.

Any internal set which contains a monad n cEC *C also contains one of the standard sets *C.

Theorem 1 8 . 1 . 2 (Cauchy's Principle) Hyper-real numbers

A E *JR:. are classified the following way

A is finite . A is

-

infinitesimal . A is infinite .

-

::lr E lR:. lxl .:::; *r Vr E lR:. [r > 0 =? lxl A is not finite.

.:::;

*r]

0 and f-L denote respectively the set of finite and infinitesimal hyper­ real numbers. For any given ( real ) locally convex space S. let fs denote a gauge, i.e. a directed set, of semi-norms defining the topology of S; fin(*S) and f-L(*S) shall denote respectively the sets of finite and infinitesimal elements in *S, i.e. , for any given x E *S,

V1 E fs r(x ) E 0 V1 E fs r (x ) E f-L

x E fin(*S) XE

f-L(*S)

(18. 1 . 1) (18. 1 . 2)

When rs is unbounded, in the sense that Vx E

S fs (x) .- { r (x ) l r E fs}

is unbounded,

(18.1.3)

it so happens that Vx E

*S [x E f-L(*S)

¢?

vst l E fs r(x)

_:::;

1] .

(18.1 .4)

Condition (18.1 .4) has the syntactical advantage of dispensing with a quanti­ fier; as we will present strongly syntactical reasoning,

we assume from now on that condition ( 18. 1 .3) is always verified. Denote uc the set of standard elements of *C, i.e.,

uc

:=

{*xl x E C}.4

If E and F are locally convex spaces, L ( k ) (E, F) denotes the space of continuous k-linear maps l : E k ---+ F; a map l E *L ( k ) (E, F) is said to be finite ( resp. infinitesimal) if l(fin(*E) k ) � fin(*F) ( resp. l(fin(*E) k ) � tt(*F) ) . I t is useful to keep in mind the following results ( [4] is a most complete source; also see [14, chap. 10] ) . 4

We sometimes identify C with "C for the s ake of simplifying notation.

18.

256

The power of Gateaux differentiability

In a locally convex space S, 1. x E f-L(*S) if and only if \f).. E 0 Ax E f-L(*S) 2. x E fin (*S) if and only if \f).. E f-L Ax E f-L(*S)

Theorem 1 8 . 1 . 3

3.

(x E *S) .

( x E *S) . Vx E f-L(*S) ::3 ).. E * ffi. \ 0 Ax E f-L(*S), therefore i t is also true that Vx E *S [x E f-L(*S)

¢?

::3 ).. E * ffi. \ 0 Ax E f-L(*S)]

Let x ::::l y mean that x is infinitely near y, i.e., denote standard part whenever appropriate, i.e . ,

y

= st (x)

:=

y E us & x

::::l

x - y E f-L( *S) ; also let st y:

ns (*S) will denote the set of near-standard vectors of ns (*S)

*S, i.e. ,

: = us + f-L( *S) .

f : *E ---+ *F is S-continuous on A c:;: *E if for all x, y E A , whenever x ::::l y ; by definition, any standard function f is S-continuous if *f is. Theorem 18. 1 .2 implies that S-continuity of internal functions is nothing else than continuity measured with standard tolerances; this and condition (18. 1 .3) imply: A function

f(x)

::::l

f(y).

Suppose A E *P(E) . An internal function f A ---+ *F is S-continuous at x E A iff Vv E U fp Vc E U ffi. ::lr E u rE ::36 E Uffi. Vy E A [r(x - y) < 6 =? v(f(y) - f(x)) < c] .

Lemma 18. 1 . 1

In particular

A function f : A c:;: E ---+ F between locally convex spaces E and F is 1. continuous (on A) iff it is S-continuous on *A n ns (*E) ; 2. uniformly continuous (on A) iff it is S-continuous on *A.

Theorem 18.1.4

And

Given locally convex spaces E and F and an internal *k­ linear function l *E k ---+ *F, the following conditions are equivalent 1. l is S-contmuous.

Theorem 1 8 . 1 . 5 :

18.2.

257

Smoothness

2. l ( fin (*E)) c:;: fin(*F) (i. e. the internal S-continuous maps are the internal finite maps). 3. l (tL(*E) ) c:;: tL(*F) . It might also be interesting to notice the following Theorem 1 8 . 1 . 6 Let E and F and ¢ : (A x fin (*E)) ---+ *F be an

be locally convex spaces, A be a subset of *E internal function which is linear in the second coordinate. If ¢ is S-continuous in the sense that Vx. y E A Vu, v E fin( *E) [x :::::J y & u :::::J v =? cjJ(x, u) :::::J cjJ(y, v)] , then, for all x E A , ¢(x, ·) is a finite map. Proof. Assume ¢ : (A x fin (*E)) ---+ *F is S-continuous, x E A c:;: *E and that u E fin (*E) ; for all t E fL , tu :::::J 0, therefore t¢(x, u) ¢(x, tu) :::::J ¢(x, 0) 0, hence, by theorem 18.1 .3, ¢(x, u) E fin(*F) as required. D Observe that, when S is normed, an element x E *S is infinitesimal (resp. finite) iff ll x ll E fL (resp. llxll E 0) and the following holds. =

=

Corollary 18. 1 . 1 Given normed spaces E and F and a k-linear function l : E k ---+ F, the following conditions are equivalent 1. l is continuous 2. Vx E *S [ llxll :::::J 0 =? ll l (x) ll :::::J O J 3. Vx E *S [ ll xll E 0 =? ll l(x) ll E 0 ] . Of course theorem 18. 1.5 and corollary 18. 1.1 essentially say that , for linear maps continuity is equivalent to continuity at zero.

18.2

Smoothness

Let E and F be complete locally convex spaces, A be an internal subset of *E, f : A ---+ *F and l ( - ) : A ---+ *L(E, F) be internal and write f (x + tv) = f(x) + tl x (v) + tL (x E A , v E *E, t E *ffi., L E *F) . (18.2. 1) :::::J

f i s GS-differentiable at x with GS-derivative lx, i f L 0 whenever v E fin (*E) and t :::::J 0; f is uniformly differentiable at a E *E with uniform derivative l () if f is GS-differentiable at x and lx is finite, for all x :::::J a. We shall also call l : A x fin (*E) ---+ *F the derivative of f and define (x, v E *E) . df(x, v) := l(x, v) : = lx (v) := dfx (v)

18.

258

The power of Gateaux differentiability

Observe that , when one takes x and v in E within equation (18.2. 1), the GS­ derivative df ( x. v) is just the Gateaux derivative ( G-derivative for short) of f at x along v. The following theorem is well known too (see [13] ) .

Theorem 18.2.1 If E and F are Banach spaces, the function f ---+

:

E

---+

F

has G-derivative df(x, y), for all x E E, and x df(x, ) : E ---+ L (E, F) is continuous for the uniform topology on L (E, F), then x ---+ df(x, ) is actually a Fnichet derivative. ·

The following is a simple generalization of proposition

·

2.4 in [15] .

Theorem 18.2.2 If the internal map f : *E

at all points of an internal set A E 1.

*P( E)

---+ *F is uniformly differentiable with derivative df, then

f is S-continuous on A

2. df : A x fin(*E) ---+ *F verifies (a) �f (A x fin(*E)) c:;: fin (*F) (b) df is S-continuous in the sense that Vx, y E A Vu, v E fin ( *E) [x :::::J y & u :::::J v =? dfx (u) :::::J dfy (v)] . Proof. Borrowing from [15] : suppose that f : *E ---+ *F is internal and uni­ formly differentiable at all points of the internal set A, that x, y E A and that x :::::J y; take A E *ffi. \ 0 such that .A(x - y) E f-L ( *E) (theorem 18.1 .3) and define t := ±; t is infinitesimal and, for some L E f-L(*F),

f(y + t.A(x - y)) - f(y) tdfy (.A(x - y)) + tL dfy (x - y) + tL E f-L(*F)

f(x) - f(y)

by theorem

18.1 .5,

and

1 is proven .

2 (a) is an immediate consequence of the fact that the maps dfx are finite and this is useful in proving S-continuity of df in (b) : take x, y E A and u, v E fin(*E) such that x :::::J y & u :::::J v; observe that df(x, u) - df (y, v)

=

dfx(u - v) + dfx (v) - dfy (v).

As the maps dfx arc finite, by theorem 18.1 .5, df(x, u - v) E f-L(*F) and it is enough to show that, under the present assumptions, dfx (v) - dfy (v) :::::J 0.

18.2.

259

Smoothness

Now, taking A and t as above and v E fin ( *E ) , there exist infinitesimal vectors L. rJ, J E *F such that

djy (x - y) + t L ( 18.2.2) f(x) - f(y) (18.2.3) tdfx(v) + tr] f(x + tv) - f(x) tdjy (v) + djy (x - y) + tJ. (18.2.4) f (y + t(v + .A(x - y))) - f(y) Summing (18.2.2)+( 18.2 .3)-(18.2.4), we obtain t(dfx (v) - djy (v)) = t ( L + rJ - 6) and dfx (v) - djy (v) :::::J 0 as required. D Moreover Stroyan also showed in [15] that uniform differentiability is a very good generalization of G-differentiability in that, among other properties, it overcomes the impossibility of topologizing L(E , F) in such a way that the eval­ uation map (x, y) f---+ l(x) : L(E, F) x E ---+ F be continuous when E and F are not nonnable locally convex spaces 5 ; we established the equivalence between uniform and Cqb differentiability of standard functions on complete locally con­ vex spaces, and discuss relations with a weaker kind of differentiability when fin ( *E ) = ns ( *E ) , in [12] . The main results in this context read as follows. Assume that U is a non-empty open subset of E, and k E N.

:

f U ---+ F, a E 17U + f-L (*E)

Definition 18.2.1 (Stroyan [15] ) The function f is k-uniformly differen­ tiable at a if there are finite maps d ( i ) f() U ---+ L( i ) ( E, F) , ( 1 <:.:: i <:.:: k) - the derivatives of f - such that, with d d ( l ) , for all x E fin ( *E ) and all t E f-L(*IR'.) , there exist rJ E f-L(*F) and infinitesimal maps T/z (2 <:.:: i <:.:: k), :

:=

so that

=

(18.2.5) f(a) + tdfa (x) + tr] f (a + tx) + l ( ( ) ) ) ( d z fa (·) + td z fa (x , · ) + tr]i ( - ) (1 <:.:: i < k) . (18.2.6) d " fa + tx(·) f is k-uniformly differentiable in U if it is k-uniformly differentiable at all a E 17 U + f-L(*E) . When f is 1-uniformly differentiable we just say that it is uniformly differentiable. The following may also be found in [15] . Theorem 18.2.3 When f is k-uniformly differentiable, not only .f itself is S-continuous but also the derivatives d ( z ) f ns ( *E ) x fin ( *E ) ---+ *F are S­ cordinuous: Vx, y E ns ( *E ) Vu, v E fin ( *E ) [x :::::J u & y :::::J v =? d (i ) fx (u) :::::J d( i ) fy (v)] (1 S: i <:.:: k) . =

:

5A

quite simple and clear explanation of this can be found in

[7,

page

2] .

18.

260

The power of Gateaux differentiability

Theorem 18.2.4 ( Taylor's formula [15]) f is k-uniformly differentiable in U if there are maps dj ( i) U ---+ L � '? (E, F) (1 <:.:: i <:.:: k), such that, whenever :

a E au + f-L(*E), x E fin(*E), t E f-L, there exists rJ E f-L(*F) such that k i j(a + tx) = f(a) + L �t df�i ) (x, · · · , x) + t k rJ. i= l z.

Relations with F-differentiability on Banach spaces are actually studied in [14, chap. 5] . On what regards standard definitions of smoothness we take from [6] , whereto we refer the reader for details, namely on convergence

structures.

Definition 18.2.2 Let N denote the filter of neighborhoods of zero in JR:., B be

a filter on E and, for any filter :F in a (real) vector space, N:F denote the filter generated by { {rb l r E N & b E F} l N E N & F E F } ; also, for k E N, let B k be the filter in E k generated by {B 1 x · · · x Bk l B1 , · · · Bk E B} and, for any filter :F in L ( k) (E, F), F(B k ) be generated by {U 1 E F l (B) I F E :F & B E B}; finally, if¢ is afunction and :F is a filter on the domain of¢, ¢(F) is generated by {¢(F) I F E :F} . B is quasi-bounded if NB converges to zero; Aqb denotes the conver­ gence structure of quasi-bounded convergence; when a filter :F converges to x E E with respect to Aqb, we write :F E Aqb (x) . 2. A filter :F in L ( k) (E, F ) qb-converges to zero if :F(Bk ) converges to zero in F whenever B is quasi-bounded in E. 3 . A junction l : U ---+ L ( k ) (E, F) is qb-continuous if l(:F) E Aqb(l(a)) whenever a E U and :F is a filter in E converging to a. 1.

4.

Recall that U is an open subset of E and f : U ---+ F. We say that f is

of class c;b if

(a) f is qb-continuous. (b) There exist maps d'i( . ) : U ---+ L ( z ) (E, F) (1 <:.:: i <:.:: k) such that i. d' i( . ) is qb- continuous, for all i = 1 , · · , k ii. For all a E U, all x E E and all i = 0, · · · , k - 1 ·

i = d + 1 fa( x, · ) lim t --> 0 �t ( d'fa+ tx - Ja) ' for the topology of pointwise convergence on L i (E, F) .

18.3.

261

Smoothness and finite points

Theorem 18.2.5 ([12] ) The function f is of class c;b , with derivatives di i( . ) , ( 1 :S i :S k) iff it is k-uniforrnly differentiable with the sarne derivatives. The following is also a consequence (for example see theorem ( 18.2.5) or of theorem 18.2.4.

[6] )

either of this

:

Theorem 18.2.6 The function f E ---+ F, between Banach spaces E and F, is of Frechet class C k , with derivatives di fc ) , ( 1 :S i :S k) iff it is k-uniforrnly

differentiable with the sarne derivatives.

18.3

Smoothness and finite points

Recall that E and F denote real locally convex spaces.

Lemma 18.3.1 Suppose A E *P(E) . The following conditions are equivalent for any internal rnap f A ---+ *F which is GS-differentiable at all x E A n fin(*E), with GS-derivative df : (A n fin(*E)) X fin(*E) ---+ *F. 1. f is uniformly differentiable at all x E A n fin (*E) 2. f is S-continuous on A n fin (*E) 3. df( (A n fin(*E)) x fin(*E)) C:: fin(*F) :

Proof. For the sake of simplicity, we assume A = *E; the proof of the general case is an easy adaptation thereof. Suppose that df : fin(*E) x fin(*E) map f : *E ---+ *F.

---+

*F is the GS-derivative of the internal

(1 =? 2) This is 1 in theorem 18.2.2 above. (2 =? 3) Assume that f is S-continuous on fin(*E), let df() : fin(*E) ---+ *L(E, F) be the GS-derivative of f. Pick x and v in fin(*E); we must show that dfx (v) is finite; suppose this is not the case, take a semi-norm 1 E fp such that r(dfx (v)) t/c 0 and let t := r(df: ( v ) ) ; t is infinitesimal and, by hypothesis, there exists L E f-L(*F) such that f(x + tv) = f(x) + tdfx (v) + tL; but then 1 r(tdfx(v)) I (f(x + tv) - f(x) - tL) � 0 =

=

which is impossible; it follows that dfx (v) must be finite as required. (3 =? 1) In the presence of GS-differentiability, 3 completes the definition of uniform differentiability. D Say that a standard function f : E ---+ F is GS, or uniformly, differentiable at x E *E with derivative dfx if respectively *f is GS, or uniformly differentiable, at x with derivative *df.

18.

262

The power of Gateaux differentiability :

Theorem 18.3.1 If a standard function f E ---+ F is GS-differentiable with standard derivative dfx at all x E fin(*E), the following conditions are equivalent 1.

f is uniformly differentiable on bounded sets with derivative df .

2. f is umformly continuous on bounded sets. 3.

For any bounded subset B of E2 , df(B) is bounded in F.

4.

df is uniformly continuous on bounded subsets of E2 .

As we said before, the proof is easy. Start by keeping in mind that

Theorem 18.3.2 A subset B of the locally convex space S is bounded iff *B C:: fin ( *S ) ; in particular, if E and F are normed and L(E: F) is endowed with the usual uniform norm, a subset B of L(E, F) is bounded iff

V¢ E *B Vx E fin(*E) ¢(x) E fin(*F).

(18.3.1)

Proof of thm. 18.3 . 1 . Equivalences (1 ¢? 2 ¢? 3 ) are immediate conse­ quences of lemma 18.3.1 and theorem 18.3.2 (note that , by theorem 18.3.2 above. *B n fin(*E) = *B if and only if B is bounded) .

(3 =? 4) Follows from theorems 18.3.2 and 18.2.2. ( 4 =? 2) If we show that condition 4 implies that dfx is a finite map when­ ever B is a bounded subset of E and x E *B, we are done; but this follows from theorem 18.1 .6. D As an application to Banach spaces: :

Corollary 18.3.1 Let E and F be Banach spaces and f E ---+ F be a S­ continuous and GS- differentiable function at all x E fin (*E) with standard

denvatzve df, then 1.

df zs actually the Fnichet derivative of f and f is of class C 1 .

2. (Mean Value) Denotmg [x, y] the line segment from x to y, Vx, Y E E 3z E [x, y] ll f(x) - f(y) ll :::; ll dfz ll · ll x - Y ll ;

(18.3.2)

therefore f is Lipschitzian on bounded sets, i.e., for each bounded subset B of E, there exists K E rn:. such that Vx, y E B ll f(x) - f(y) ll :::; K ll x - Y ll .

(18.3.3)

263

18.4. Smoothness and the nonstandard hull

For all infinitely near finite vectors x, y, z E *E , there exists an infinites­ imal L E *F such that

3.

f(x) - f(y)

=

dfz (x - Y ) + ll x - Y IIL

( 18.3.4)

Proof. 1 . As f is S-continuous on fin(*E) . it is uniformly continuous on bounded sets and hence, by theorem 18.3. 1 , of Frechet class C 1 . 2 . Condition (18.3.2) is true in view of part 1 above; therefore condi­ tion ( 18.3.3) is true (by theorems 18.3.1 and 18.3.2) because fin(*E) is *convex. 3 . We borrow from [14, page 97] : the Frechet derivative x r--+ dfx is S-continuous at finite points by 2b in theorem 18.2.2 and we may apply Trans­ fer of the Mean Value condition ( 18.3.2) to the function

g

='

V r--+

J ( V ) - dfz ( V - ) Z

,

whose derivative v r--+ dgv = dfv - dfz is infinitesimal whenever therefore verifies, for some c E [x , y] , hence c :::::J z,

II J ( x ) - J ( y ) - dfz (x - y ) ll ll x - Y ll 18.4

=

ll g(x) - g(y) ll

s;

v :::::J

l dgc ( l l xx -- yY l l ) I :::::J

z, and

0.

D

Smoothness and the nonstandard hull

This section evolves along main ideas due to Manfred Wolff. Recall that · :::::J is an equivalence relation on *S, with equivalence classes x := x + f-L(*S) and let S be the nonstandard hull of *S with semi-norms i' or •

S

:=

fin( *S ) ;

�

& ry (x)

:=

st ( r ( x ) ) (x E fin(*S) ) .

Nonstandard hulls are complete spaces, but may vary with the chosen model of analysis A: actually Banach spaces with invariant nonstandard hulls are finite dimensional, but this is not the case with locally convex non-normable ones; spaces with invariant non-standard hulls were characterized in [5] and were named HM-spaces by Keith Stroyan (relevant data is summarized in detail in [14, chap. 10] ; also see [10, sec. 3.9] for nonstandard hulls of metric spaces) . Any internal function between locally convex spaces. f : A C:: *E r--+ *F that is S- continuous on fin(*E) n A and such that J (fin(*E) n A) C:: fin(*F) has a natural non-standard hull j : C:: E ---+ P too defined by

A

A ](x)

{x l x E fin(*E) n A } j(x) (x E A n fin(*E)) ;

18.

264

The power of Gateaux differentiability

note that j is not a standard function in the sense we have been considering, for it certainly is not a priori an element of the model of analysis A, nevertheless an application of Nelson's Algorithm ( [8] , [11]) will provide us with definitions of differentiability for functions without reference to nonstandard extensions. From now on f : *E ---+ *F denotes an internal GS-differentiable function with derivative df such that

f ( fin(*E))

C:::

(18.4. 1 )

fin(*F)

and

(18.4.2)

whenever this is a good definition, namely when df is S-continuous on fin(*E) and x, v E fin(*E) . For any GS-differentiable function h : E ---+ F ,

t:.. ( h x , v, t) 18.4.1

:=

( � (h(x + tv) - h(x) - dh(x v) ) ) ,

(x, v E *E; t E * ffi. \ {0}) .

Strong uniform differentiability :

Definition 18.4 . 1 An internal function f *E ---+ *F is Strongly Uni­ formly Differentiable (SUD for short) if it is uniformly differentiable on fin(*E) and f(fin(*E)) C::: fin(*F) . Let P and In these terms, ously verified

Q f

denote respectively gauges of semi-norms for E and F. is SUD when the two following conditions are simultane­

\i(x, v) E *E 2 [(x, v) E fin(*E) 2 =? df(x, v) E fin(*F)] \i(x, v) E *E2 \it E * ffi. [(x, v) E fin( *E) 2 1\ 0 cj. t E f-L =? t:.. (j , x , v, t) E f-L ( *F ) j ; these expand respectively to

\i(x, v) E *E2 [vstp E *P 3 8t m E *N p(x) + p(v) � m =? \i8t q E *Q3 8t n E *N q ( df(x , v)) � n] \i(x, v) E *E 2 \it E 'R vstp E *P 3 8t m E *N p(x)+p(v) � m 1\ \ist n E *N 0 < i t I � � =? \ist q E *Q q(t:.. ( f, x, v, t)) � 1 ;

[

]

which reduce respectively to the following, where we leave the domains implicit,

\i(x, v) vst q 3 8t (p, n) vst m [p(x) + p(v) � m =? q (df(x, v)) � n] \i(x, v, t) \i8t q 3 8t (p, n) \i8t m p (x) + p(v) � m

[

1\ 0 < l t l

�

� =? q(t:.. ( j , x , v , t)) � 1] ;

18.4.

265

Smoothness and the nonstandard hull

Nelson's algorithm then gives

\j8t q \18t m 3 st fzn p x N V(x, v) 3(p, n) E P x N [p(x) + p(v) � m(p, n) =? q (df(x, v)) � n ] \18t q \18t m 3 st fin p x N V(x, v, t) 3(p, n)

[

p(x) + p(v) � m(p, n)

1\

0 < ltl

(18.4.3)

PxN

E

1

� ; =?

q( !:::. (j, x, v, t)) � 1 ] .

(18.4.4)

Formulas (18.4.3 ) and (18.4.4) together define strong uniform differentia­ bility of an internal function and we actually proved the following

Theorem 18.4. 1 An internal GS-differentiable function f : *E ---+ *F is S UD

iff

\j8t q \18t m 3 st fin P x N V(x, v, t) 3(p, n)

[ [p(x) + p(v)

�

m(p, n)

1\

0 < ItI

�

E

PxN

� ] =?

[ q (df(x, v))

�

n

1\

q (!:::. ( j, x, v, t) )

�

]

(18.4.5)

1] :

When f is standard, transfer of ( 18.4.5) provides the definition, 18.4.2 below, of SU differentiability for functions between locally convex spaces in any "universe" A:

Definition 18.4.2 A standard function h : S ---+ T between locally convex spaces S and T, with unbounded gauges of semi-norms respectively I: and 8 , is Strongly Uniformly Differentiable, with derivative dh ( ) : S ---+ L(S, T) , (S UD for short) when, leaving implicit that a E I:, T E e, m : I: X N N , x E S, v E S and t E JR \ { 0}, --->

VT Vm

::J f z n c

[ [a (x) + a(v)

X

N c:;: I:

�

X

m(a, n)

e

V(x, v, t) 3 (p, n)

1\

0 < ltl

�

Ec

X

N

� ] =?

[T (df(x, v)) � n

1\

T( !:::. (h, x, v, t))

�

]

(18.4.6)

1] .

NB: variants of this formula where any inequality � is taken to be

strict, i.e., to be < , are equivalent.

18.

266

18.4.2

The power of Gateaux differentiability

The non-standard hull

Theorem 18.4.2 Let E and F be locally convex spaces and f : *E ---+ *F be

an internal GS-differentiable function with derivative df such that f ( fin(*E))

( 18.4. 7)

� fin(*F) .

f is SUD iff j : E ---+ F is uniformly differentiable with derivative dj : E2 ---+ F given by d](x , v) := djr;;:; ) . Proof. Let us first assume that the internal function f : *E ---+ *F is SUD with derivative df, and takes finite vectors of *E into finite vectors of *F. Define j3

Q := { I] I q E Q } ;

:= {P I p E P}

although there might exist continuous semi-norms not of the type ,-y, P and Q arc unbounded directed families of semi-norms, so that condition ( 18. 1 .4) still holds with obvious adaptations. Also recall that

](x) = f(x) and observe that equation

defines dj well, in view of theorems 18. 1 . 5 and Take q E Q and rrL : Q x N ---+ N . Define

M(p, n)

:=

m(p, n)

18. 1.6 and lemma 18.3. 1 .

(p E P, n E N) ;

M is a standard function. therefore we may deduce from exists a standard finite set P x N � *P x 'N such that

(18.4.5)

that there

V(x, v, t) E *E2 x 'R 3(p, n) E P x N

[p (x) + p(v) < M(p, n) 1\ 0 < l t l

:<::

�

=?

]

q(df(x, v)) :<:: n 1\ q( !:::. (f, x, v, t)) < 1 . P and N being standard and finite, N

:= N

'

p

define

:= {P I p E P} ,

267

18.4. Smoothness and the nonstandard hull

and take (x, v, t) ; whatever the representatives x and v, there is a correspond­ ing pair (p, n) E P x N, which may be vary with x and v, but always veri­ fies ( 18.4.3) ; suppose that

�; ( 18.4.8) n all the�bers involved here are standard, p(x) :::::J p(x) & p(v) :::::J p(v) as well as q(df(x, v)) q(df(x, v)); in particular, it follows p(x)

+ p(v) < M(p, n)

1\ 0

< ltl

�

:::::J

p(x) + p(v)

M(p, n )

m(p, n)

< so that

q(df(x, v))

n

< an thus

-------

(j(df(x, v))

<

n

that is

(j(d}(x, v ) )

<

n

and the first factor in the consequent of ( 18.4.5) follows for j as required; for the remaining factor of the consequent, observe that, actually

p(x) + p(v) < M(p, n )

1\ 0

< l tl

1

� -;

n

implies

q

( � (f(x + tv) - j(x) - dj(x, v) ) )

too; t and n are standard, 0 #- t and 18.3. 1 , thus 1 t finally

( f (X + tv) - j (X )

-

dj ( X, V )

)

= q(t:.(J, x, v, t)) <

1

f as well as df are S-continuous by lemma :::::J

(

1 ' t j (X + t v )

A

- j (X)

-

-------

dj ( X, V )

)

;

(j(t:.(}, , x, v, t) ) � 1 . The proof that j is SUD with derivative dj is finished. Now suppose that j is itself uniformly differentiable with derivative d} : E2 ---) F. Take q E ()"Q and a standard function m : *(P X N) ---) *N; since ( ) is 1 - 1 M (p, n) := m(p, n ) ·

268

18. The power of Gateaux differentiability

defines a function from P x N into N and we may apply (18.4.6) appropriately written in terms of j, and obtain the corresponding finite set P x N � P x N; pick x , v E E, t E ffi. \ {0} and use ( 1 8 . 4 . 6 ) , with < for <:.:: , in order t o obtain (p, n) E P x N so that

[iJ (x) + fj(v)

<:.::

M(fj, n) 1\ 0 < l t l

<:.::

[<1 ( d](x, v))

�]

=?

< n 1\ q(l� (j, x, v, t)) < 1 ] ;

therefore, if

<:.:: -

fj ( x) + fj ( v )

<:.:: - ,

then hence or

1 n

p( x ) + p(v) < m(p, n) 1\ 0 < l t l <:.::

M (fj, n) 1\ 0 < l t l

( 18.4. 9)

( 18.4. 1 0)

1 n

q ( d](x, v)) < n 1\ q (!:::. ( ], x, v, t)) < 1 q ( dn;:;)) < n 1\ q(!:::. (j, x, v, t)) < 1 .

I t immediately follows that

q (df( x , v) )

<:.:: n,

because both q (ifr;;;:; ) ) and the n above are standard; moreover and when t >f, 0 q(!:::. ( f, x , v, t)) <:.:: 1 ;

f) is linear ( 18.4. 1 1 )

it so happens that t � q (!:::. (f, x , v , t ) ) i s always an internal function and we actually have shown that

[

Vq E uQ 3n E u N o < l t l < which is the same as

Vq E ()"Q [o # t � o

=*

� =? q(!:::. (f, x , v, t))

<:.::

1

]

q(!:::. (f, x , v, t ) ) � o]

and thus (18.4. 1 1 ) also holds when t � 0. Again because ( ) is 1- 1 , we may define p : = {P I p E P} -

and we have shown that ( 18.4.5) holds.

D

269

References

References [1] S . ALBEVERIO, J- E . FENSTAD, R. H 0EGH-KROHN and T. LINDSTR0M ( eds. ) , Nonstandard Methods in Stochastic Analysis and Mathematical Physics. Academic Press, 1 986. [2] LEIF 0. A RKERYD, NIGEL .J. CUTLAND and C. WARD HENSON (cds . ) , Nonstandard Analysis, Theory and Applications, Kluwer, 1997. [3] C. WARD HENSON, A Gentle Introduction to Nonstandard Extensions, in Nonstandard Analysis, Theory and Applications, Arkeryd, Cutland & Henson (eds. ) , Kluwer, 1997. [4] C. WARD HENSON and L . C . MOORE, "The Nonstandard theory of topo­ logical vector spaces", Trans. of the AMS, 172 ( 1 972) 405-435. [5] C . WARD HENSON and L . C . M ooRE , " Invariance of The Nonstandard Hulls of Locally Convex Spaces", Duke Math . .T . . 40 93- 205. [6] H . H . KELLER, Differential Calculus in Math 417, Springer-Verlag, 1 974.

in Locally Convex Spaces, Le

e

. Notes

[7] ANDREAS KRIEGL and PETER W . MICHOR, The Convenient setting of Global Analysis, Math. Surveys and Monographs 53, AMS ( 1 997) 405-435. [8] FRANCINE D IENER and KEITH STROYAN, Syntactical Methods in In­ finitesimal Analysis, in Cut land ( ed. ) , Nonstandard Analysis and its Ap­ plications, CUP, 1 988. [9] T oM LINDSTR0lvl , An invitation to Nonstandard Analysis. in Cutland (ed. ) , Nonstandard Analysis and its Applications, CUP, 1 988. [10] PETER LOEB and MANFRED WOLFF (eds . ) , the Working Mathematician. Kluwer, 2000.

Nonstandar-d Analysis for

[ 1 1] E. NELSON, " Internal Set Theory: a new approach to Nonstandard Anal­ ysis", Bull. AMS, 83 ( 1 977) . [12] VfTOR N EVES, " Infinitesimal Calculus in HM spaces", Bull. Soc. Math. Belgique, 40 ( 1 988) 1 77-198. [13] JACOB T. ScHWARTZ, 1969.

Nonlinear Functional Analysis, Gordon & Breach,

270

18. The power of Gateaux differentiability

[14] KEITH D. STROYAN and W . A . J . LUXEI\IBURG : ory of Infinitesimals, Academic Press, 1976.

Introduction to the The­

[15] KEITH D . STROYAN, " Infinitesimal Calculus in Locally convex Spaces: 1 . Fundamentals", Trans. Amer. Math. Soc . , 240 ( 1978) , 363-383.

Nonstandard Palais- Smale conditions Natalia ��Iartins* and Vftor N eves**

Abstract

the Palais-Smale con cli tiou (PS) be­ low, some of them generalizations, but still sufficient to prove Mountain Pass Theorems, which arc quite import ant in Critical Point Theory. \!lie

19. 1

present uonst.anclarcl versious of

Preliminaries

For notation, other preliminaries and basic references on Nonstandard Analysis we suggest section 1 in article [13] in this volume, namely we assume that we have two set-theoretical structures A and B - the former a. model of "the relevant" Analysis - and a 1-1 function * ( · ) : A -. B which satisfies the Transfer and Polysatmation Principles. Specific notation and results follow. Recall that *JR denotes the set of hyperreal numbers and 0 the set of finite hyperreal numbers; if x E "'ll.t is infinitesimal we write x � 0. If x, y E *JR are such that x - y � 0, we say that x is infinitely close to y and write 'E � y. Also remember that finite hyperreal numbers have a standard part: .

Theorem 1 9 . 1 . 1 (Standard Part Theorem)

E

0 ihe1·e exists a ·u.niq'lLe r E lR sv,ch that :r; � t·; ·r is called the standard part of :r and is denoted by st ( x ) or 0x. Moreover, for all x, y E 0, st(x + y) = st (x) + st(y) , st(xy) = st(x)st(y) and if x � y then st (x) � st(y) .

rf x

*Departamento de Matematica, Universidade de Aveiro. nataliam@mat . ua . pt

Work for !:his article was partially supported by the R&D unit Center for Rc­ :>em·ch in Optimi?.atiou and Control ( CEOC) of t.hc University of Jlwciro and grm1t. POCTijMAT/41683/2001 of the Portuguese Foundation for Science m1d Technology

·�

(FCT)

via

FEDER.

vneves@mat . ua . pt

272

19. Nonstandard Palais-Smale conditions

Next we restrict some definitions to normed spaces; just recall that a normed space is a locally convex space whose topology is defined by a sin­ gle norm and also recall that, for any set A E A, "A :=

{*a : a E A},

and often A and "A are identified for simplicity.

Definition 19. 1 . 1 Let (E, 11 ·11) be a real normed space and x E *E. 1. x E fin(*E), i. e., x is finite if llxl l E 0;

2. x

E f-L(*E) , x E f-L(*E);

i. e.,

x

is infinitesimal if llxl l

3. x E ns(*E) , i.e., x is near-standard x :::::J a, which means x - a :::::J 0; 4. x E pns(*E), exists a E 17E

:::::J 0

in *JR; x

if there exists a

:::::J 0

E 17E

means

such that

i. e., x is pre-near-standard if for all positive c E lR there such that ll x - a ll < c.

It follows from the above definitions that ns(*E) c;; pns(*E) and ns(*E) fin(*E) . In general, ns(*E) of: pns(*E) and ns(*E) of: fin(*E).

c;;

Theorem 1 9 . 1 . 2 Let (E, 11 · 11) be a real normed space. 1. E is complete if and only if pns(*E) ns (*E); =

=

2. E is finite dimensional if and only if fin(*E) ns(*E); 3. A c;; E is compact if and only if *A c;; ns(*E) and st(*A) = A . The nonstandard extension of N, *N, is the set of hypernatural numbers and *N= denotes the set of infinite hypernatural numbers. It is also useful to keep in mind the following properties of sequences and functions in a real normed space (E,

11 · 11) .

Proposition 19. 1 . 1 Suppose ( xn ) n EN is a sequence in E. Then 1. ( X n ) n E N is bounded if and only if Vn E *N= X n E fin(*E);

2. ( xn ) nEN converges to a E E if and only if Vn E *Noc X n :::::J a; 3. ( x n ) n EN has a convergent subsequence if and only if ::lm Xrn E ns(*E).

E *N=,

Theorem 1 9 . 1 . 3 Let E and F be two real normed spaces and f E ---+ F . The function f is continuous o n a E E if and only if

Vx E *E [ x :::::J a =;. f(x)

:::::J

f(a)] .

273

19.2. The Palais-Smale condition

19.2

The Palais-Smale condition

:t\'Iany results in Critical Point Theory involve the following condition, orig­ inally introduced in 1 964 [10] by Palais and Smale:

If E is a real Banach space, the C 1 junctional j satisfies the Palais-Smale condition iffor all ( Xn ) n EN E EN ,

Definition 1 9 . 2 . 1

:

E ---+ lR

(f(xn )) n EN is bounded and nlim -HXJ j'(x n ) = 0 =? ( Xn ) n EN has a convergent subsequence.

(PS)

Suppose (E, 1 1 · 1 1 ) is a real Banach space with continuous dual E' and duality pairing ( ·, · ) . Let C 1 ( E, lR) denote the set of continuously Frechet differentiable functionals defined on E; a is critical point ( resp. almost critical point) of f if f'(a) = 0 (resp. f'(a) :::::J 0) , i.e . , if (j'(a), v) = 0 (resp. (j'(a), v) :::::J 0) for all v E E (resp. v E fin(*E) ) : j(a ) is a critical value of f if j -1 (J(a)) contains critical points. If f : E ---+ lR is Frechet differentiable we will denote by K the set of all critical points of f, that is,

K = {x E E : j ' (x) = O} and, for each that is,

c

E R

Kc

will denote the set of critical points with value

Kc = {x E E : .f ' (x) = 0 1\ f(x) = c} = K n f -1 (c) .

The following is easy to prove.

Su,ppose that f E C1 (E, IR) satisfies (PS) . Then 1. If f is bounded, K is a compact set. 2. For each a, b E lR such that a ::; b, {u E E : a :s; j(u) :s; b 1\ j'(u) = O} = f -1 ( [a, b] ) n K

Proposition 19.2.1

is a compact set. Note that

f satisfies and

(PS) and

K is compact

K is compact and f is bounded

c;l?

as can be seen with the following two examples.

c;l?

f is bounded

f

satisfies (PS ) ,

c,

19.

274

Nonstandard Palais-Smale conditions

Example 1 9 . 2 . 1 The real function f( x ) = x3 ( x E IR) satisfies (PS), is compact and f is not bounded . Example 19.2.2 Let

{ 2;2-

x2 if X E if X tt

[ - 1 , 1] [- 1 , 1]

f is bounded, K = {0} but f does not satisfies

(PS ) .

f (X)

=

K = {0}

Example 19.2.3 The real functions exp ( x ) , cos ( x ) , sin ( x ) and all the con­ stant functions defined on lR do not satisfy (PS ) . Now we will present an important class of functionals that satisfies ( PS ) . Proposition 19.2.2 C1 (E, IR) is coercive fies (PS ) .

If E is a finite dimensional real Banach space and f E (i. e. f(x) ----+ +oo whenever ll x ll ----+ +oo), then f satis­

Proof. Let (x n ) n EN C:.: E be such that (f(x n ) ) n EN is bounded and limn _, = f'(x n ) = 0. Then, for all n E *Noc, f(x n ) E 0 ( Proposition 19. 1 . 1 ) . Since f is coercive, for all n E *Noc , X n E fin(*E) . E is finite dimensional then , by Theorem 19.1 .2, fin ( *E ) = ns ( *E ) and therefore, for all n E *N= , X n E ns ( *E ) . Finally, Proposition 19. 1 . 1 implies that (x n ) n EN has a convergent subsequence. D

19.3

Nonstandard Palais-S male conditions

The following definition often shortens statements

Suppose f E C1 (E, IR) . We say that a sequence (un ) n EN is a Palais-Smale sequence for f zf

Definition 19.3.1

(j(un )) n EN is bounded and nlim ->CXJ j'(un ) = 0. Therefore,

f satisfies the Palais-Smale condition ( PS ) if every Palais-Smale se­ quence for f has a convergent subsequence.

275

19.3. Nonstandard Palais-Smale conditions

Suppose E is a real Banach space and f E C 1 (E, IR) . Next we present some nonstandard variations of (PS ) : (PSO)

( un ) n EN is a Palais-Smale sequence -U::lm E *N= Um E ns (*E)

(PS1 )

f (u) E 0 1\ f'(u)

f(u) E 0 1\ f'(u) :::::J 0 -Uu E fin(*E) 1\ st ( f (u ) ) is a critical value of f

(PS2)

(PS3)

(PS4)

(un ) n EN

( Un) n EN is a Palais-Smale sequence -Uis bounded 1\ Vn E *N= st (f(un ) ) is a critical value of f

(un ) n EN

( un ) n EN is a Palais-Smale sequence -Uis bounded 1\ ::l n E *N= st (f (un ) ) is a critical value of f

Proposition 19.3.1 If f E C 1 (E, IR) (PS1 )

:::::J 0 =? u E ns(*E)

<=?

[f (u) E

0 1\ j' (u)

:::::J

0

then

=?

u E ns(*E)

1\ st (f(u))

is a critical value

]

of f .

Proof. The implication -¢== is trivial. For the proof of the other implication, suppose that u E *E is such that f (u) E 0 and f'(u) :::::J 0. By (PS1 ) , there exists a E 17E such that u :::::J a and, therefore. from the continuity of f and f' it follows that (Theorem 1 9 . 1 .3) f (a) too, so that f (a)

=

:::::J f ( u)

and f' (a)

:::::J f' ( u) :::::J 0

st (f(u)) and f (a) is a critical value of f.

Nonstandard versions of (PS) and (PS) itself are related as follows.

D

276

19. Nonstandard Palais-Smale conditions

Theorem 19.3.1 (PS1)

For any real Banach space =;.

(PS)

¢?

(PSO)

=;.

(PS2)

E we have ¢?

(PS3)

=;.

(PS4) .

Proof. It is clear that (PS) ¢? (PSO), (PS1) =;. (PSO) and (PS3) therefore (PS 1 ) =;. (PS) ¢? (PSO) and (PS3) =;. (PS4)

*E

=;.

(PS4) ,

are true. First we prove that (PSO) =? (PS2 ) . Let u E such that f(u ) E 0 and For each f ' (u) :::::J 0. Fix M E JR+ such that l f (u) l < M. Suppose u � n E N, define the standard set

Hn { x E E : :=

Since, for each

fin(*E).

�

IJ ( x ) l < M 1\ ll f' ( x ) ll <

1\ l l x l l > n } ·

n E N,

*Hn = { x E *E : l f( x) l < M 1\ ll f' ( x ) ll < � 1\ ll x l l > n } . we conclude that *Hn # 0 and the Transfer Principle says that Hn # 0. For each n E N, take Xn E Hn. Then ( x n)n E N is a Palais-Smale sequence but, for all n * N oc , Xn � fin(*E), and hence. for all n * N = , Xn � ns(*E). This i s a contradiction with (PSO) and therefore u E fin(*E). u

E

E

E

Now we will prove that if f(u) critical value of f. Let a = st (f (u) ) and, for each n

Fn {X E E : :=

Since, for each

E

0 and

f ' (u) :::::J 0, then st (j(u)) i s a

E N, define

I J ( x ) l < M 1\ I I J' ( x ) ll <

�

1\ I a - f( x ) l

<

� }.

n E N,

*Fn {X E *E : IJ ( x ) l < M 1\ ll f' ( x ) ll < � 1\ lo: - f(x) l < �} then , *Fn # 0 and therefore Fn # 0 . For each n E N take Xn E Fn. Then ( x n)n E N is a Palais-Smale sequence u

E

=

and from (PSO) we conclude

Since f

E C 1 (E, IR) , a :::::J f( xm ) :::::J j( a ) and 0 :::::J j' ( xm )

:::::J

j' ( a )

.

277

19.3. Nonstandard Palais-Smale conditions Therefore a = j(a) , f'(a) is proven. Lets prove that ( P 52) Then,

=

0,

a

is a critical value of f and (PSO)

=? (PS2)

=? ( P 53) . Let ( un ) n EN be a Palais-Smale sequence.

From (PS2) we conclude that

Vn E *N= [ un E fin(*E) 1\ st (j(un ) ) so we may conclude that

is a critical value of f ]

( un ) n EN is bounded and

Vn E *N= st(j(u n ) )

is a critical value of

f.

Finally we will prove that ( P 53) =? ( P 52) . Let u E *E such that f ( u) E 0 and f'(u) :::::J 0. Let M E JR+ such that l f(u) l < M. Suppose u � fin(*E) ; we construct a Palais-Smale sequence ( Xn )nEN using the sets H71 , as in the proof of the first part of the proof that ( P SO) =? ( P 52) , in such a way that ( Xn ) n EN is not bounded, which contradicts (PS3 ) . Suppose now that f ( u) E 0 and f' ( u) :::::J 0 . We need t o prove that a = st (j ( u)) is a critical value of f. .Just as in the second part of the proof of (PSO) =? (PS2 ) , we can build a Palais-Smale sequence (x n ) n EN such that for all n E *N= , j(x n ) :::::J o:. From (PS3) we conclude that for all n E *N= , D st(j(xn ) ) = a is a critical value of f. More can be said when

E is separable:

When E is a separable Banach space, (PS) and (PS1 ) are equivalent; in other words: if E is a separable Banach space, a C1 junction f : E ---+ lR verifies the Palais-Smale condition if and only if almost critical points where f is finite are near-standard. Theorem 19.3.2

Proof. Suppose f satisfies (PS) and u E *E is such that f(u) E 0 and f'(u) :::::J 0. If u � ns(*E) , it follows from Theorem 19. 1 . 2 that u � pns(*E), that is, ::lc E JR+ Vy E uE ll u - Y ll > c. Let V : = { Vp : p E N} be dense in E. We will construct a Palais-Smale sequence (xn ) n EN in E such that for all N E *N = , X N � ns(*E) which contra­ dicts (PS ) . Let M E JR+ b e such that l f(u) l < M. For each n E N, define Cn

:=

{ x E E : IJ(x) l < M 1\ llf'(x) ll < � 1\ Vp E N [p :::; n =? llx-vv ll > c ] } ·

278

19. Nonstandard Palais-Smale conditions

Since, for each n E N, u E *C71 , we conclude that *Cn # 0 and therefore Cn # 0 . For each n E N, take Xn E crl' Let N E *N = and v E 17E . Since v = E, there exists P o E N such that ll v - Vp0 II < � . Since we conclude that x N � ns(*E) which contradicts (PS ) .

D

It is not clear whether this equivalence is true in general. The following result is consequence of the fact that if E is finite dimensional, then E is separable and fin(*E) = ns(*E ) . Theorem 19.3.3 ( PS 1 )

If E is finite dimensional ¢?

(PS)

¢?

(PSO) ¢? (PS2) ¢? (PS3) ¢? (PS4) .

Another easy observation:

Any C1 functional in a Banach space which verifies (PS4) and admits a Palais-Smale sequence, has at least one critical point.

Proposition 19.3.2

Proof. Suppose that ( un ) nEN is a Palais-Smale sequence for the functional f E C 1 (E, IR) . Then there exist m E *N= such that st ( f ( um) ) is a critical value of f. Hence, there exists a E E such that j(a) st (j(um ) ) and f'(a) 0. D =

=

Next we present an example which shows that

(PS2) f? ( PS 1 ) . Example 19.3. 1 Let H be an infinite dimensional Hilbert space and define j : H ----+ lR x f---+ f ( x ) = g ( ll x ll 2 - 1 ) where g : lR ----+ lR i s given by

g(t)

�

{

0 t 2 exp

_

__L t2

if t :<.:: 0 if t > 0

Observe that g is a C1 function and

g ' ( t) :::::: 0 Also,

¢}

h:H x

[ t :<.:: 0

v

t

----+ lR f---+

llxll2 - 1

::::::

0 ].

( 1 9.3 . 1 )

19.3.

279

Nonstandard Palais-Smale conditions

is a C1 functional and \fa E H

Vx E H

(h'(a) , x)

=

•

2a x

where • denotes the inner product in H. Therefore, f is a C1 functional. We will prove that f does not satisfy ( P 51) but satisfies ( P 52) . By Theorem 19.1.2 we can take u E *H such that u E fin(*H) \ ns(*H) and l l u l l = 1 . Hence, j(u) = 0 and f'(u) = 0, which shows that f does not satisfy (P51 ) . Note that v tf_ fin(*H) =? f(v) tf- 0 ·

·

and

f'(v) :::::J 0

<=?

<=?

ll f'(v) l l :::::J 0 Vx E fin(*H) (f'(v), x)

=

(2v x)g'(l l v l l 2 - 1 ) :::::J 0. •

(19.3.2)

Next we will prove that

f'(v) :::::J 0 =? [ l lv l l

�

1

V

llvll :::::J 1 ] .

(19.3.3)

f'(v) :::::J 0 and v # 0, by (19.3.2) either 2v ll�l l :::::J 0, and thus ll v ll :::::J 0, or g '(llvll2 - 1) :::::J 0, so that, by ( 19.3. 1 ) , llvll � 1 or llvll :::::J 1; in all possible cases ( 19.3.3) holds. •

If

Since

[ llv ll

<:::

1

and 0 is a critical value of

j,

we may conclude that

f(v)

E 0 1\

proving that

V

ll v ll :::::J 1 ] =? f(v) :::::J 0

j'(v) :::::J 0 =? v E fin(*H) f

does satisfy

1\

st(f(v))

0 is a critical value of f

( P 52) .

Remark 19.3. 1 Other consequences of Example 1 . I f we assume H to b e separable, Theorem

(P52) 2.

=

Moreover, the fact that that the sets

19.3 . 1 :

19.3.2 shows that

f? (PS ) .

f E C1 (E, IR) and satisfies (P52), does not imply (a

are compact (see Proposition

19.2. 1 ) .

<::: b)

19.

280

Nonstandard Palais-Smale conditions

19.1.1

It follows easily from Proposition lent to

that condition (PS3) is equiva­

( un ) n EN

(un ) n EN

is a Palais-Smale sequence -Uis bounded 1\ all convergent subsequences of converge to a critical value of f

(j(un ) ) nEN

and (PS4) is equivalent to

( un ) n EN

( un ) n EN

is a Palais-Smale sequence -Uis bounded 1\ there is a subsequence of (!( un ) ) n EN which converges to a critical value of f.

Moreover, in the finite dimensional case, these two standard conditions are equivalent .

19.4

Palais-Smale conditions per level

In this section we present a weaker compactness condition for C1 functionals introduced in 1980 [3] by Brezis, Caron and Nirenberg. In the survey book [7] the reader can find more variants of the (PS ) condition. c

Suppose f E C1 (E, IR) and E R We say that (un ) n E N is a Palais-Smale sequence of level {for f) if

Definition 19.4.1

c

lim f(un ) n---j . CXJ

=

lim f'(un ) c and n---j . CXJ

=

0.

f satisfies the Palais-Smale condition of level (PS )c, if every Palais­ Smale sequence of level has a convergent subsequence. c,

c

Remark 19.4.1 Suppose that

f satisfies

(PS ) , then

f E C1 (E, IR) .

f satisfies

Then

(PS) c for all

c

1.

If

2.

If f satisfies (PS) c, then the set of critical points of value compact.

E

R c,

Kc , is

Example 19.4. 1 The function exp(x) : lR ____, lR satisfies (PS ) c for all c except for c = 0. The real functions sin( x) and cos( x) defined in lR satisfy (PS )c for all c except for c = 1 and c = - 1 .

19.5. 19.5

281

Nonstandard variants of Palais-Srnale conditions per level

Nonstandard variants of Palais-Smale conditions per level

As above, E is a real Banach space, Obvious adaptations of conditions following conditions per level:

(PSO)c

( P 51 )

( un ) n EN

c

(PS2)c Note that if

and

c

and

ER (PS2)

is a Palais-Smale sequence of level -U::lm E *N= Um E ns(*E)

f(u) :::::J

c

1\

1\

c

=

c

1\ -U-

c

f'(u) :::::J 0

st (f(u) )

is a critical value of f

( un ) n EN is a Palais-Smale sequence of level

hence, the adaptation of conditions alent to the standard condition:

provide the

f'(u) :::::J 0 =? u E ns(*E)

f(u) :::::J u E fin(*E)

f E C1 (E, IR) (PSO) , (PS1)

c,

then

(PS3) and (PS4) to this context are equiv­

is a Palais-Smale sequence of level c -U( un ) n EN is bounded 1\ c is a critical value of f.

( un ) n E N

19.3.1, 19.3.2 and 19.3.3 can be easily proven. Theorem 1 9 . 5 . 1 For any real Banach space E we have (PS1)c =} (PS)c B (PSO)c =} (PS2)c• Theorem 19.5.2 Suppose E is a real separable Banach space and f E C 1 (E, IR) . Then f satisfies (PS)c if and only if f satisfies (PS1 )c. Theorem 19.5.3 If E has finite dzmension, then (PS1)c
=

19.

282

19.6

Nonstandard Palais-Smale conditions

Mountain Pass Theorems

Some important results in Critical Point Theory, such as the Mountain Pass Theorems and some variants of Ekeland's Variational Principle, can be obtained using a deformation technique. This technique was introduced in 1934 by Lusternik and Schnirelman [8] and consists in deforming a given C 1 functional outside the set of critical points. In 1983 Willcm proved in [14] the Quantitative Deformation Lemma; a very technical lemma that involves the concept of pseudo-gradient vector field (sec [14] or [9] ) . For

S s;;; E , a E JR+

and

c E IR,

r : = {x E E : f(x) :S c} where

we usc the following notations and

Sa := {x E E : dist(x, S) :S a}

dist(x, S) = inf { l l x - Y ll : y E S}.

Lemma 19.6.1 (Quantitative Deformation Lemma) S S: E, c E IR, c, 6 E JR+ be such that

\iy E f -1 ( [c - 2c, c + 2r::l ) n S25 x

Let f E C1 (E, IR) ,

[ ll f' (y) ll 2 8; ] .

Then there exists rJ E C( [O, 1] E, E) such that 1. rJ(O, y) y; 2. rJ(t, y) = y if y It' f -1 ( [c - 2r:: , c + 2r:: ] ) n S2s; =

3.

rJ(1, r+E n S)

s;;;

r- E ,.

for all t E [0, 1] , rJ(t, ) : E ----+ E is a homeomorphism; 5. \it E [0, 1] , \iy E E ll rJ (t, y) - Y l l :S 6 ; 6. for each y E E, f ( rJ( y)) is non increasing;

4.

·

·,

7. \it E ]0, 1] , \iy E r n Ss f(rJ(t, y))

< c.

An easy consequence of the Quantitative Deformation Lemma is the fol­ lowing variant of Ekeland's Variational Principle (see [9, page 14] ) . Corollary 19.6.1 Let f E C 1 (E, IR) b e bounded from below. Then, for any c E JR+ , there exists u E E such that

f(u)

:<:: inf

xE E

f(x) + c and ll f '( u) ll

< vfc.

19.6.

283

Mountain Pass Theorems

Applying the Transfer Principle to Corollary Corollary 19.6.2 Let f a point u E *E such that

19.6.1 we obtain

E C1 (E, IR) be bounded from below. Then there exists

f(u) :::::J inf f(x) and j'(u) :::::J 0. x EE Now we can deduce =

Let f E C 1 (E, IR) be bounded below and c infxE E f(x) . If f satisfies (PS2)c then c is a critical value of f. Proof. By Corollary 19.6.2 we conclude that there exists u E *E such that f(u) :::::J c and f'(u) :::::J 0. Since f satisfies (PS2)c, c st(f(u)) is a critical value of f. D Theorem 19.6.1 is a generalization of the classical result ( see [7, page 16] ) : Theorem 19.6.1

=

Theorem 19.6.2 Let f E C1 (E, IR) be bounded f satisfies ( PS )c then c zs a cntical value of f.

below and c

=

infxEE f(x) .

If

We now state the Mountain Pass Theorem introduced by Ambrosetti and Rabinowitz in 1973 [1]. Theorem 19.6.3 (Mountain Pass Theorem, Ambrosetti-Rabinowitz) Let E be a real Banach space and f E C 1 ( E, lR) . Suppose that 1.

there exists e E E and r E JR+ such that llell > r and max{f(O), f( e ) } < b inf f(x); :=

[[ x [[=r

:

2. f = {r E C( [0, 1], E) : 1 (0) = 0 A r(1) = e } and c = inf max f(r(t) ); 1Er tE[O,l] 3. f satisfies (PS) . Then c :2: b and c is a critical value of f . We usually say that if f : E ---+ lR satisfies condition 1 of Theorem 19.6.3, then f satisfies the mountain pass geometry ( with respect to 0 and e) . Remark 19.6.1 Conditions 1 and 2 of Theorem 19.6.3 are not enough to imply that c is a critical value of f as we can see with the following example ( sec [7, page 36] ) . The function f : IR2 ---+ lR defined by f( x, y) = x2 + ( x + 1 )3y 2 satisfies the mountain pass geometry with 0 = (0, 0), e = ( -2, 3) and r = ! · (0, 0) is a strict local minimizer and is the only critical point of f. Therefore, there is no z E IR2 such that f(z) = c > 0 and f'(z) = 0.

19.

284

Nonstandard Palais-Smale conditions

In 1980 Brezis, Caron and Nirenberg obtained in [3] a generalization of the Mountain Pass Theorem of Ambrosetti-Rabinowitz for functionals satisfying the (P S) c condition: Theorem 19.6.4 (Mountain Pass Theorem, Brezis- Coron-Nirenberg) Let E be a real Banach space and f E C1 ( E, lR) . Suppose that 1.

there exists e E E and r E JR+ such that li e I I max{f(O) , f(e)}

2.

=

r {r E C( [O, 1] , E) : 1(0)

=

<

0 !\ 1(1)

b

:=

=

>

r and

inf f(x); ll x ll =r

e} and

c :=

inf max

1Er tE[ O .l]

f(r(t)) ;

f satisfies (PS)c · Then 2: b and is a critical value of f . 3.

c

c

This result and the Mountain Pass Theorem of Ambrosetti-Rabinowitz are easy consequences of the Quantitative Deformation Lemma, since it is possible to prove that for every C 1 functional that satisfies conditions 1 and 2 of both theorems. there exists a Palais-Smale sequence of level c (see [9, page 19]). We now deduce the following generalization of the Mountain Pass Theorem of Brczis-Coron-Nirenberg. Theorem 19.6.5

pose that 1.

Let

E

be a real Banach space and f E

there exists e E E and r E JR+ such that li e I I max{f (O), f(e)}

2. f 3.

Then

=

{r E C( [O, 1] , E) : 1(0)

f satisfies c

2:

=

<

0 !\ 1(1)

b =

:=

>

C1 (E, IR) .

Sup­

r and

inf f(x); ll x ll =r

e } and

c :=

inf max

1Er tE[O,l]

f(r(t) ) ;

(PS4) c · c

b and is a critical value of f.

Proof. From the Quantitative Deformation Lemma there exists a Palais-Smale sequence of level c . Since f satisfies (PS4) c, c is a critical value of f. D

285

References

References [1] A. A!VI BROSETTI and P . RABINOWITZ, " Dual vatiational methods in crit­ ical point theory and applications", J. Funct . Anal., 14 (1973) 349-38 1 . [2] LEIF 0 . ARKERYD, NIGEL J . CUTLAND and C . WARD HENSON (eds . ) , Nonstandard Analysis, Theory and Applications, Kluwer, 1997. [3] H. B REZIS, J . M . C oRON and L. N IRENBERG, "Free vibrations for a non­ linear wave equation and a theorem of P. Rabinowitz", Comm. Pure Appl. Math. , 33 ( 1980) 667-684. [4] NIGEL J. CUTLAND (ed.) , CUP, 1988.

Nonstandard Analysis and its Applications,

[5] NIGEL J . CUTLAND, VfTOR NEVES, FRANCO OLIVEIRA and J OSE S ousA PINTO (cds.), Developments in nonstandard mathematics, Long­ man, 1995. [6] IVAR EKELAND "Nonconvex minimization Problems", Bulletin (New Se­ ries) of the American l\Iathematical Society, 1 ( 1979) .

An In­ troduction to Minimax Theorems and their Applications to Differential Equations, Kluwer Academic Publishers, 200 1 . L . LUSTERNIK and L . SCHNIRELMANN, Methodes topologiques dans le problemes variationnels, Hermann, Paris, 1934. .JEAN MAWHIN. Crztical Point Theory and Applicatwns to Nonlinear Dif­ ferential Equations, VIGRE Minicourse on Variational Methods and Non­

[7] MARIA DO RosARIO GROSSINHO and STEPAN AGOP TERSIAN,

[8] [9]

linear PDE, University of Utah, 2002 . [10] R . S . PALAIS and S . SMALE, "A generalized Morse theory", Bull. Amer. Math. Soc., 70 (1964) 165-172.

Minimax Methods in Critical Point Theory with Applicatwns to Dzfferentwl Equations, CBl\IS Regional Conference, 65,

[ 1 1] PAUL H. RABINOWITZ,

AMS, Providence, Rhode Island, 1986. [12] K . D . STROYAN and W . A . .J . LUXEMBURG, Infimtesimals, Academic Press. 1976.

Introduction to the theory of

[13] V. NEVES, The power of Gateaux differentiability, this volume. [14] M . WILLEM, Lectures on Critical Point Theory, Trabalho de Matermitica 199, Fund. Univ. Brasilia, Brasilia. 1983.

Averaging for ordinary different ial equat ions and functional differential equat ions Tewfik Sari* Abstract A nonstauclard approach to averaging theory for ordiuary di fferential equations and functional differential equations is developed. We define a notion of perturbation and we obtain averaging results under weaker conditions than the results in the literature. The classical averaging the­

orems apptoximate the solutions of the system by th e solutions of the averaged system, for L ipschitz continuous vector fields, and when the so­ lutions exist on the same interval as the �;olutions of the averaged system. vVe extend these resnlt.s to perturbations of vector fields which are uui­

tormly continuous in the spatial variable with l'E'Spect to the time variable and without any restriction on the interval of existence of the solution.

20 . 1

Introduction

In the early seventies, Georges R.ceb, who learned about Abraham Robin­ son's Nonstandar-d A nalysis (NSA) [29[ , was convinced that NSA gives a lan­ guage which is well adapted to the study of perturbation theory of differen­ tial equations (see [6, p. 374[ or [251). The a,'domatic presentation Internal Set Theo1·y (IS T) [26[ of NSA given by E. Nelson corresponded more to the Reeb's dream and was in agreement with his con vi cti on "Les entie'f's na:i.fs ne remplissent pas N ". Indeed, no formalism can recover exactly all the actual phenomena, and nonslanda:rd objecls which may be considered as a. formal­ ization of non- naive objects are already elements of our usual (standard) sets. We clo not need any use of sta.-rs and enlarg ements. Thus, the Reebian school adopted IST. For more informations about Recb's dream and convictions sec the Recb's preface of Lut z and Gaze's book [25] , St.ewart,'s book [40] p. 72, or Lobry's book [23[. The Reebian school of nonstandard perturbation theory of differ-ential eqv.a,­ tions produced various and numerous studies and new results as attested by a * univcrsiic de Haute Alsace, l'vlulhousc, Fra.11ce. T , Sari@uha . fr

20.2.

Deformations and perturbations

287

lot of books and proceedings (see [2, 3, 4, 7, 8, 9, 10, 12, 23, 25, 30, 37, 42] and their references) . It has become today a well-established tool in asymptotic theory, see, for instance, [17, 18, 20, 24, 39] and the five-digits classification 34E18 of the 2000 Mathematical Subject Classification. Canards and rivers (or Ducks and Streams [7] ) are the most famous discoveries of the Reebian school. The classical perturbation theory of differential equations studies defor­ mations, instead of perturbations, of differential equations (see Section 2.1). Classically the phenomena arc described asymptotically, when the parameter of the deformation tends to some fixed value. The first benefit of NSA is a natural and useful notion of perturbation. A perturbed equation becomes a simple nonstandard object , whose properties can be investigated directly. This aspect of NSA was clearly described by E. Nelson in his paper Mathematical Mythologies [30] , p. 159, when he said "For me, the most exciting aspect of non­

standard analysis is that concrete phenomena, such as ducks and streams, that classically can only be described awkwardly as asymptotic phenomena, become mythologized as simple nonstandard objects. "

The aim of this paper is to present some of the basic nonstandard techniques for averaging in Ordinary Differential Equations (ODEs) , that I obtained in [32, 36] . and their extensions, obtained by M. Lakrib [19] , to Functional Dif­ ferential Equations (FDEs) . This paper is organized as follows. In Section 20.2 we define the notion of perturbation of a vector field. The main problem of perturbation theory of differential equations is to describe the behavior of tra­ jectories of perturbed vector fields. We define a standard topology on the set of vector fields� with the property that f is a perturbation of a standard vector field fo if and only if f is infinitely close to fo for this topology. In Section 20.3 we present the Stroboscopic Method for ODEs and we show how to use it in the proof of the averaging theorem for ODEs. In Section 20.4 we present the Stroboscopic Method for FDEs and we show how to use it in the proofs of the averaging theorem for FDEs. The nonstandard approaches of averaging are rather similar in structure both in ODEs and FDEs. It should be noticed that the usual approaches of averaging make use of different concepts for ODEs and for FDEs: compare with [5, 31] for averaging in ODEs and [13, 14, 15, 22] for averaging in FDEs.

20.2 20. 2 . 1

Deformations and perturbations Deformations

The classical perturbation of differential equations

theory of differential equations x = F(x, c ) ,

studies families

(20.2. 1 )

20.

288

Averaging

where x belongs to an open subset U of JRn , called phase space, and c belongs to a subset B of JRk , called space of parameters. The family (20.2.1) of differential equations is said to be a k-parameters deformation of the vector field P0 (x) = P(x, co), where co is some fixed value of c. The main problem of the perturbation theory of differential equations is to investigate the behavior of the vector fields P(x , c) when c tends to c0 . The intuitive notion of a perturbation of the vector field Po which would mean any vector field which is close to Po does not appear in the theory. The situation is similar in the theory of almost periodic functions which, classically, do not have almost periods. The nonstandard approach permits to give a very natural notion of almost period (see [16, 28, 33. 41] ) . The classical perturbation theory of differential equations considers deformations instead of perturbations and would be better called defoTmation theory of dijjeTential equations. Ac­ tually the vector field P(x, c) when c is sufficiently close to co is called a perturbation of the vector field P0 (x ) . In other words, the differential equation

x

=

Po (x)

(20.2.2)

is said to be the unperturbed equation and equation (20.2.1), for a fixed value of c, is called the perturbed equation. This notion of perturbation is not very satisfactory since many of the results obtained for the family (20.2. 1) of dif­ ferential equations take place in all systems that are close to the unperturbed equation (20.2.2). Noticing this fact, V. I. Arnold (see [1] , footnote page 157) suggested to study a neighborhood of the unperturbed vector field P0 (x) in a suitable function space. For the sake of mathematical convenience, instead of neighborhoods, one considers deformations. According to V. I. Arnold, the situation is similar with the historical development of variational concepts, where the directional derivative ( Gateaux differential) preceded the derivative of a mapping (Frcchet differential) . Nonstandard analysis permits to define a notion of perturbation. To say that a vector f is a perturbation of a stan­ dard vector field fo is equivalent to say that f is infinitely close to fo is a suitable function space, that is f is in any standard neighborhood of fo . Thus, studying perturbations in our sense is nothing than studying neighborhoods, as suggested by V. I. Arnold.

20.2.2

Perturbations

Let X be a standard topological space. A point x E X is said to be infinitely close to a standard point x0 E X, which is denoted by x c:::- x0, if x is in any standard neighborhood of x0. Let A be a subset of X . A point x E X is said to be nearstandard in A if there is a standard x0 E A such that x c:::- x0.

20.2.

289

Deformations and perturbations

Let us denote by

N SA

=

{x E X : :3st xo E A x

c:::-

xo},

the external-set of nearstandard points i n A [34] . Let E b e a standard uniform space. The points x E E and y E E are said to be infinitely close, which is denoted by x c:::- y, if (x, y) lies in every standard entourage. If E is a standard metric space, with metric d, then x c:::- y is equivalent to d(x , y) infinitesimal.

Let X be a standard topological space X. Let E be a standard uniform space. Let D and Do be open subsets of X, Do standard. Let f : D ---+ E and fo : Do ---+ E be mappings, fo standard. The mapping f is said to be a perturbation of the mapping fo, which is denoted by f fo, if N SDo C D and f(x) fo (x) for all x E N SDo . Let :Fx,E be the set of mappings defined on open subsets of X to E : Definition 1

c:::-

c:::-

:Fx , E = { (!, D) : D open subset of X

and

f : D ---+ E} .

Let us consider the topology on this set defined as follows. Let The family of sets of the form

(fo, Do) E :Fx,E .

{(!, D) E :Fx.E : K C D Vx E K (j(x), fo (x)) E U}, where K is a compact subset of Do, and U is an entourage of the uniform space E, is a basis of the system of neighborhoods of (!0, D0 ). Let us call this topology the topology of uniform convergence on compacta. If all the mappings are defined on the same open set D, this topology is the usual topology of uniform convergence on compacta on the set of functions on D to E.

Assume X is locally compact. The mapping f is a perturbation of the standard mapping fo if and only if f is infinitely close to fo for the topology of uniform convergence on compacta.

Proposition 1

Proof. Let f : D ---+ E be a perturbation of fo : Do ---+ E. Let K be a standard compact subset of D0. Let U be a standard entourage. Then K C D and f(x) c:::- fo (x) for all x E K. Hence (j(x), fo (x)) E U. Thus f c:::- fo for the topology of uniform convergence on compacta. Conversely, let f be infinitely close to fo for the topology of uniform convergence on compacta. Let x E N SDo . There exists a standard xo E Do such that x c:::- xo . Let K be a standard compact neighborhood of x0, such that K C Do (such a neighborhood exists since X is locally compact) . Then x E K C D and (j(x) , j0 (x)) E U for all standard entourages U, that is N sDo C D and f ( x) c:::- fo ( x) on N sDo . Hence f is a perturbation of fo. D

20.

290

Averaging

The notion of perturbation can be used to formulate Tikhonov's theorem on slow and fast systems whose fast dynamics has asymptotically stable equi­ librium points [24] , and Pontryagin and Rodygin's theorem on slow and fast systems whose fast dynamics has asymptotically stable cycles [39] . In the fol­ lowing section we use it to formulate the theorem of Krilov, Bogolyubov and Mitropolski of averaging for ODEs. All these theorems belong to the singular perturbation theory. In this paper, by a solution of an Initial Value Problem (IVP) associated to an ODE we mean a maximal (i.e. noncontinuable) solu­ tion. The fundamental nonstandard result of the regular perturbation theory of ODEs is called the Short Shadow Lemma. It can be stated as follows [36, 37] . Let g : D ---+ IRd and g0 : Do ---+ IRd be continuous vector fields, D, Do C 0 IR+ x IRd . Let ag and a be initial conditions. Assume that g0 and a8 are standard. The IVP =

dXjdT g(T, X), X(O)

=

a0

(20.2.3)

is said to be a perturbation of the standard IVP =

=

(20.2.4) dXjdT go (T, X), X(O) a8, 0 if g g0 and a ag. To avoid inessential complications we assume that equation dX/ dT g0 (T, X) has the uniqueness of the solutions. Let ¢0 be the solution of the IVP (20.2.4) . Let I be its maximal interval of definition. Then, by the following theorem any solution of problem (20.2.3) also exist on I and is infinitely close to ¢o . Theorem 1 (Short Shadow Lemma) Let problem (20. 2. 3) be a perturba­ tion of problem (20. 2.4). Every solution ¢ of problem (20. 2. 3) is a perturbation of the solution c/Jo of problem (20. 2.4), that is, for all nearstandard t in I, ¢(t) is defined and satisfies ¢(t) ¢o (t). Let us consider the restriction 1jJ of ¢ to NsI. By the Short Shadow Lemma, for standard t E I, it takes nearstandard values 1/J(t) ¢0 (t) . Thus its shadow, which is the unique standard mapping which associate to each standard t the standard part of 1jJ ( t) , is equal to ¢0 . In general the shadow of ¢ is not equal to ¢0 . Thus, the Short Shadow Lemma describes only the "short time behaviour" "='

"='

=

"='

"='

of the solutions.

20.3

Averaging In ordinary differential equations

The method of averaging is well-known for ODEs. The fundamental re­ sult of this theory asserts that , fm small c > 0, the solutions of a nonau­ tonomous system

x = f (tjr:: , x, r:: ) ,

where

x = dxjdt,

(20.3. 1)

291

20.3. Averaging in ordinary differential equations are approximated by the solutions of the averaged autonomous system iJ

=

F (y) ,

where

F(x)

=

1 lim --+ T =T

1 f(t, x, T

0

0)

dt.

(20.3.2)

The approximation of the solutions of (20.3. 1) by the solutions of (20.3.2) means that if x ( t , c ) is a solution of (20.3. 1 ) and y(t) is the solution of the averaged equation (20.3.2) with the same initial condition, which is assumed to be defined on some interval [0, T] , then for c c:::: 0 and for all t E [0, T] , we have x(t, c ) c:::: y(t). The change of variable z ( T ) = x(u) transforms equation (20.3. 1) into equation 1 (20.3. 3) z = c f ( T, z , c ) . where z ' = dz j dT . Thus, the method of averaging can be stated for equation (20.3 .3), that is, if c is infinitesimal and 0 ::; T ::; T/c then z ( T, c) c:::: y(cT) . Classical results were obtained by Krilov, Bogolyubov, Mitropolski, Eck­ haus, Sanders, Verhulst (see [5, 31] and the references therein). The theory is very delicate. The dependence of f ( t, x, c ) in c introduces many complications in the formulations of the conditions under which averaging is justified. In the classical approach, averaging is justified for systems (20. 3 . 1 ) for which the vector field f is Lipschitz continuous in x. Our aim in this section is first to formulate this problem with the concept of perturbations of vector fields and then to give a theorem of averaging under hypothesis less restrictive than the usual hypothesis. In our approach, averaging is justified for all perturbations of a continuous vector field which is continuous in x uniformly with respect to t. This assumption is of course less restrictive than Lipschitz continuity with respect to x.

20.3.1

KBM vector fields

Let U0 be an open subset of IRd . The continuous vector field fo : IR+ x U0 ---+ JRd is said to be a Krilov-Bogolyubov-Mitropolski (KBM) vector field if it satisfies the following conditions Definition 2

1.

The function x ---+ fo ( t, x) is continuous in x uniformly with respect to the variable t.

2. For all x E Uo the limit F(x) 3.

=

=

limr--+ = � J0T fo(t, x)

dt exists.

The averaged equation y(t) F(y(t)) has the uniqueness of the solution with prescribed initial condition.

20.

292

Averaging

Notice that, in the previous definition, conditions (1) and (2) imply that the function F is continuous, so that the averaged equation considered in condition (3) is well defined. In the case of non autonomous ODEs, the definition of a perturbation given in Section 20.2 must be stated as follows. Definition 3 Let U0 and U be open subsets of IRd . A continuous vector field f : IR+ x U ---+ IRd is said to be a perturbation of the standard continuous vector x

field fo : IR+ Uo ---+ JRd if U contains all the nearstandard points in Uo , and f(s, x) fo (s, x) for all s E IR+ and all nearstandard x in Uo. c:::-

Theorem 2 Let fo : IR+ x U0 ---+ IRd be a standard KBM ao E Uo be standard. Let y( t) be the solution of the IVP

y (t)

=

F(y(t)),

y(O)

=

vector field and let (20.3.4)

ao,

defined on the inteTval [0, w[, 0 < w < oo. Let f IR+ U ---+ JRd be a perturbation of fo. Let c > 0 be infinitesimal and a ao. Then every solution x(t) of the IVP (20.3.5) x(O) = a, x(t) f (tjr:: , x(t)) ' is a perturbation of y ( t), that is, for all nearstandard t in [0, w [, x( t) is defined and satisfies x(t) y(t). c:::-

:

x

=

c:::-

The proof, in the particular case of almost periodic vector fields, is given in Section 20.3.4. The proof in the general case is given in Section 20.3.5.

20.3.2

Almost solutions

The notion of almost solution of an ODE is related to the classical notion of c- almost solution.

function x( t) is said to be an almost solution of the standard differential equation x G(t, x) on the standard interval [0, L] if there exists a finite sequence 0 to < < t N + 1 L such that for- n 0, , N we have

Definition 4 A

=

=

c:::-

·

·

·

tn+ l tn, x(t) and x(tn+ l ) - x(tn) tn+ l - tn

=

=

·

c:::-

x(tn) for t E [t n , t n+ l ] ,

c:::-

G(tn , x(tn )).

·

·

The aim of the following result is to show that an almost solution of a standard ODE is infinitely close to a solution of the equation. This result which was first established by .J. L. Callot (sec [1 1 , 27]) is a direct consequence of the nonstandard proof of the existence of solutions of continuous ODEs [26] .

293

20.3. Averaging in ordinary differential equations

Theorem 3 If x ( t) is an almost solution of the standard differential equation x = G(t, x) on the standard interval [0, L] , x(O) y0, with y0 standard, and the IVP y = G(t, y), y(O) = y0, has a unique solution y(t), then y(t) is defined at least on [0, L] and we have x(t) y(t), for all t E [0, L] . "='

"='

D

Proof. See [ 1 1 , 36]

Let us apply this theorem to obtain an averaging result for an ODE which does not satisfy all the hypothesis of Theorem 2. Consider the ODE (see [ 1 1 , 27. 36] )

tx . . X. ( t ) = Sln c

(20.3.6)

The conditions (2) and (3) in Definition 2 are satisfied with F(x) = 0. Thus, the solutions of the averaged equation arc constant. But condition ( 1 ) of the definition is not satisfied, since the function f(t, x) = sin (tx) is not continuous in x uniformly with respect to t. Hence Theorem 2 docs not apply. In fact the solutions of (20.3.6) are not nearly constant and we have the following result: Proposition 2 If c > 0 is infinitesimal then, in the region t 2 x > 0 solutions of (20. 8. 6) are infinitely close to hyperbolas tx = constant . In region x > t 2 0, they are infinitely close to the solutions of the ODE

the the

2 2 (20.3. 7) x = G(t, x) , where G(t, x) = vlx - t - x . t Proof. The isocline curves h = { ( t, x) : tx = 2k7rc} and I� = { ( t, x) : tx = (2k + �) 7rc} define, in the region t 2 x > 0, tubes in which the trajectories are trapped. Thus for t 2 x > 0 the solutions are infinitely close to the hyperbolas tx = constant. This argument does not work for x > t 2 0. In this region, we consider the microscope

T

=

t - tk

X

=

c ' where ( tk , X k ) are the points where a solution h· Then we have

X - Xk c

x( t)

' of (20.3.6) crosses the curve

X(O) = 0. By the Short Shadow Lemma (Theorem 1 ) , X(T) is infinitely close to a solution of dXjdT = sin (x k T + t k X). By straightforward computations we have

X -X tk + l - tk

k k+l '------

":::

G(t k , X k ) .

Hence, in the region x > t 2 0, the function x( t) is an almost solution of the ODE (20.3. 7) . By Theorem 3, the solutions of (20.3.6) are infinitely close to the solutions of (20.3.7). D

20.

294

20.3.3

Averaging

The stroboscopic method for ODEs

�+

In this section we denote by G : x D ----+ function, where D is a standard open subset of function such that 0 E I C Definition 5 We say that respect to G if there exists

�+.

�d.�d

�d

a standard continuous Let x : I ----+ be a

x satisfies the Strong Stroboscopic Property with > 0 such that for every positive limited t0 E I with x(to) nearstandard in D, there exists t1 E I such that f-L < t1 - to 0, [to , t1] c I, x(t) x(to) for all t E [to, t1], and f-L

c::::

c::::

x(t l ) - x(to) G ( t0, x t0 )) . ( t1 - to The real numbers t0 and t1 are called successive instants of observation of the �

--------'-'------'---'- _

stroboscopic method. Theorem 4 (Stroboscopic Lemma for ODEs) Let a0 E D be standar-d. Assume that the IVP y(t) G (t, y(t)), y(O) a0, has a unique solution y defined on some standard interval [0, L] . Assume that x(O) a0 and x satisfies the Strong Stroboscopic Property with respect to G. Then x is defined at least on [0, L] and satisfies x(t) y(t) for all t E [0� L] . =

=

c::::

c::::

Proof. Since x satisfies the Strong Stroboscopic Property with respect to G, it is an almost solution of the ODE x = G(t, x). By Theorem 3 we have x(t) c:::: y(t) for all t E [0, L] . The details of the proof can be found in [36] . D The Stroboscopic Lemma has many applications in the perturbation theory of differential equations (see [11, 32, 35, 36� 38 , 39]). Let us use this lemma to obtain a proof of Theorem 2.

20.3.4

Proof of Theorem 2 for almost periodic vector fields

·,

Suppose that fo is an almost periodic in t, then any of its translates fo (s + x0) is a nearstandard function, and fo has an average F which satis­ fies [16, 28, 33, 41]

s+Tfo (t, x) dt, 1 T-->rx> T s s E �+ · F 1 1 s+ T F(x) T s fo (t, x) dt,

F(x) uniformly with respect to

=

1

lim -

Since

c:::: -

is standard and continuous, we have

(20.3.8)

20.3.

295

Averaging in ordinary differential equations

IR

for all s E + , all T ::::::' oo and all nearstandard x in U0 . Let x : I ---+ U be a solution of problem (20.3.5) . Let t0 be an instant of observation: t0 is limited in I, and x0 = x(to) is nearstandard in U0 . The change of variables

X transforms

(20.3.5)

=

_x---'-(t o + cTc) _- x_o ' c _ _ __

_

into

dX/dT

=

f(s + T, xo + eX) ,

s

where

=

tojc.

By the Short Shadow Lemma ( Theorem 1 ) , applied to g(T, X) f(s + T, xo + eX) and go (T, X) fo ( s + T, xo) , for all limited T > 0, we have X(T) ::::::' J0Y fo (s + r, xo)dr. By Robinson's Lemma this property is true for some unlimited T which can be chosen such that cT 0. Define t1 t0 + cT. =

=

::::::'

Then we have

=

1

l

+ y 1 s Y ::::::' -1 0 fo( s + r, xo) dr = T s fo (t, xo) dt ::::::' F(xo). T T Thus x satisfies the Strong Stroboscopic Property with respect to F. Using the Stroboscopic Lemma for ODEs ( Theorem 4) we conclude that x(t) is infinitely close to a solution of the averaged ODE (20.3.4). x(t l ) - x(to) t1 - to

----'----'---'----'- =

20.3.5

X ( T)

--

Proof of Theorem 2 for KBM vector fields

Let fo be a KBM vector field. From condition (2 ) of Definition 2 we deduce + that for all s E + , we have F(x) = limy_, = � fss Y fo (t, x) dt, but the limit is not uniform on s . Thus for unlimited positive s , the property ( 20.3.8 ) docs not hold for all unlimited T, as it was the case for almost periodic vector fields. However, using also the uniform continuity of fo in x with respect to t we can show that (20.3.8 ) holds for some unlimited T which are not very large. This result is stated in the following technical lemma [36] .

IR

IR

IRd

Let g : + x M ---+ be a standard continuous function where M is a standard metric space. We assume that g is continuous in m E M uniformly with respect to t E + and that g has an average G(m) limy_, = � J0Y g(t, m) dt. Let c > 0 be infinitesimal. Let t E + be limited. Let m be nearstandard in M . Then there exists a > c, a 0 such that, for all limited T 2 0 we have + 1 s YS g(r, m) dr ::::::' T G ( m ) , where s = tjc, S = o:jc. Lemma 1

S

1 8

IR

::::::'

IR

=

20.

296

2

The proof of Theorem found also in [36] .

Averaging

needs another technical lemma whose proof can be

Let g IR+ JRd JRd and h IR+ JRd be continuous functions. Suppose that g(T, T X) h(T) holds for all limited T E IR+ and all limited X E JRd , and J0 h ( r) dr is limited for all limited T E IR+ . Then, any solution X(T) of the IVP dXjdTT g(T, X), X(O) 0, is defined for all limited T E IR+ and satisfies X(T) "=' J0 h(r) dr. Lemma 2

:

:

----+

x

"='

=

----+

=

Proof of Theorem 2. Let x : I ----+ U be a solution of problem (20.3.5). Let t0 E I be limited, such that x0 = x(t0) is nearstandard in U0. By Lemma 1 , applied to g = fo, G = F and m = x(to), there is a > 0, a "=' 0 such that for all limited T 2 0 we have

T 1 rs + S fo (r, xo) dr "=' TF (xo), s .}s

where

s = to/c, S = o:jc.

(20.3.9)

The change of variables

X(T) transforms

(20.3.5)

=

x (to + o:T) - xo 0:

into

dXjdT

=

f(s + ST, xo + o:X).

By Lemma 2, applied t o g(T, X) = f(s and (20.3.9) , for all limited T >

+ ST xo + o:X) and h(T) = fo (s + 0, we have T s +TS X(T) "=' fo (s + Sr, xo) dr = -1 fo (r, xo) dr ":' TF (xo). 0 s s Define the successive instant of observation of the stroboscopic method t1 by t1 = to + a. Then we have ST, xo),

lo

1

x(t l ) - x(to) = X ( 1) "=' F(xo). t1 - to Since t1 - to = a > c and x(t) - x(to) = o:X(T) "=' 0 for all t E [to, t1] , we have proved that the function x satisfies the Strong Stroboscopic Property with respect to F. By the Stroboscopic Lemma, for any nearstandard t E [0 , w[, x(t) i s defined and satisfies x(t) "=' y(t). D

20.4.

297

Functional differential equations

20.4

Functional differential equations

Let C = C ([ - r, 0] , IRd ) , where r > 0, denote the Banach space of continuous functions with the norm 11 ¢ 11 = sup{ II ¢(B) II : e E [ - r, O] } , where 1 1 · 11 is a norm of JRd . Let L 2 t0. If x : [-r, L] ---+ IRd is continuous, we define Xt E C by setting Xt (B) = x(t + B), e E [ - r, OJ for each t E [0, L] . Let g : JR + x C ---+ lRd , (t, u) f--+ g(t, v. ) , be a continuous function. Let ¢ E C be an initial condition. A Functional Differential Equation (FDE) is an equation of the form

x(t) = g (t, xt) ,

xo = ¢.

This type of equation includes differential equations with delays of the form =

x(t) G(t, x(t), x(t - r) ) , where G : IR+ IRd IRd ---+ JRd . Here we have g(t, u ) G(t, u(O) , u( - r)). The method of averaging was extended [13, 22] to the case of FDEs x

x

=

the form where

of

(20.4. 1)

c is a small parameter.

In that case the averaged equation is the ODE

(20.4.2) where F is the average of of the form

f.

It was also extended

x(t)

=

f (t/c, x t ) .

[14]

to the case of FDEs

(20.4. 3)

In that case the averaged equation is the FDE

(20.4.4) Notice that the change of variables x(t) = z(t/c) does not transform equa­ tion (20.4. 1) into equation (20.4.3) , as it was the case for ODEs (20.3.3) and (20.3.1), so that the results obtained for (20.4. 1) cannot be applied to (20.4.3) . In the case of FDEs of the form (20.4. 1) or (20.4.3), the clas­ sical averaging theorems require that the vector field f is Lipschitz continuous in x uniformly with respect to t. In our approach, this condition is weak­ ened and we only assume that the vector field f is continuous in x uniformly with respect to t. Also in the classical averaging theorems it is assumed that the solutions z(T, c) of (20.4. 1) and y(T) of (20.4.2) exist in the same interval [0, T /c] or that the solutions x(t, c) of (20.4.3) and y(t) of (20.4.4) exist in the same interval [0, T] . In our approach, we assume only that the solution of the averaged equation is defined on some interval and we give conditions on the vector field f so that, for c sufficiently smalL the solution x(t, c) of the system exists at least on the same intervaL

298

20. Averaging

20.4.1

Averaging for FDEs in the form z'(r)

=

E:f (r, Z-r )

c is a small parameter zo c/J . The change of variable x(t) z(t/c) transforms this equation in (20.4. 5) x(t) f (tjc, Xt ,c ) , x(t) c/J(t/c), t E [ - cr, 0] , where Xt , E E C is defined by Xt ,E (B) x( t + cB) for B E [ - r, 0] . Let f IR+ x C -+ JRd be a standard continuous function. We assume that (H1) The function f : u f (t, u) is continuous in u uniformly with respect to the variable t. (H2) For all u E C the limit F(u) limr-Hx> � J0T f (t, u) dt exist s. We identify IRd to the subset of constant functions in C , and for any vector c E JRd , we denote by the same letter, the constant function u E C defined by u(B) c, e E [ - r, 0] . Averaging consists in approximating the solutions x(t, c) of (20.4. 5) by the solution y(t) of the averaged ODE (20.4. 6) y(t) F (y(t)) , y(O) ¢(0) . According to our convention , y(t) , in the right-hand side of this equation. is the constant function u t E C defined by ut (e) y(t), e E [-r, O] . Since F is We consider the IVP, where

=

=

=

=

=

:

f--+

=

=

=

=

=

continuous, this equation is well defined. We assume that

( H 3)

The averaged ODE (20.4.6) has the uniqueness of the solution with pre­ scribed initial condition.

(H4) The function f is quasi-bounded in the variable u uniformly with respect to the variable t, that is, for every t E IR+ and every limited u E C, f (t, u) is limited in JRd . Notice that conditions (H1), (H2) and ( H 3) are similar to conditions (1 ) , (2) and (3) of Definition 2 . In the case of FDEs we need also condition (H4) . In classical words, the uniform quasi boundedness means that for every bounded subset B of C, f (IR+ x B) is a bounded subset of IRd . This property is strongly related to the continuation properties of the solutions of FDEs (see Sections 2 . 3 and 3. 1 o f [1 5] ) . Theorem 5 Let f : IR+ x C -+ IRd be a standard continuous function satisfying

the conditions (Hl ) - (H4). Let cjJ be standard in C. Let L > 0 be standard and let y : [0, L] -+ IRd be the solution of (20.4. 6). Let c > 0 be infinitesimal. Then every solution x(t) of the problem {20. 4 . 5) is defined at least on [- cr. L] and satisfies x(t) "::: y(t) for all t E [0 , L] .

20.4.

299

Functional differential equations

20.4. 2

The stroboscopic method for ODEs revisited

In this section we give another formulation of the stroboscopic method for ODEs which is well adapted to the proof of Theorem 5. Moreover , this formulation of the Stroboscopic Method will be easily extended to FDEs ( see Section 20.4.4) . We denote by G : IR x IRd ---+ IRd , a standard continuous function. Let x : I ---+ JRd be a function such that 0 E I C IR .

+

+

We say that x satisfies the Stroboscopic Property with respect to G if there exists f-L > 0 such that for every positive limited t0 E I, satisfying [0, t0] C I and x(t) is limited for all t E [0, t0] , there exists t1 E I such that f-L < t1 - to "::: 0, [to, t1] c I, x(t) "::: x(to) for all t E [to , t1], and

Definition 6

x(t l ) - x(to) t l - to

----'-----'------'.----'-

_

�

G ( t0, x ( to )) .

The difference with the Strong Stroboscopic Property with respect to G con­ sidered in Section 20.3.3 is that now we assume that the successive instant of observation t1 exists only for those values t0 for which x(t) is limited for all t E [0, t0] . In Definition 5, in which we take D = IRd , we assumed the stronger hypothesis that t1 exists for all limited to for which x(to) is limited.

Let ao E D be standaTd. Assume that the IVP y(t) G (t, y(t)), y(O) ao, has a unique solution y defined on some standard interval [0, L] . Assume that x(O) "::: ao and x satisfies the Stroboscopic Property with respect to G. Then x is defined at least on [0, L] and satisfies x(t) y(t) for all t E [0, L] .

Theorem 6 (Second Stroboscopic Lemma for ODEs) =

=

c:::-

Proof. Since x satisfies the Stroboscopic Property with respect to G, it is an almost solution of the ODE x = G(t, x). By Theorem 3 we have x(t) "::: y(t) for all t E [0, L] . The details of the proof can found in [19] or [21] . D Proof of Theorem 5. Let x : I ---+ JRd be a solution of problem (20.4.5). Let t0 E I be limited, such that x(t) is limited for all t E [0, t0] . By Lemma applied to g = f, G = F and the constant function m = x(to), there is a > a "::: 0 such that for all limited T 2: 0 we have

1 S

ls+TS f(r, x(to)) dr "::: TF(x(to)), s

where

s = to/c, S = o:jc.

(20.4. 7)

Using the uniform quasi boundedness of f we can show ( for the details see or [21]) that x(t) is defined and limited for all t "::: t0 . Hence the function e

X ( , T)

=

x(to + aT + cB) - x(to) , B E [-r, O] . T E [0, 1] , 0:

1, 0,

[19]

20.

300

is well defined. In the variable

X ( - , T)

(20.4.5)

system

Averaging

becomes

ax (0, T) = f(s + ST, x(to) + aX( - , T)). oT Using assumptions (H1 ) and (H4) together with (20.4. 7), we obtain after some computations that for all T E [0, 1] , we have

X (O, T)

"'='

{T f(s + Sr, x(to)) dr = 1 1s+TSf (r, x(t0)) dr

Jo

S

8

"'='

TF(x(t0)).

Define the successive instant of observation of the stroboscopic method t1 = to + a. Then we have

t1

by

x(t i ) - x(to) X (O, 1 ) F(x(to)). t l - to Since t1 - to a > c and x(t) - x(to) aX(O, T) 0 for all t E [to, t1 ] , we have proved that the function x satisfies the Stroboscopic Property with respect to F. By the Second Stroboscopic Lemma for ODEs, for any t E [0, L] , D x(t) is defined and satisfies x(t) y(t). "'='

=

=

"'='

=

"'='

20.4.3

Averaging for FDEs in the form x (t)

We consider the IVP, where

x(t)

=

c

=

is a small parameter

f (tj c , Xt ) ,

xo

=

f (tjE:, Xt )

(20.4.8 )

¢,

We assume that f satisfies conditions ( H1 ) , ( H2 ) and ( H4 ) of Section Now, the averaged equation is not the O D E (20.4.6), but the FDE

y(t) = F ( Yt ) ,

20.4. 1.

(20.4.9)

Yo = ¢.

Averaging consists in approximating the solutions x(t, c ) of ( 20.4.8 ) by the solution y(t) of the averaged FDE (20.4.9). Condition (H3) in Section 20. 4. 1 must be restated as follows

(H3)

The averaged FDE ( 20.4.9 ) has the uniqueness of the solution with pre­ scribed initial condition.

d Theorem 7 Let f : IR+ C ---+ IR be a standard continuous function satisfying the conditionsd (Hl)-(H4) . Let ¢ be standard in C. Let L 0 be standard and let 0 be infinitesimal. y : [0, L] IR be the solution of problem (20.4.9). Let x

---+

>

c >

Then every solution x(t) of the problem {20.4.8) is defined at least on [ - r, L] and satisfies x(t) y(t) for all t E [-r, L] . "'='

20.4.

301

Functional differential equations

2 0.4.4

The stroboscopic method for FDEs

Since the averaged equation (20.4. 9) is an FDE, we need an extension of the stroboscopic method for ODEs given in Section 20.4.2. In this section we denote by G : IR x C ---+ IRd , a standard continuous function. Let x : I ---+ IRd be a function such that [ - r, OJ C I C IR .

+

+

We say that x satisfies the Stroboscopic Property with respect to G if there exists fJ· > 0 S1Lch that for every positive limited to E I, satisfying [0, to] C I and x(t) and G(t , Xt) are limited for all t E [0 , to] , there exists t 1 E I such that fJ < t1 - to 0, [to, t1] C I, x(t) x(to) for all t E [to, t1] , and

Definition 7

c:::-

c:::-

x(t x(to) ------'- l )---'- - -----'- ---'- - G ( to, X to ) . t1 - to �

Notice that now we assume that the successive instant of observation t1 exists for those values t0 for which both x ( t) and G ( t, xt ) are limited for all t E [ 0, t0 ] . In the limit case r = 0 , the Banach space C is identified with IRd and the function X t is identified with x(t) so that , G(t , X t ) is limited, for all limited x(t) . Hence the "Stroboscopic Property with respect to G " considered in the previous definition is a natural extension to FDEs of the "Stroboscopic Property with respect to G " considered in Definition 6.

Let ¢ E C be standard. A s­ sume that the IVP y(t) G (t, Yt ), Yo ¢, has a unique solution y defined on some standard interval [ -r, L] . Assume that the function x satisfies the Stroboscopic Property with respect to G and x0 c:::- ¢. Then x is defined at least on [-r, L] and satisfies x(t) y(t) for all t E [-r, L] .

Theorem 8 (Stroboscopic Lemma for FDEs) =

=

c:::-

Proof. Since x satisfies the Stroboscopic Property with respect to G, it is an almost solution of the FDE x = G(t, Xt ) · For FDEs, we have to our disposal an analog of Theorem 3. Thus x ( t) c:::- y ( t) for all t E [0, L] . The details of the proof can found in [19] or [21] . D Proof of Theorem 7. Let x : I ---+ JRd be a solution of problem (20.4.8) . Let t0 E I be limited, such that both x(t) and F(xt ) are limited for all t E [0, t0] . From the uniform quasi boundedness of f we deduce that x( t) is S-continuous on [0, t0] . Thus Xt is nearstandard for all t E [0, t0] . By Lemma 1 , applied to g f, G F and m Xt 0 , there is o: > 0, o: c:::- 0 such that for all limited T 2 0 we have =

=

1

S

1s +TS f (r, X 8

=

t 0 ) dr

c:::-

T F ( Xt0 ) ,

where

s

=

to/c , S

=

o:jc.

(20.4.10)

20.

302

Averaging

Using the uniform quasi boundedness of f we can show ( for the details see or [2 1]) that x(t) is defined and limited for all t ,--- t0. Hence the function

[19]

X ( e , T ) = x(to + aT + B) - x(to + B) , B E [-r, 0] , T E [0, 1] , a is well defined. In the variable X ( · , T) system (20.4.8) becomes ax · oT (0, T) = f(s + ST, X t0 + aX( , T) ). Using assumptions ( H 1) and (H4) together with (20.4.10), we obtain that for all T E [0, 1] , we have

{T

1 X(O, T) ,--- J f (s + Sr, xt 0 ) dr = s o

1s+ TS f (r, x ) dr ,--- TF (xt ) . 0 t0 s

Define the successive instant of observation of the stroboscopic method t1 = to + a. Then we have

t1

by

x(t l ) - x(to) X (O, 1) "-' F ) (x to t1 - to Since t1 - to a > c and x(t) - x(to) aX (O, T) 0 for all t E [to, h ] , we have proved that the function x satisfies the Stroboscopic Property with respect to F. By the Stroboscopic Lemma for FDEs, for any t E [0, L] , x ( t ) is defined and satisfies x(t) ,--- y(t) . D =

=

=

,---

References

[1]

V . I . ARNOLD ( ed. ) , Dynamical Systems Sciences, Vol. 5, Springer-Verlag, 1994.

V, Encyclopedia of Mathematical

[2]

H . BAR REAU and .J . HARTHONG (editeurs ) , dard, Editions du CNRS, Paris, 1989.

La mathematique Non Stan­

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E. BENOIT ( Ed. ) , Dynamic Springer-Verlag, Berlin, 1991.

Proceedings, Luminy

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I . P . VAN DEN BERG , Nonstandard Asymptotic in Math. 1249, Springer-Verlag, 1987.

Bifurcations,

Analysis,

1990,

Lectures Notes

[5] N . N . BOGOLYUBOV and Yu. A . I\IITROPOLSKI, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.

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J . W . DAUBEN , Abraham Robinson, The Creation of Nonstandard Anal­ ysis, A Personal and Mathematical Odyssey, Princeton University Press, Princeton, New Jersey, 1995.

[7]

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F. DIENER and G . REEB,

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Nonstandard Analysis in Practice,

Analyse non standard,

Hermann,

1989.

standard et repnisen­

[10] M . DIENER and G. WALLET (editeurs) , Mathematiques finitaires et anal­ yse non standard, Publication mathematique de l'Universite de Paris 7, Vol. 31-1, 31-2, 1989. [11]

J. L. CALLOT and T. SARI , Stroboscopie et moyennisation dans les sys­ tcmes d'cquations diffcrcntielles a solutions rapidement oscillantes, in

Mathematical Tools and Models for Control, Systems Analysis and Sig­ nal Processing, vol. 3, CNRS Paris, 1983.

[12]

A . FRUCHARD and A. TROESCH (editeurs), Colloque Trajectorien a la memoire de G. Reeb et J. L. Callot, Strasbourg-Obernai, 12-16 juin 1995, Prepublication de l'IRtvi A, Strasbourg, 1995.

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J . K . HALE, "Averaging methods for differential equations with retarded arguments and a small parameter", .J. Differential Equations, 2 (1966)

57- 73.

[14] [15]

J . K . HALE and S . M . VERDUYN LUNEL, "Averaging in infinite dimen­ sions", J. Integral Equations Appl . , 2 (1990) 463-494. .T . K . HALE and S . M . VERDUYN LUNEL , lntmduction to Functional Dif­ Applied Mathematical Sciences 99, Springer-Verlag,

ferential Equations, New York, 1993.

[16]

L . D . KLUGLER, Nonstandard Analysis of almost periodic functions, in Applications of Model Theory to Algebra, Analysis and Probabzhty, W.A.J. Luxemburg ed. , Holt, Rinehart and Winston, 1969.

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M. LAKRIB, "The method of averaging and functional differential equa­ tions with delay", Int. J . Math. Sci. , 26 (2001) 497-511.

[18]

M . LAKRIB, " On the averaging method for differential equations with delay", Electron. J. Differential Equations, 65 (2002) 1-16.

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Stroboscopie et moyenmsation dans les equations differen­ tielles fonctionnelles a retard, These de Doctorat en Mathematiques de

[19] 1\ I . LAKRIB,

l' Universite de Haute Alsace, Mulhouse, 2004. [20] 1\ I . LAKRIB and T. SARI, "Averaging results for functional differential equations", Sibirsk. Mat. Zh. , 45 ( 2004) 375-386; translation in Siberian Math. J . , 45 (2004) 3 1 1-320. [2 1] 1\ I . LAKRIB and T. SARI , Averaging Theorems for Ordinary Differential Equations and Retarded Functional Differential Equations. http : //www . math . uha . fr/ps/20050 1 lakrib . pdf [22] B . LEHMAN and S . P . WEIBEL . "Fundamental theorems of averaging for functional differential equations", J . Differential Equations, 152 ( 1 999) 160-190. [23] C. LOBRY, 1 989.

Et pourtant . . . ils ne remplissent pas N !,

Aleas Editeur, Lyon,

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with

[26] E . NELSON, "Internal Set Theory", Bull. Amer. 1\Iath. Soc . , 83 ( 1 977) 1 165- 1 1 98. [27] G. REEB, Equations differentielles et analyse non classique ( d'apres .J . L. Callot) , in Pmceedings of the 4th International Colloquium on Differ­ ential Geometry ( 1 978) , Publicaciones de la Universidad de Santiago de Compostela, 1 979. [28] A. RoBINSON , " Compactification of Groups and Rings and Nonstandard Analysis". Journ. Symbolic Logic, 34 ( 1 969) 576-588. [29] A. ROBINSON. 1 974.

Nonstandard Analysis,

American Elsevier, New York,

[30] .J .-l\1. SALANSKIS and H . SINACEUR (eds.) , Le Colloque de Cerisy, Springer-Verlag, Paris, 1992.

Labyrinthe du Continu,

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T. SARI, Petite histoire de la stroboscopie, in Colloque Trajectorien a la Memoire de J. L. Callot et G. Reeb, Strasbourg-Obernai 1995. Publication IRMA, Univ. Strasbourg (1995) , 5-15.

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T . SARI, Stroboscopy and Averaging, in Colloque Trajectorien a la Me­ moire de J. L. Callot et G. Reeb, Strasbourg-Obernai 1995, Publication IRMA, Univ. Strasbourg, 1995.

[37]

T. SARI, Nonstandard Perturbation Theory of Differential Equations, Ed­ inburgh, invited talk in International Congres in Nonstandard Analysis and its Applications, ICMS, Edinburgh, 1996. http : //www . math . uha . fr/sari/papers/i cms 1996 . pdf

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T . SARI , Averaging in Hamiltonian systems with slowly varying parame­ ters, in Developments in Mathematzcal and Experimental Physics, Vol. C, Hydrodynamics and Dynamical Systems, Proceedings of the First Mexican Meeting on Mathematical and Experimental Physics, El Colegio Nacional, Mexico City, September 10-14, 2001 , Ed. A. Macias, F. Uribe and E. Diaz, Kluwer Academic/Plenum Publishers, 2003.

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T . SARI and K . YADI, "On Pontryagin-Rodygin's theorem for convergence of solutions of slow and fast systems", Electron. J. Differential Equations, 139 (2004) 1-17.

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I. STEWART,

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K . D . STROYAN and W . A . .J . LUXEMBURG , infinitesimals, Academic Press, 1976.

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I I I8

1987.

The Problems of Mathematics,

Oxford University Press,

Introduction to the theory of

rencontre de Geometrie du Schnepfenried, Feuilletages, Geometrie symplectzque et de contact, Analyse non standard et applications, Vol. 2, 10-15 mai 1982 , Asterisque 109-110, Societe Mathematique de France, 1983.

Path-space measure for stochastic differential equation with a coefficient of polynomial growth Toru Nakamura

A a-- additive measure over

*

Abstract

a spa.ce of paths

is constructed to give the so­ lution to the Fokker-Planck equation associated with a st.oehastic differ­ ential equation with coefficient fuuction of polynomial growth by making use of nonstauda.rd analysis.

21.1

Heuristic arguments and definitions

Consider a stochastic differential equation,

dx(t)

=

f (x(t)) dt + db(t),

(21 . 1 .1 )

where f ( x) is a real-valued function and b(t) a Browni an motion vvith vari an ce 2Dt for a time interval t. vVe wish to construct a m easme over a space of paths for (21 . 1 .1 ) . To my knowledge, the coefficient. fund.ion has been assumed to have at most linear growth, that is [.f(x) [ .::::; const· [ x [ for sufficiently lar ge x, otherwise some paths explode to infinity in finite tunes. As an example, let .f(x) = [ x [ l+6 (8 > 0) and define explosion t ime e for each continuous path x(t) by limt-->e-o x(t) = ±oo. Then, it is proved that P(e oo) < 1 and more strongly P( e = oo ) 0. For general case, see Feller's test for explosion in [1]. Despite the explosion, we shall consider f(x) of polynomial growth of an arbitrary order and define a measure over a space of paths. vVe use nonstandard analysis because it has a very convenient theory, Loeb measure theory [2, 3] , which enables us t.o construct. a standard O"-additive measure in a simple way. ·

=

=

*Department of Mat.hemat.ics. S undai Preparatory School, Kanda-S urugadai. Chiyoda­ kn, Tokyo 1 0 1-0062, .Japan.

21.1.

307

Heuristic arguments and definitions

Let us interpret (21 . 1 . 1) as a law for a particle momentum x = x(t) at time t with the force f ( x( t)) for drift and the random force db( t) / dt acting on the particle. By a time t > 0 some particles may disappear to infinity along the exploding paths, but others still exist with finite momentum so that they should make up a "probability" density of a particle momentum. We wish to usc the word "probability" though its total value may be less than 1 because of the disappearance of particles to infinity. Especially when f ( x( t)) is a repulsive force, that is, its sign is the same as that of x(t) , a particle can get larger momentum compared to the case where f (x(t)) is absent. However, once it gets very large momentum, it can hardly come back to the former one because the drift force f (x(t)) acts on the particle so as to increase its momentum. Thus, the particles which have gone to infinity by a time t > 0 could not contribute to the probability density at t. This consideration suggests that we can introduce a cutoff at an infinite number in momentum space if it is necessary in order to define a measure over a space of paths so that the probability density should be constructed by a path integral with respect to the measure. Eq. (21.1.1) gives the forward Fokker-Planck equation for the probability density U(t, x) of a particle momentum x at time t, 0

0 2 D o 2 U(t, x) - { J(x) U(t, x) } . (21 . 1 .2) ot ox ox We assume that the drift coefficient f(x) and the initial function U(O. x) satisfy

U(t. x)

=

the following conditions:

(A1)

For some natural number large x.

n

E N,

l f(x) l

:<::

const · l x l n

for sufficiently

(A2) f(x) E C 2 (IR) . (A3) f(x) 2 /(4D) + f'(x)/2 is bounded from below. We denote the min {f(x) 2 /(4D) + f'(x)/2 1 x E IR}.

bound by

c =

(A4) U(O, x) E C 2 (IR) and its support is a bounded set . Rewrite (21.1 .2) into a difference equation with infinitesimal time-spacing c and momentum-spacing 6 V'fDc using a forward difference quotient 0 1 0 U(t, x) ==? - { U(t + r:: , x) - U(t, x) } t c =

for time-derivative, and central ones 0

ox

1 U(t, x) ==? 26 { U(t, x + 6) - U(t, x - 6) }

21.

308

and

Path-space measure for stochastic differential equation

oEJ2x2 U(t, x) =? 1)21 { U(t, x + 6) + U(t, x - 6) - 2 U(t, x) }

for space-derivative. The result is

U(t + c, x) � { 1 + .f (x - 6)Jc/ (2D ) } U(t, x - 6) ( 21.1.3 ) + � { 1 + .f (X + 6) (- JcI (2D ) ) } u ( t ' X + 6) ' which indicates that the coefficients H 1 + .f ( .T - 6)Jc/( 2D )} should be as­ signed to the infinitesimal line-segment with end points ( t, x ( t) ) ( t, x ( t + c) - 6) and ( t + c. x(t + c) ) of a *-polygonal path x( s), and !{1 + .f (x + 6) (- Jc/ ( 2D ) ) } to the segment with end points ( t, x(t) ) (t, x(t + c) + 6) and ( t + c , x(t + c )) . Taking into account the approximation 1 .f (x(t) ) (±v 2Dc ) - 1 .f (x(t) ) 2 c log [ 1 + .f (x(t ) ) (± Jc/ ( 2D ) )] � 4D 2D 1 .f (x(t)) { x(t + c) - x(t)} - 1 .f (x(t)) 2 c , 4D 2D =

=

=

�

=

let us interpret the coefficients as

� { 1 + .f (x(t) ) (±Jc/(2D) ) }

� � exp [lt+c 2�.f (x( s) ) db ( s) - 1 t+c 4�.f ( x( s) ) 2 ds] .

The first integral on the right-hand side is the Ito-integral. Thus, we define *-path w, *-measure f-L for each w, and U(t, by

x)

Definition 1 (1)

(2)

v be [t/c] with Gauss ' parenthesis. For each internal function {0, 1 , · , v - 1} { - 1 , 1} and y E IR, define Xk by Xk y + ��:� o: ( i )6, and w by the *-polygonal path with vertices (0, y), ( c , x l ), (vc , xv)· Let a

:

·

·

=

---+

·

·

·

,

Define * -measure f-L by f-L (w)

=

1v exp [ rt 1 .f (w ( s) ) db (s ) - rt 1 .f (w ( s) ) 2 ds ( 21.1. 4) Jo 2D Jo 4D 2 ]

with the first integral in the exponent being the Ito integral, and U (t. x) L U(O, w(O) ) tL(w) ( 21.1.5 ) =

w

where the sum is taken over all w satisfying w( vc )

=

x.

21.2.

309

Bounds for the *-measure and the *-Green function

We note that the *-measure mula

[4].

21.2

( 21.1. 4)

corresponds to the Girsanov for­

Bounds for the *-measure and the *-Green function

l\Iaking use of the I to formula t

1 f(w( s) ) db( s) -1 { F(w(t) ) - F(w(O)) } - � {t f' (w(s) ) ds 2D 2 }0 }0 2D where F'(x) f(x), we obtain a bound for f-L as f-L(w) 21v exp [ 2� {F(w(t) ) - F(w(O)) } - lot{ 4� J(w ( s) ) 2 + �j' (w(s)) }ds] (21.2.1) :::; 21v exp [ 2� {F(w (t)) - F(w(O) )} - ct] . The constant c in the last line was given in the assumption ( A3 ) . Let us fix the end points of w at finite numbers y and x, and consider a {

=

=

=

space of *-paths,

w(c[t/ cl) x } , and define the *-Green function for the interval [0, t ] by (21.2.2 ) Q(t, x : 0, y ) 216 L f-L (w) . wEP(t,x:O,y) Then, U(t, x) defined in ( 21. 1.5) is written as U (t, x) L U(O, y)Q(t, x : 0, y) 26. ( 21.2.3) y The infinitesimal spacing corresponding to dy is not 6 but 2 6 in ( 21.2.3 ) because only the paths that start every other point y could reach the end point x . Since 26 exp [- (x - y) 2 ] ( 1 + O (c l/2 ) ) "" 2_ _ (21.2. 4) v 4 l/ 4 D nD 2 2 t ( � t) wEP(t,x:O,y) P(t, x : 0, y )

=

{w

I

w(O) y =

and

=

=

=

'

the *-Green function is bounded as

F(y) Q(t, x : O, y) :::; exp [ F(x) 2D - 2D - ct ] 1 exp [ (x4-;�) 2 ] ( 1 + 0 (c l/2 )) , (21.2.5) (4nD t) 1 /2 x

21.

310

Path-space measure for stochastic differential equation

and hence

(21.2.6)

U(O , y )

Since the support of is assumed in (A4) to be a bounded set, the right-hand side of (21.2.6) is near-standard. To define a standard function as the standard part of we introduce two time scales of different orders, one for the Brownian motion, r:: , and the other, T, for the changes in the drift term in (21 . 1 . 1) ; we choose c = 0(T3), for example. The spacing c is finer and stands for the time-spacing of *-random walks, and T is long enough to cover many steps of the *-random walks, yet short enough for the change in the drift term to be small. Define a standard function t, as the standard part of the value of at the coarse-grained lattice point of time !;_:

U,

U

U

U ( x)

U(t,

x) = st U(i, x)

where

i = T [t/T] ,

(21.2.7)

which we expect to be the solution to the Fokker-Planck equation

21.3

(2 1 . 1 .2) .

Solution t o the Fokker-Planck equation

x)

In order to prove that U(t, in (21.2.7 ) is the solution to (21 . 1 .2 ) , or more concretely to estimate h in (2 1.3.7 ) below, we should truncate the *-paths at an infinite number as

A PA(i, x : O, y) = { w E P(t, x : O, y) l \is E [O, i] l w ( s)I < A } . The magnitude of A will be determined later in (21 .3.5) . Then the correspond­ ing *-Green function YA and UA arc defined by ( 21.3.1 ) YA(t, x : 0, y) = 261 L fl (w) wEPA (t,x: O , y)

and

UA(L x) = L U(O, y)QA(i, x : 0, y) 26. y

( 21.3.2 )

Let us first calculate the difference between ( 21 .2.2) and ( 21.3.1 ) . Consider a *-path 0, y) and put min { s 0, \ Define a *-path by turning upside down the section of the path for

w E P(t, x : y) PA(t, x : w'

t(w) =

I l w ( s) l = A}. w( s)

311

2 1 .3 . Solution t o the Fokker-Planck equation

0 0. P(t, x : O, y) \PA(t, x : O, y)

2A - y -2A - y

the interval :<.:: s :<.:: t (w ) so that w' should start or at time s = The section of w'(s) for t(w ) :<.:: s :<.:: t is that of w (s) for the same interval. In this way, we can define a one-to-one correspondence from wE to w' E Then by (2 1 .2.4) ,

P(t, x : 0, 2A - y) UP(t, x : 0. - 2A - y).

and hence

F(y) ] 1 l9(t, x : O, y ) - 9A(t, x : O, y) l :<; exp [ F(x) 2D - 2D - ct (4nDt) l/2 x (2 1 .3 . 3) { exp [ (2A - y - x) 2 ] + exp [ (- 2A - y - x) 2 J } x ( 1 + ( 1 ;2 ) ) 4D t

c

4D t

,

which implies

(21 .3.4)

x.

for any finite Now, we are ready to prove that wish to evaluate the standard part of

for an infinitesimal parameter as

A

a

=

� {U( t +

U(t, x)

a, X )

is the solution to (21 . 1 .2) . We

- U ( t, ) } X

kT (k E *N) given arbitrarily. Choose the truncation

(21.3.5 )

0,

where (3 > a standard constant, is just introduced to make the argument of logarithm dimensionless . Then by (21 .3.4) ,

UA (t, x)

for any standard natural number n, meaning that can be identified with up to negligible error. Therefore we shall hereafter deal with the truncated instead of Then the difference quotient we should calculate is

U(t, x) UA

U.

21.

312

Path-space measure for stochastic differential equation

where

In order to calculate the sum in h , we wish to expand the summand as power series in � except for the exponential function, which is possible if � is sufficiently small. Since

(21.3.9) satisfies

YA (f

+ a, x : f, x + �) :S exp [ 2� { F(x) - F(x + �) } - ca] �2 1 exp [ 4D a ] ' ( 4n D a ) 1 / 2 x

(21.3.10)

-

the summand in (21.3. 7) contains exp [- 4b ] as a factor which enables us to u restrict � to be sufficiently small, 1 � 1 in rough estimation. I n reality, it is restricted to =

for some small

a

such as

O (a 112 ),

1/10 as shown in the following.

Note that

(21.3.11) for some m E N by the truncation at A (A1) . Then by (21.3. 10) ,

=

O ( l log

,Ba l) and the assumption

21.3.

313

Solution t o the Fokker-Planck equation

(D /{3) ! ({Ja) !-a ,

e llog /3ul m ) (4Du . ] o (e- l log/Jul m ) .

in the last line the infinitely large number If I� I :2: In fact, can be controlled by the half of the factor e - e I

1 [ 8Da ] - [ 8({3a)2 a

exp - £ < exp Hence, the sum 2.:::�; in canst · 1�;"12(D I

(21 .3.7)

=

for such � is estimated as

�2 4( nDa) 112-;-c- [ 8Da ] d� o (aae - 8(P") 2a ) o(an ) /3)1/2 (/3u ) 1/2-a

J

1

-

exp

- --

I

=

=

for any n E N, so that it can be neglected. Now, we have only to consider � of infinitesimal order equal or less than so that we can expand

F(x) - F(x + �)

=

1 , a2-a -U(x) - I�2 f'(x) + O (a2-3a ). 3

2.

Next, we wish t o replace the integral in

(21 .3.9)

(21 .3.12)

by

l t+u f(w( s ) ) 2 ds + 1 lt+u f' (w(s) ) ds '"-' a J (x) 2 + a J' (x) . (21 .3.13) 4D ? 4D 1

2

t

t

It is justifiable if paths going very far at some time neglected. Let us take an infinitesimal

where a' is small but a ' > a , for example, Then, the sum over which satisfies bounded by

{ [ ( 4nDa)l l 2 exp

a

=

a

(2A' - �) 2 ] + exp [ ( 2A' + �) 2 ] } 4D a 4D a O (a- l l 2 e -(t3:)2a ) o (an ) =

_

can be

=

(21 .3.14)

=

for any n E N in the same way as used the fact � = = consider satisfying

w

[t , t + a]

2/1 0 if we take 1/1 0 . l w ( s) - x l :2: A' for some s E [t, t + a] is

w

canst

'

sE

(21.3.3), and hence negligible.

O (a! -a ) o (A') because

a

'

>

a.

l w ( s ) - xl :::; (D / f3) ! (f3a) !-a'

Here, we have Thus, we have only to

21.

314

for all

s

E

Path-space measure for stochastic differential equation

[t , t + a], and hence

[ 4D f(x) 2 - �2 f'(x) J 1 exp [- - 1 {f ( w ( s)) 2 - f ( x) 2 } ds 4D Ho- � 1 { f' (w(s)) ds - f'(x) } ds ] exp [- � f ( x) 2 - � j' ( x) J ( 1 + (a�- a' ) ) . 4

= exp - _5!___ x

=

Putting

:t

o-

(21.3. 15) =

0

(21 .3. 12), (21.3. 15) "'

� wEPA (t+ ,x:t,x+ l; ) o-

into

H

x

and

_1:_ 2 v' -

[-__r__ ] ( 1 + O (a3/ 2 ) ) exp / 4Da ( 4nD a) l 2 26

(2 1.3.9), we obtain e + a, x : f, x + �) - ( 4nDa1 ) l/2 exp [- 4Da J= { 2 2 Da � 2 - 2Da f(x) 2 o ( a ) = - � f(x) - � + f (x) + + } 4 D 8D2 2D 2 1 4�Da J · 4( nDa) 1 /2 exp [- -

YA (f

1

x

(21.3.16)

x

Lastly, we usc the expansion

(21.3.17) with for h . For h , we usc the expansion taken one step further,

with

21.3.

315

Solution to the Fokker-Planck equation

because (21.3.16) contains a factor of order O(a � -a ) , whereas there is no such factor in h . We shall not give the proof of these expansions, for it requires elementary but a little long estimations. Estimations similar to this case can be found in [5, 6, 7] . Putting (21.3. 16) and (21.3. 17) into (21 .3.7) and replacing the sum by integral which is permitted because 6 = o(T 31 2 ), we obtain h

J

=

{ - � j(x) - �2 + 2Da f (x) + �2 - 2Da f(x) 2 + o(a) }

� � � � ( D 1 ;9) 112 (iJu) 1/2-a x

4D

2D

1

8D 2

� 2 d� {uA (t, x) + �Do UA (t, x) } (4nD1a) 112 exp [ - 4Da J

{ ( - 2D� f(x) - �2 +4D2Da f (x) + �2 8D- 2Da 2) 2 j(x) UA (t . x) 1 2 1 � f(x)D ( x) o(a) } �2 d�. . -UA t + o [ 1 2D 4Da J (4nDa) 1 2 J

1

�� � � ( D 6) 1/2 (;Ju ) 1/2-a

exp

-

Since the domain of integration can be extended to *JR by the factor �2 1 (47rDu ) I /2 exp [ 4Du l '

-

h

=

- 2Da f(x) 2 ) UA ( x) t J { ( - 2D� J(x) - �2 +4D2Da f (x) + f,2 8D 2 1 e f(x)Do UA (t , x) + o(a) } exp [ - e J d� - 2D 4Da (4n Da) 1 1 2 a { f'(x)UA (t, x) + f(x)Do UA (t , x) } + o(a) . 1

* !R

(21 .3.19)

= -

Similarly, putting

(21 .3. 18)

into

(21 .3.8),

we obtain

(21 .3.20) Therefore,

� {UA (t + a, x) - UA(t x) } DDJ UA (t, x) - { f' (x)UA (t, x) + j(x)Do UA (t, x) } + o(1 ) =

(21 .3.21)

holds for any infinitesimal a kT, which means that U ( t, x) st UA (t , .1: ) is differentiable with respect to t and its t-derivative is the standard part of =

=

316

2 1 . Path-space measure for stochastic differential equation

the right-hand side of (2 1 .3.21 ) . Though we omit the proof, it is proved that U(t, x ) is twice differentiable with respect to x and their values are and Taking the standard part of the both-hand sides of (2 1 .3 . 2 1) , we obtain

(21.3.22) namely U (t, x ) is the solution to the Fokker-Planck equation (2 1 . 1 .2) . Let us finally note that, by Loeb measure theory, a standard a-additive measure over a space of paths can be derived from the *-measure f-L we have constructed in this paper so far.

References

[1]

H . P . M cK E AN , JR., London. 1969, (§ 3 . 6) .

Stochastic Integrals, Academic Press,

[2]

P . A . LOEB, "Conversion from nonstandard to standard measure spaces and applications in probability theory", Trans. Amer. lVIath. Soc . , 2 1 1 ( 1975) 1 13-1 22.

[3]

K. D. STROYAN and .J . l\1 . BAYOD , Foundations of infinitesimal stochastic Studies in Logic and the Foundations of Mathematics, North­ Holland. Amsterdam, 1986.

[4]

I. V. GrGRSANOV, " On transforming a certain class of stochastic processes by absolutely continuous substitution of measures", Theor. Prob. Appl . , 5

New York and

analysis,

( 1960) 285-301 .

[5]

T . NAKAMURA, " A nonstandard representation of Feynman's path inte­ grals", J. Math. Phys. , 32 (1991) 457-463.

[6]

T . NAKAMURA, "Path space measures for Dirac and Schri:idinger equations: Nonstandard analytical approach", J. Math. Phys., 38 (1997) 4052-4072.

[7]

T. NAKAMURA, "Path space measure for the 3+ 1-dimensional Dirac equa­ tion in momentum space", J. Math. Phys . , 41 (2000) 5209-5222.

Optimal control for N avier-Stokes equations Nigel J. Cutland

*

and Katarzyna Grzesiak

**

Abstract We survey receut results on existence of optimal controls for stochastic

Navier-Stokes equations in 2 and 3 dimensions using Loeb space methods.

22.1

Introduction

this paper we give a brief survey of recent results 1 concerning the exis­ tence of optimal controls for the stochastic Navier-Stokes equations (NSE) in a bounded domain D in 2 and 3 space dimensions; that is, D C JRd with d 2 or 3. The controlled equations in th eir most general form are as follows ( sec the uext section for details) : In

=

·n (t) = no +

{t { .fo

+ .lot

g

-

IJ

A v, ( ., )

-

B ( 'U ( s)) + f ( s, n( s) , O (s , v.) ) } d.s

(22.1. 1)

(s, u(s)) dw(s)

Here the evolving velocity field ·u = u(L, w) is a stochastic process with values in the Hilbert space H � V1(D) of divergence free functions with domain D; t his gives the (random) velocity 'lt(l, x, w) E JR.d o f the fluid a t any time t and point. x E D. The most general kind of control 0 that we consider act.s through the external forcing term f, and takes the form 0 : [0, T] x 1-l ----7 M wh ere 1-l is the space of paths in H and the control space Jv[ is a compact metric space. · Mathematics Department, University of York UK. ,

nc507@york . ac . uk

**Department of Finance, \;\/yzsza Szkola Biznesu, National-Louis University, Nowy Sacz. Poland. j ermakowicz@wsb-nlu . edu . pl

1The results reported here are developed f1·om the second author's PhD thesis [16] writ.t.en under the supervision of the first. author. Full details may be found in the papers 191 and 11 Oj.

318

22. Optimal control for Navier-Stokes equations

In certain settings however it is necessary to restrict to controls of the form : [0, T ] ----+ lvl that involve no feedback, or those where the feedback only takes account of the instantaneous velocity u( t). The terms vA, B i n the equations are the classical terms representing the effect of viscosity and the interaction of the particles of fluid respectively; the term B is quadratic in u and is the cause of the difficulties associated with solving the Navier-Stokes equations (even in the deterministic case g = 0) . The final term in the equation represents noisy external forces, with w denoting an infinite dimensional Wiener process . The methods involve the Loeb space techniques that were employed in [4, 6] to solve the stochastic Navier-Stokes equations with general force and multi­ plicative noise (that is, with noise g (s, u(s)) involving feedback of the solu­ tion u) , combined with the nonstandard ideas used earlier in the study of optimal control of finite dimensional equations (see [7] for example) . For the 3-d case it is necessary to utilize the idea of approximate solutions developed in [ 1 1 ] for the study of attractors. We assume a fixed time horizon for the problem, and then the aim is to establish the existence of an optimal control that minimizes a general cost J that has a running component and a terminal cost, modelled by

e

J(B , u) = JE

(lT h (t, u(t) , B(t, u)) dt + h (u(T) ) ) .

(22 . 1 .2)

The existence of an optimal control (in some cases, a generalized or relaxed control) can be established in a number of settings. The results for d = 2 are somewhat stronger than for d = 3, which is a reflection of the well-known distinction between these two cases even for the uncontrolled equations: in di­ mension d = 2 there is uniqueness of solutions and the solutions are strong ( es­ sentially this means that the field u(t, x ) is differentiable in x for each t) whereas for d = 3 the solutions that exist for all time are weak and uniqueness is a ma­ jor open problem2 . Consequently even the formulation of the optimal control problem is more difficult, and optimal controls are obtained using our methods for more restricted classes of controls compared to the results for d = 2. The plan of the paper is as follows. First (Section 22.2) we provide some details of the Hilbert space setup for the equations and their solution, and then recall the basic nonstandard ideas concerning controls. The results for dimen­ sion d = 2 are then outlined in Section 22.3, and in the final section we do the same for d = 3. We omit proofs, referring the interested reader to [9] and [10] , although in some cases where new ideas are introduced we provided sketches. Alternative approaches to control theory for stochastic Navier-Stokes equa­ tions have been studied in [3, 13, 18] , where the controls are assumed to be 2

In fact this is o n e o f the Millennium problems. Uniqueness

but for d

=

�s

known for strong solutions

3 �uch �olution� can only be obtained for �hart timco.

319

22.2. Preliminaries stochastic processes

e

:

[o, T]

x

n

____, H

adapted to the information filter given by complete observations of the fluid velocity. In [3] for example, the author assumes that there is a constant p > 0 such that l e(t, w) l < p for all t E [O, T] and w E D, while in [13, 18] it is assumed that the controls satisfy the condition IE

(loT l e(t) l 2 dt)

< 00 .

In these papers the control is subject to the action of a linear operator. The stochastic minimum principle is derived providing a necessary condition for an optimal control, and a dynamic programming approach is presented to give a sufficient condition. It is shown that the minimum value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation associated with the problem. In [13] the authors obtain smooth solutions to the H.TB equation which justifies the dynamic programming approach.

22.2 22.2.1

Preliminaries Nonstandard analysis

For optimal control theory, nonstandard analysis is a natural tool to use because one can always find a "nonstandard" optimal control - this is simply eN for any infinite N, where (en)nEN is a minimizing sequence of controls. The task then is to see whether e N can be transformed into a standard optimal control of some kind. For finite dimensional DEs and SDEs this idea was developed in [7] and related papers. In the study of the Navier-Stokes equations (particularly the stochastic ver­ sion) Loeb space techniques have proved very powerful, providing for example the first general existence proof for the stochastic Navier-Stokes equations in dimensions up to 4 (see [6] ) and more recently new results concerning the existence of attractors (see [5] , [ 1 1] and [ 1 2] ) . I n the work reported here the nonstandard techniques used in these two areas are combined. We work in a standard universe V V(S ) where S is a base set that contains all the objects of interest, and take an N 1 -saturated extension *V(S ) C V(*S) . For ease of reference we gather together in an Appendix the most important facts about the nonstandard representation of the spaces used in the study of the NSE equations.

=

22.2.2

The stochastic Navier- Stokes equations

The classical form of the uncontrolled stochastic Navier-Stokes equations with zero boundary condition on a time interval 0 :<:: t :<:: T is as follows:

22 . Optimal control for Navier-Stokes equations

320

{

du = { v �u - < div u = 0 u lan = 0 u(O, x)

=

u. \7 > u - \7p + j(t, u) } dt + g ( t, u)dw(t)

uo (x)

These equations are considered in a bounded domain D C IRd ( d = 2 , 3) which is fixed throughout the paper, with the boundary oD of class C 2 . Here u : [0, T] X D X n ----+ JR3 is the random velocity field, v is the viscosity, p is the pressure, and f represents external forces; u0 is the initial condition. The diffusion term g together with the driving Wiener process w represents additional random external forces, or noise. Underlying this model is a filtered probability space 0 = (D, :F, (:Ft ) t >o , P) . For the conventional Hilbert space formulation, write L 2 ( D ) = ( L 2 ( D )) d and let V = { u E Crf' (D, IRd ) : div u = 0 } . Then H is the closure o f V i n L 2 ( D ) with the norm given by

(u, v)

lul2

=

(u, u) , where

J ui ( x )v i ( x)dx. L 2= 1 d

=

D

and V is the closure of V in the norm

l ui + ll u ll where ll u ll 2

(

)

=

((u, u)) and

ou o ((u, v)) = L ox ' oxv · J J ]=1 d

H and V are real Hilbert spaces, V dense in H. The dual space to V 1s denoted by V' with the duality extending the scalar product in H and V c H = H' c V ' . Write A for the Stokes operator on H (the self-adj oint extension of the pro­ jection of -�) which is densely defined in H; it can be extended to A : V ----+ V' by Au [v ] = (( u, v)) for u, v E V. The operator A has an orthonormal ba­ sis of eigenfunctions { ek h E N = [ C H with eigenvalues 0 < A k / oo . For u E H write u = 2...::: ukek. Write Hn for the finite dimensional subspace Hn = span{ e 1 , e 2 , . . . , en } and Prn for the projection onto Hn . A family of spaces Hr for r E lR is defined as follows: for r 2 0 Hr

=

=

{ U E H : kL Ak Uk < 00} =1

321

22.2. Preliminaries with the norm given by

=

l ui ; = L >-.'k uk . k= l and H- r is the dual of Hr . We may represent H - r by H- r

=

{ (uk)kEN :

=

L x ;;r u k < 00} k=l

In terms of this family we have H0 = H, H 1 = V, and H - 1 = V' with the norms l u i = l u l o and ll u ll = l u l l · The quadratic function B is given by (B(u), y) = b(u, u, y) , where

b(u, v, y)

=

d J u� (x) 80�j (x)y1 (x) dx i

L

2,J = l D

whenever the integral is defined. This describes the nonlinear inertia term in the equation. The trilinear form b has many properties [19] including the following that we need here.

b(u, v, v) 0, l b(u, v, y) l :::; c l u l i ll u ll � lvl i llvll � II Y II =

(22.2. 1) (22.2.2)

There are additional properties that hold only i n dimension d = 2. The inequality (22.2.2) gives the second of the following properties. Proposition 22. 2 . 1

For u E V

I Au lv' = ll u ll 1 3 I B(u) lv' :::; cl u l2 ll u P In the above setting the evolution form of the Navier-Stokes equations (without explicit control) in the space V' (the vector \lp 0 in this space) is given by =

lot -vAu(s) - B (u(s)) + f (s, u(s))} ds t + lo g (s , u (s)) dw(s)

u(t) = uo +

{

(22.2.3)

with the initial condition u0 E H. The first integrals are Bochner integrals in V'. The driving noise process w is an H-valued Wiener process with covari­ ance Q , a fixed non-negative trace class operator (see [6, 14] for details) , and

322

22. Optimal control for Navier-Stokes equations

the stochastic integral is the extension of the Ito integral to Hilbert spaces due to Ichikawa [17] . I n the study of the Navier-Stokes equation there are several types of solu­ tion. The most general is a weak solution, but in the case d = 2 we also have strong solutions. The definitions are as follows (see [6] ) : Definition 22.2.2 ( a ) An adapted process u : [0. T] X n ----+ H is a weak solution to the stochas­ tic N avier-Stokes equations (22.2.3) if (i) for P-a.a.

w the path u( -, w)

has

u(-, w) E L = ( [ O , T] ; H) n L2 ( [0, T] ; V ) n C( [O , T] ; Hweak ) , (ii) for all (iii)

tE

[0, T] the equation (22.2.3) holds as an identity in V',

u satisfies the energy inequality IE

By

(

T

)

l u(t) l 2 + l{o ll u (t) ll 2 dt < oo. t E[O, T] sup

solution henceforth in this

paper we mean

(22 . 2 . 4)

weak solution.

(b) A weak solution is strong if (i) for P-a.a.

w the path u(-, w)

has

u( · , w) E L = ([ o , T] ; V ) n L 2 ( [0, T] ; H2 ) n C([ O . T] ; Vweak ) , u(- , w ) E C( [O , T] : H)) ; w the path u ( , w ) has

(which implies that (ii) for P-a.a.

(22.2.5)

sup

tE[O .T]

-

T

ll u (t) ll 2 + l{o Au(t) 2 dt < oc .

(22 . 2 . 6)

When controls are introduced into the forcing term f the equation (22.2.3) takes the form

u ( t)

=

lot -vAu (s) - B (u( s)) + t + lo g ( s , u(s)) dw(s) .

uo +

{

f ( s , u( s ), B(s, u))} ds

In the next section we discuss the types of control that we consider.

323

22.2. Preliminaries

The following general existence result for solutions to the sNSE was first proved in [4] ( see also [6] for an exposition ) , using Loeb space methods. First define Km = { v : I v i i <:.:: m } C::: V, with the strong topology of H . I n the theorem below, continuity o n each Km turns out t o b e the appropriate condition for the coefficients f, g; this is weaker than continuity on V in either the H-norm or the weak topology of V.

Suppose that uo E H and

Theorem 22.2.3

f:

[0,

oc ) x V ----+ V' ,

g:

[0, 00) X v

----+ L ( H, H )

are jointly measurable functions with the following properties (i) f ( t, · ) E C(Km, V�cak) for all m, (ii) g(t, ) E C(Km, L ( H, H ) weak ) for all m, (iii) l f (t, u) l v' + l g(t, u) IH ,H <:.:: a(t) (l + l ui) where a E L 2 ( 0, T ) for all T. Then equation (22.2. 3) has a solution u on a filtered Loeb space. ·

In dimension d = 2 a stronger existence theorem ( and uniqueness, given Lipschitz coefficients ) has been established - this will be noted when required in Section 22.3.

22.2.3

Controls

The simplest controls considered in this paper are those with no feedback, as follows. A:o is customary in optimal control theory, it is often necessary to extend this class to its natural completion, which is the class of generalized or relaxed controls. These, and the topology on them are as defined thus ( as in [7] ) . The compact metric space lvl is fixed for the whole paper. Definition 22.2.4

(i) The class C of ordinary controls is the set of measurable functions e : [0, T] ----+ AI, where M is a fixed compact metric space. (ii)

The class D of relaxed controls is the set of measurable functions cp [O, T] ----+ M1 (M), where M1 (M) is the set of probability measures on M. (We regard C C::: D by identifying a E M with the Dirac measure 6a ) .

:

The weak or narrow topology on It is given by means of the set JC of bounded measurable functions z : [0, T] x M ----+ lR with z(t, ) continuous for all t E [0 , T] . The action of B E It on z E }( is defined by ·

324

22. Optimal control for Navier-Stokes equations

B(z)

T

=

j z(t, B(t)) dt. 0

Then the topology on hoods the sets

11: is defined by specifying as subbase of open neighbour­

{B : I B(z ) l < c} c>O , zE IC· The effect of a relaxed control o n a function z E JC is obtain by first extending the domain of each z E JC to [0, T] x M1 (M) as follows: for a probability measure f-L E M 1 ( M) define z(t , f-L )

=

j z(t, m) df-L (m) .

M

The weak (or narrow) topology is then extended to ::D by defining the action of cp E ::D on z E JC to be

where for brevity we use C{Jt := cp (t) . The fundamental results about controls that we shall usc arc summarized below; for details consult for example [7] , [15] , [2] or [20] (noting that a relaxed control is a particular case of a Young measure). Theorem 22.2.5

The set of relaxed controls ::D is compact.

This means that for an internal ("nonstandard") control (*ordinary or *relaxed) there is a well defined standard part 0 E ::D. (In [7] it is shown how this can be defined explicitly using Loeb measures.) Note that for a control cp E ::D we have 0 ( * cp) = cp. For the next result we define a uniform step control to be an ordinary control e such that there is a partition of [0, T] into intervals of constant length with e constant on each partition interval. Theorem 22.2.6

The set of uniform step controls 11:5 is dense in ::D .

To handle controls in a nonstandard setting the next definition and re­ sult is basic. Definition 22.2. 7 Z : * [0, T] x *M ---+ *R is a bounded uniform lifting of z E JC if Z is an internal *measurable function such that (i) Z is finitely bounded

22.3. Optimal control for

(ii)

for a.a.

T E * [0, T] ,

d

=

325

2

for all

a

E *M : Z(T,

o:

) :::::J

z(0T, 00:).

Let