An Introduction
to the
Geometry of Stochastic Flows
An Introduction
to the
Geometry of Stochastic Flows
Fabrice Baudoin Université Paul Sabatier, France
Imperial College Press
Published by
Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by
World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
AN INTRODUCTION TO THE GEOMETRY OF STOCHASTIC FLOWS Copyright © 2004 by Imperial College Press All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-481-7
Printed in Singapore by World Scientific Printers (S) Pte Ltd
Au vent d'autan...
Preface
The aim of the present text is to provide a self-contained introduction to the local geometry of the stochastic flows associated with stochastic differential equations. The point of view we want to develop is that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry whose main tools are introduced throughout the text. By using the connection between stochastic flows and partial differential equations, we apply this point of view to the study of hypoelliptic operators written in 116rmander's form. Many results contained in this text stem from my stay at the Technical University of Vienna where I had the great pleasure to discuss passionately with Josef Teichmann. I learnt a lot from him and I thank him very warmly. I also would like to take this opportunity to thank Nicolas Victoir for reading early drafts of various parts of the text and for his valuable suggestions. F. Baudoin, Toulouse, June 2004
VII
Contents
Preface 1.
Formal Stochastic Differential Equations 1.1 1.2 1.3 1.4 1.5
2.
vii
Motivation The signature of a Brownian motion The Chen-Strichartz development formula Expectation of the signature of a Brownian motion Expectation of the signature of other processes
Stochastic Differential Equations and Carnot Groups 2.1 The commutative case 2.2 Two-step nilpotent SDE's 2.3 N-step nilpotent SDE's 2.4 Pathwise approximation of solutions of SDEs 2.5 An introduction to rough paths theory
3.
Hypoelliptic Flows
1 1 3 8 14 17 21 22 24 33 45 46 49
3.1 Hypoelliptic operators and H6rmander's theorem 3.2 Sub-Riemannian geometry 3.3 The tangent space to a hypoelliptic diffusion 3.4 Horizontal diffusions 3.5 Regular sublaplacians on compact manifolds 3.6 Stochastic differential equations driven by loops Appendix A Basic Stochastic Calculus A.1 Stochastic processes and Brownian motion ix
50 57 60 70 79 85 97 97
Book Title
A.2 Markov processes A.3 Martingales A.4 Stochastic integration A.5 Itô's formula AM Girsanov's theorem A.7 Stochastic differential equations A.8 Diffusions and partial differential equations A.9 Stochastic flows A.10 Malliavin calculus A.11 Stochastic calculus on manifolds Appendix B
B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8
Vector Fields, Lie Groups and Lie Algebras
Vector fields and exponential mapping Lie derivative of tensor fields along vector fields Exterior forms and exterior derivative Lie groups and Lie algebras
The Baker-Campbell-Hausdorff formula Nilpotent Lie groups Free Lie algebras and Hall basis Basic Riemannian geometry
99 100 103 106 106 108 109 110 111 113 117 117 119 120 121 123 126 126 127
Bibliography
133
Index
139
Chapter 1
Formal Stochastic Differential Equations
The goal of this first chapter is to establish the Chen-Strichartz formula which, in a way, is a cornerstone of this book. This formula is universal and determines very precisely and explicitly the local structure of any stochastic flow. To derive this formula, it is quite convenient to work in an abstract and formal setting, in which we do not have to care about convergence questions. The reader which is not so familiar with the theory of stochastic differential equations and vector fields is invited read the Appendices A and B which are included at the end of the book.
1.1
Motivation
Let us consider a stochastic differential equation on Rn of the type
Xr ° = x0
d
E
t
vi ( X:o) o c113„ t > 0,
(1.1)
where:
(1) xo
E
Rn ;
are C' bounded vector fields on Rn; (3) o denotes Stratonovitch integration; (4) (Bt ) t >0 = Bi, Bcti) t > 0 is a d-dimensional standard Brownian motion. (2)
V1,...,Vd
(
1
2
An Introduction to the Geometry of Stochastic Flows
Let f : Rn ----* JR be a smooth function and denote by (X° )>o the solution of (2.1) with initial condition xo E Rn . First, by Itô's formula, we have d
t
f (vif)(x;°) 0 dB si , t
f (XT°) = f (x0) +
O.
Now, a new application of Itô's formula to Vi f(X;) leads to d
d
E
f (Xr) = f (X0) +E(Vif)(SO)B
i=1
t
ff
0 0
s
(vi vi.f)(x:.0) 0 dB4
dB si .
We can continue this procedure to get after N steps f (Xr) =
ASO
(vi i ...vik f)(xo)
E E k=1
/=(Zl 9- 2.k )
f
odB/ +
Ro),
A k [04]
for some remainder term RN, where we used the notations:
(1) Ak[0,t] =
e [0, t] k , ti <
< tk};
(2) If I = (i1,...ik) E {1,...,d} k is a word with length k,
f
odB I =
Ak [03]
f
odB" t i o •••o dBikf-k
If we dangerously do not care about convergence questions (these questions are widely discussed in [Ben Arous (1989b)1), it is tempting to let N +oo and to assume that RN O. We are thus led to the nice (but formal!) formula
+00
f
(xr) = f (x0) + E k=zl
I=(ij ,...ik )
(Vl ...Vikf)(xo)fkadBI.. [0,t]
(1.2)
We can rewrite this formula in a more convenient way. Let (Dt be the stochastic flow associated with the stochastic differential equation (2.1). There is a natural action of (I)t on smooth functions: The pull-back action given by
(vif ) (xo ) = fa (p t)( x0 ) = f(xr). (
Formal Stochastic Differential Equations
3
The formula (1.2) shows then that we have the following formal development for this action +00
= Id +
E E vi,...vik fAk At] odBi.
(1.3)
k=1
Though this formula does not make sense from an analytical point of view, at least, it shows that the probabilistic information contained in the stochastic flow associated with the stochastic differential equation (1.1) is given by the set of Stratonovitch chaos 0.a f k A]t odBi. What is a priori less clear is that the algebraic information which is relevant for the study of stD;` is given by the structure of the Lie algebra generated by the Vi's, and this is precisely this aspect we want to stress in this chapter which is devoted to the study of formal objects like +00
Id +
E E vi,...vik fAk [0,t] odBi.
k=1 /=(ii,...ik)
Such objects and their relations with flows seem to appear the first time in the works of K.T. Chen [Chen (1957)], [Chen (1961)].
1.2
The signature of a Brownian motion
Let us denote by R[[X I , Xd]] the non-commutative algebra of formal series with d indeterminates.
Definition 1.1 The signature of a d-dimensional standard Brownian motion (Bt ) t >0 is the element of R[[X I , ..., X]] defined by +00
S(B) t = 1 +
E E xi,...x ik fAk [0,t] odBi, t > 0.
k=1 /=(ii,...ik)
Remark 1.1
We define the signature by using Stratonovitch's integrals because we keep in mind the connection with stochastic flows which appeared with formula (1.3). Nevertheless, it is possible to define a signature by using ltd 's integrals. The link between these two signatures is given in Proposition 1.2 below.
Remark 1.2
In the same way, it is of course also possible to define the signature of a general semimartin gale.
4
An Introduction to the Geometry of Stochastic Flows
Observe that the signature hence defined is the solution of the formal stochastic differential equation d S(13)t =1-1-
t
E f S(B),X i o dB, t > 0.
(1.4)
i=1 Such linear equations appear in the study of Brownian motions on Lie groups. Indeed, let G be a Lie group with Lie algebra g.
Definition 1.2 A process (Xt ) t >0 with values in G is called a (left) Brownian motion on G if: (1) (X)>o is continuous; (2) for each s? 0, the process (X;'Xt+3) t>0 is independent of the process (X) o<< 5 ; (3) for each s > 0, the processes (X»Xt+5 ) t>0 and (Xt)t>0 are identical in law. In a general way, one can construct Brownian motions on Lie groups by solving differential equations. Let us consider V1, ...,Vd G g. As explained in Appendix B, E g can be seen as left invariant vector fields on G, so that we can consider the following stochastic differential equation
xt=k+f E i=1 0
vi (xs )odBis ,
t>0,
(1.5)
where (Bt ) t> 0 is a standard Brownian motion on Rd. For instance, if G is a linear group of matrices, equation (1.5) can be rewritten
xt
+ Ef xs vi 0 dBis . 0
It is easily seen that there is a unique solution (Xt ) t>0 to the stochastic differential equation (1.5), and this solution is a (left) Brownian motion on G. The process (X t ) t >0 is called a lift of (B)>o in G. It is interesting to note that, conversely, each Brownian motion on G is solution of a stochastic differential equation
xt
xo + f vo(x s )ds+Ef vi(xs)odBis , t>0, 0 i=1 0
Vd are left-invariant vector fields on G; for further details where 170 , on this, we refer to [Hunt (1958)] and [Yosida (1952)].
5
Formal Stochastic Differential Equations
With this in mind, we interpret now Rgi , Xdil as the universal enveloping algebra of the free Lie algebra with d generators fd . So, with this interpretation, at the formal level the signature of (Bt ) t>0 can be interpreted as a lift of (Bt ) t>0 in the formal object exp(fd ). On the other hand, the first section of this chapter has shown that the pull-back action on functions of the stochastic flow (cl't)t>o associated with the stochastic differential equation
El vipuo) d
Xf° = xo +
t
dB si , t > 0,
0
solves formally the stochastic differential equation
(11 = Id +
f
0
0 dBis
,
so that ( 41)t>o can formally be seen as a lift of (B)> o in the formal object exp Va)) where Z(V1 , Vd ) is the Lie algebra generated by VI , Vd. Therefore, since fd is a universal object in the theory of Lie algebras, the signature appears as a universal object in the theory of stochastic flows. In particular, if we do not care about convergence questions, any algebraic formula concerning the signature of (Bt ) t>0 can be applied to study the stochastic flow associated with any stochastic differential equation driven by (Bt ) t>0 . As it will be seen in the next chapters, one of the most illuminating example in this direction is certainly the universality of the Chen-Strichartz formula; an other example is given by the expectation of the signature (see the end of the chapter), a purely algebraic object, which explains in a different way than the usual one, the Markov property shared by any process that solves a stochastic differential equation driven by Brownian motions. We have a fundamental flow property for the signature which stems directly from the following key but simple relations, known as the Chen's relations since the seminal work [Chen (1957)1.
Lemma 1.1
Lin[0,t]
For any word (i 1 ,., i) E ti,
E
k=O L [O, s]
where we used the following notations:
dr and any 0< s < t, fs,ti OdB (ik+1
" . " in)
An Introduction to the Geometry of Stochastic Flows
6
(I)
L
Odkil " . " ik) = k
[8, tl
OdBtil ... dBtikk .9
(2) if I is a word with length 0, then f6,90,ti odBI = 1. Proof.
It follows readily by induction
on n by noticing that
t
f
odB (i ''.. •'in - ' )
=
d.131:.
An [0
Proposition 1.1
For 0 < s < t,
+00
xi,
S(B) t = S(B) 5 (1+ E E
f
A k [34]
k=1 /=(ii ,•••ik)
Proof.
We
odB/) .
have, thanks to the previous lemma, +0.
S(B) 3 (1
+ E E x„
f
k=1 / +00
=1 + k,k'=1
odBI)
At k [s,]
xi, .••xik xe„.-xi/k, fAk,s, odB/ f Ak, [o,s] odBr E /,/'
+0.
=1
+ E E x„...xik f
odB i
Ak [0,1
k=1 /
=S(B) t .
0
Remark 1.3
Observe that if I E {1, for any 0 < s < t:
d} k is a word with length k then
(1) fAkfs,ti odB i is independent from (Bu)u<s; (2)
fAk [57 t]
OdB i = 1"
Lk
OdB I .
Therefore, we can roughly conclude that: (I) for each s > 0, the process (S(B); 1 8(B) t+5) t>0 is independent of the process (S(B)u)o
7
Formal Stochastic Differential Equations
(2) for each s > 0, the processes (S(B); 1 S(B) t+3 ) t>0 and (S(B) t)t>0 are identical in law. By using the relation between Stratonovitch's and Itô's integral (see Appendix A), it is possible to give a formula for the signature of a Brownian motion which only involves Itô's iterated integrals. Proposition 1.2
We have
+00
S(B) t = 1 +
EE
dB I , t > 0, Lom
k=1
where we used the following notations:
(1) 1 X0 = —
2
d
EX?
,
B=t;
i= 1
(2)
L k At]
d13 1 =
dB tili
0
Proof.
Let I = G Stratonovitch's integrals, we have
{1, ..., d } k .
From the definition of
t
odB I
OdB il " . " ik-1 f0 VA k
dB ti k
[0 ,tk]
ft
1 2
•
OdBil •• Jik -2
k-z, k
jAk -2 [0,tk _d
dtk_i,
where Tik-174
=
0 if
= 11f ik_ i =
ik ik•
Consider now the smallest set I of words which satisfies the following properties:
(1) / E i; (2) if J =-then
ji) E I and if j,, = jrn+i 0 0 for some 1 < m < I — 1, G
An Introduction to the Geometry of Stochastic Flows
8
By iterating the previous formula, we get 1
odBi
f
dB i
A 1.1 1[0 ,
J EX
where I J denotes the length of the word J. The expected result follows readily. Remark 1.4 get
Observe that if we write equation (1.4) in Itô's form, we d
S(B) t =1+ J0 tS(B) 3
(E d X?) ds
t
f S(B),X i d.B's , i=1 o
i.1 z
which explains intuitively formula of Proposition 1.2.
1.3
The Chen-Strichartz development formula
This section is devoted to the proof of the Chen-Strichartz development formula. The formula we give is actually a restatement of a result of [Chen (1957)1 and [Strichartz (1987)1, and can be seen as a deep generalization of the Baker-Campbell-Hausdorff formula (see Appendix B). The Chen-Strichartz formula is an explicit formula for in S(B) t . In particular, it appears that ln S(B) tis a Lie element, a result which is far from being obvious at a first look. As it is illustrated in this book, the geometric consequences of this development are rather deep. Before we go into the heart of this formula, let us first try to understand a simple case: the commutative case. We denote ek the group of the permutations of the index set {1, ..., k} and if a G ek, we denote for a word I = (i1,...,ik), a I the word ic (k )). Now, let us observe that if X1 , ..., X d were commuting, we would have d
S(B) t = exp (E X i.131)
Indeed in that case, by symmetrization, we get +00
S(B) t
+
E xi, k.i /.(ii,•.•,ik)
1 (i7 ,,e6k
f odB, A kfo,t,
I)
9
Formal Stochastic Differential Equations
Now observe that
Lk
creek
fo,ti odBcf‘ I 144 = —Btik
E
which implies, d
= exp (E Xi BI) . Of course, in the general case, this formula does not hold anymore. But, we still have a nice formula for in S(B) twhich involves iterated functionals of the commutators X i iX— Xi Xi . We define the bracket between two elements U and V of R[[X i , •••, Xdii by
[U,V] = UV VU, and it is easily checked that this bracket endows R[[X I , Xd]j with a Lie algebra structure. If I = (i1 ,...,ik ) e 1, d}k is a word, we denote by XI the commutator defined by {
= If a E ek, we denote e(a) the cardinality of the set { j E 11, Theorem 1.1
k 11, a(j)> a(j + 1)}.
We have t > 0,
S(B) t = exp (
where: (:
)
Ai(B)t = „E eek k2 ek (-; Proof.
Lk At]
odB' i .
We shall proceed in several steps.
Step 1. First, we write
+00 S(B) t =1+E
fAk [0,t ]
dW8k
10
An, Introduction to the Geometry of Stochastic Flows
where we used the notation d
dw =
E X i dB i .
Now we have,
(-1)k-1
S(B) t = exp (ln S(B) t ) = exp
(S(B) t — 1) k
)
.
k=1
Therefore we get,
exp Zt , with -1- 00
Zt
=
( _ pk-1
E"
+00 n=i
k=1
LT. At]
odw s ,„.. cicosn
.
(1.6)
For each positive integer r, consider all ways of writing
r = Pi + + Pm, rn = +pi, for j > 1. We
for pi positive integers, and set qo = 0 and qj = pi + can now expand out +co (_ ) k_1 +oo
E
o
n=i /An [0,t]
k=1
to obtain, thanks to Lemma 1.1, r
+co
zt=--EEE ( ' )m—l f odwsi ...
0
dw„,
r..1. m=1 PJ
where the integral is taken over the region given by the inequalities
0 < s i < < s qi < t, 0 < sq ,.+1 < < Sq2 < t •..
O < Sqm _ +1 < ..• < Sqm < t•
(1.7)
11
Formal Stochastic Differential Equations
Step 2. By applying now the generalized Baker-Campbell-Hausdorff formula (B.4) of Appendix B, we obtain -Foo r Zt
(
r=1 m=1 pi
1) m
mr
f[...[odw,„od,2J.-J,°dwarl,
(1.8)
where the integral is taken over the same region. The domain determined by the inequalities (1.7) can be written as the union of simplices obtained from the simplex Ar [0, t] by permuting the variables, actually
f [...[ocku31 ,ocko32 ]...],odw5r 1 =
E
[".[0&03),odw,(2)].•.]
where the inner sum is taken over the permutations a
5 °1141) .9 ,7( r )
e r that satisfy
< o- (qi +1), j = 0 , • • • ,m — 1.
o-(qi + 1) < a (qi + 2) <
Therefore, by regrouping the terms in (1.8), we obtain that Zt is equal to T
E r=1 cree, m=1
( -1 7n )m r -1
d(r,m, a) f
JA, [OM
[...[ocko scr(1) ,odw,, (2) ]...],odw„ (r) ],
where d(r, m, a) is the number of ways of choosing positive integers p i , , pm with p i + + pm = r satisfying o- (qi + 1) < < o - (qi +1) , j = 0 , m 1.
Step 3. We claim now that r E
m=1
1) -im 77L7'
dfr m, a) =
(a) —1y ( r 2(7-
— 1) e(a)
Indeed, a straightforward combinatorial argument shows that
d(r,m, o-) =
(r — e(a) — 1 — e(a) — 1 '
so that we need to sum
E (-1)m -1 (r _ e(a) — 1 m--=e(a)+1
Mr
— e(a)— 1 )
An Introduction to the Geometry of Stochastic Flows
12
But let us observe that for n > 0 and k > 0,
n
(n\
(-1)i U)
(-1)i
(7) f xi+k_i dx
i=o
3
o
= f (1 _ x )n x k—i dx (k — 1)!n!
(k + n)! Therefore, by setting k = e(a) + 1, m = k j, and n = r — k we obtain
(
m--=1
1)m
mr
1
d(r,m, a) =
(-1) e(a)
r2 ( r — 1
e(cr) Step 4. Putting things together, we get therefore +00
(-1)e(a)
zt = E E
f
[...[odws.„( , ) , odw, c, (2) ]...[, odws ,) ]. ,„ (
Ar[0,t]
r2
e(; 1 ))
r (
Now, observe that [OdWsa(i)
, [..., [ OdWsa(r _ 1) , Ocil.4/ 3aer) [...]
= (-1) T-1 [...[Ock0Ba(r) , odW3aer _ 1) [...1, OdWsa(1) 1,
and that if for 0' G e r , a* denotes the permutation defined by .7*(k) = a(r +1 — k), then e(e) = r — 1 — e(a). Therefore, we also have +00
Zt =
EE
(-1) e (a)
r2
(r — 1 f,60-[o,t] e(a) )
By expanding out
[odwsa( ) , [... [ ockusa(r _ 47E8, r 2
(r e(a)
)
,
odcoscr(r)]...]
13
Formal Stochastic Differential Equations
Into (-1 Er
r2
E
) 47)
(r — 1)
L ko,t []
odB/,
e(o-) we obtain finally the claimed formula. Remark 1.5 are:
Observe that the first terms in the Chen-Strichartz formula
(1) d
E
A.,-(B)tx,..EBOCi ; k=1
(2)
-12 E
E
[Xi, X i ]
f
o dB:19 131 o
1
it is interesting to note the above Stratonovitch integrals are also Itô integrals, that is
B o dB!,
B-7,9 o dB: = fo ;dB3 .8 ,9 — B:79 d.13.
Jo
Remark 1.6 Actually, the Chen-Strichartz formula holds for the signature of any semimartingale: this is indeed a pathwise result. Remark 1.7 The formal development for the action on functions of the stochastic flow stl)i associated with a stochastic differential equation of the type (1 .3) reads therefore
(Irt' = exp
E E
( k>1 I=.(ii ,.,.,tk)
A1(B)V1 )
.
It is also possible to obtain a formal development for the action of (Dt on smooth tensor fields. Indeed an iteration of the formula given in the Proposition A.6 of Appendix A leads, due to the Lie algebra homomorphism property of the Lie derivative, to
dot` = exp( E
E Ai(B)t,cv, k>ii....(t,,...,to
14
1.4
An Introduction to the Geometry of Stochastic Flows
Expectation of the signature of a Brownian motion
It is interesting to note that it is possible to derive in a purely algebraic manner the semigroup Pt associated with the solution of a stochastic differential equation driven by Brownian motions . Definition 1.3
The element of R[[X i ,
Xd]] defined by
+00
EE
Pt = 1 +
oc/B / ) , t > 0,
Xii ...Xik E (f Ak[o,t]
k=1
is called the expectation of the signature of the Brownian motion (B)> o . Proposition 1.3
We have d
1 2 i=1
Proof.
t > 0.
By using proposition (1.2), we get +00
Pt = +
E E
xi,...xikE (f
dB'
)
t > 0,
k[0t]
where: 1 Xo = —
d
Y .‘ ,
2
and Bt° = t.
In the previous sum, the only terms whose expectation does not vanish are the terms
L
dB' k
[O,t]
where I is a word which contains only O. Therefore, +00
Pt = 1 +
EXt Lk [OM dti...dtk.
k=1
Since
dti...dtk
1
f
tk
sho tt j k dti...dtk = k.
15
Formal Stochastic Differential Equations
we get Pt= 1
1,7 k tk .A. ° e tX0 • +'— +E k! k=1
If we think (S(B) t ) t >0 as the solution of the formal stochastic differential equation
Remark 1.8
t
d
S(B) t = 1 +E f S(B) 8 X io dB,
(1.9)
i=1
and Pt as E (S(B) t ), then the above formula is rather intuitive. Indeed, by writing the Itô's form of (1.9), and by taking the expectation, we obtain the equation Pt
d
t
=1+f 0
ds,
z
2
which directly implies
1
exp (t i_Pt x 2 i=i Remark 1.9
)
.
In the commutative case, we have
S(B) t = exp (E Xi./31) z=i and the fort! ula for Pt reduces to the well-known Laplace transform formula d
E exp (E X,J3)) (
exp
i=1
1
d
2
We stress the fact that the last formula only holds in the commutative case.
Observe that the semigroup property of Pt , that is Pt-Fs
Pt -Ps,
could have been directly derived from the identity +00
S(B) t 1 ( +E
E xil ...xik
k=1 /=(ii,—ik)
odB/) fak[t,t-1-8]
.
16
An Introduction to the Geometry of Stochastic Flows
Indeed, since fakft,t+si oc/B/ is independent of (B,L ) u< t , we deduce +co
Pt+8 = Pt (1 +
E E xi, ...Xik E
Lkft,t+sj o
c/B i )) .
Now observe that, due to the stationarity of the increments of a Brownian motion,
E (L kft,t+si odB i ) = E (f
odB I )
Akfo,s]
so that
+00 Pt+s = Pt (1 +
EE
(f
odBi)) — Pt-Ps.
A k[o,s]
k=1 /=“1,.../k)
We have already pointed out that the signature is a universal object in the theory of stochastic flows, so let us see the analytic counterpart of the purely algebraic formula +00
1±
d
EE
xi, ...x„ E
odB i ) = exp fak (0,t]
k=1
2
t
x?) .
In the first section, we have seen that for the action on smooth functions of the stochastic flow 4) associated with the stochastic differential equation d
Xf° = so +
t
El vi( x:0)
dB,
i=1 we had formally +00
=
Id +
EE
Vi i ...Vi k
odB I .
[0,t]
k=1
Therefore,
+00 E(cFt') = Id + k=1
E
14,1... 141cE
OdB i = )
(Lk[0,t]
e pE id=, v2 i -2
.
17
Formal Stochastic Differential Equations
By coming back to the definition of 4 1 we deduce that if f : Rn smooth function,
R is a
E(f(XPO) = (el t1vt2 .f) (xo), which exactly says that (X7°) t>0 is a Markov process with generator eitEL1 v z2 . In the same way, by using the formal development of cbt on smooth tensor fields which reads +00
+EE
4) t* =
k=1
—Cif, .
[otti
odB I
-ik
where L denotes the Lie derivative, we obtain that if K is a smooth tensor field on IV,
ER (NIC)(xo)] = (el tE L'
K) (xo).
Of course, all this is only formal, but should convince the reader of the relevance of the formal calculus on the signature.
1.5
Expectation
of the
signature
of other processes
As already observed, the notion of signature can be defined for other processes than Brownian motions and there is a corresponding notion of expectation for the signature. Let us for instance mention the example of the signature of a fractional Brownian Motion. A d-dimensional fractional Brownian motion with Hurst parameter H> is a Gaussian process
t > 0, where B 1 , ..., Bd are d independent centered Gaussian processes with covariance function 1 /R(t,s)= 2H + t2H —
It el)
It can be shown that such a process admits a continuous version whose paths are locally p-H61der for p < H. Therefore, if H> 1, the integrals
L dB' k [o,t1
18
An Introduction to the Geometry of Stochastic Flows
can be understood in the sense of Young's integration; see [Young (1936)1 and [Zhh le (1998)1. We define then the signature of (Bt ) t> 0 by +00
s(B) t
= i +E E
fom ,BI,t> 0,
k=1 and associate with S(B) the family of operators
+00
Pt
:=I+E E
dB') , > O.
Xii ...Xtk E
k=1
The increments of (Bt ) t>0 are not independent (they are however stationary), and (Pt)t>0 is therefore not a semigroup. Nevertheless, as shown in [Baudoin and Coutin (2004)1, when t 0, Id
1 Pt = 1 + t2H (E X i? 2
d
E
t4H
+0(t6H),
ij,k,1=1
where,
1 1s s 2H -1 - -14 ,/(5i,i [ 71 - 2110 (2H,2H - ) 11 + tii' ku3 4H(4H -1) + H(2H -1) 6 k8i,1[3(2H,2H 2) + 8 4H - 1 2H - 1 ' ' 1
with 0(x, y) = fol ti-i (1 - t) -I dt, and 5t,j is the Kronecker's symbol. A development for Pt which leads to development in small times of expressions of the type E (f(Xr)), where (Xts° )t>0 denotes the solution of the equation d
X0
Ef
t vipc:0)dBis
,
t > 0,
i=1 0
which is understood in Young's sense (see [Nualart and Rascanu (2002)1 for theorems concerning the existence and the uniqueness for the solution of such an equation). Observe that when H 1, then the above development tends to d 1 pt = 1 + - t (E
2
i=i
d 1 + - t2 (E xi2)
8
2 0(t3 ),
J=1
which is the development of the Pt corresponding to the Brownian motion.
Formal Stochastic Differential Equations
19
Also observe that the fourth order operator
E
X kXi
i,j,k,1=1
can not be simply expressed from d
Ex?. ,t=1.
Such a discussion can obviously be generalized to any stochastic differential equation driven by Gaussian processes whose paths are more than I locally Holder continuous and this is actually an interesting open question to decide what is the smallest sub-algebra of RM., Xd11 that contains Pt. Let us finally mention another type of processes for which the expectation of the signature can be explicitly computed. Let us consider the process
t > 0, where (Bt ) t>0 is a d-dimensional standard Brownian motion and a a non negative random variable independent of (Bt ) t>0 which satisfies JE (o-k ) < +co, k > O. In that case, the expectation of the signature of (Z)> o is easily seen to be given by +00
Pt
E kE ( 0.k)
tk
d Ex?
,
i=1
k=0 2 •
and observe that, like in the Brownian case, the smallest algebra containing Pt is given by R [E Xd. For instance, by taking for a an exponential law with parameter
1, that is P(o- E dx) = e— '111,0 (x),
we get Pt=
1
Chapter 2
Stochastic Differential Equations and Carnot Groups
Let us consider a stochastic differential equation d
t
Xr ° = x0 +Ef Vi (X;°) o dB
,
t> 0,
(2.1)
i=1 where x o E Rn, Vd are C°° bounded vector fields on Rn and (B)> o is a d-dimensional standard Brownian motion. Since (Xf° )t>0 is a strong solution of (2.1), we know from the general theory of stochastic differential equations that (Xr )t>0 is a predictable functional of (B t )t>0 (see Appendix A). In this chapter, we would like to better understand this pathwise representation. For this, the best tool is certainly the Chen-Strichartz formula which has been proved in the first chapter. Indeed, if we denote (4't)t>o the stochastic flow associated with equation (2.1), then the Chen-Strichartz formula shows that for the action of (I) on smooth functions
E
exp
(B)tvi ,
k>1
where =[14i,[14,2,—,[Vik_i,Viki...1,
and
fakfom odBa -1.1
Ai(B) t = ee k k2
k— 1) e(a) 21
.
22
An Introduction to the Geometry of Stochastic Flows
Even though this is only a formal development, it clearly shows how the dependance between B and Xx° is related to the structure of the Lie algebra e generated by the vector fields Ws. If we want to understand more deeply how the properties of this Lie algebra determine the geometry of Xx°, it is wiser to begin with the simplest cases. In a way, the most simple Lie algebras are the nilpotent ones. In that case, that is if L is nilpotent, then the sum
is actually finite and we shall show that the solutions of equation (2.1) can be represented from the lift of the Brownian motion (B) >o in a graded free nilpotent Lie group with dilations. These groups called the free Carnot groups are introduced and their geometries are discussed. When the Lie algebra is not nilpotent, this representation does not hold anymore but provides a good approximation for Xxo in small times. We conclude the chapter with an introduction to the rough paths theory of [Lyons (1998)], in which Carnot groups also play a fundamental role.
2.1
The commutative case
In this section, we shall assume that the Lie algebra L = is commutative, i.e. that [Vi , V3 ] = 0 for 1 < i j < d. This is therefore the simplest possible case: it has been first studied by [Doss (1977)1 and [Siissmann (1978)1. The main theorem is the following: ,
Theorem 2.1
There exists a smooth map
F: Rn x Rd such that, for xo E Rn, the solution (Xr)t>0 of the stochastic differential equation (2.1) can be written
XT° = F(x 0 ,Bt ), t > O. Proof. For i = 1, ..., d, let us denote by (etvi)tER the flow associated with the ordinary differential equation dx = Vi(st)• dt
23
SDE's and Carnot Groups
Observe that since the vector fields Vi's are commuting, these flows are also commuting (see Appendix B). We set now for (x, y) E Rn x R d , F(x,y) =
o o eYd vd) (x).
By applying Itô's formula, we easily see that the process (e S td vdx 0)
t>o
is
solution of the SDE
d (e Bctivd (so )) = Vd (e B ;Lvd (x o )) o dB. A new application of Itô's formula shows now that, since Vd and Vd_i are commuting,
d (eB1Vd-1 (eBd-lva i r e/41/(1x° ) ) t
(er4Vd X0)
di3`1-1 + Vd (e1Vd_1 ( e Std ild xo)) 0 c1.14 .
We deduce hence, by an iterative application of Itô's formula that the process (F(s o ,Bt )) t >0 satisfies d
dF(X0,Bt)
= Evi(F(x 0 ,Bt)) o dig
Thus, by pathwise uniqueness for the stochastic differential equation (2.1), we conclude that Xf° = F(xo ,B t ), t > O. Computing the function F which appears in the above theorem is not possible in all generality. Indeed, the proof has shown that the effective computation of F is equivalent to the explicit resolution of the d ordinary differential equations dx t dt
which is of course not an easy matter. There is however a case of particular importance where it is possible to solve explicitly the stochastic differential equation (2.1): this is the case n = d = 1. Indeed, in that case, (2.1) can be written dX t = v(X) odB,
24
An Introduction to the Geometry of Stochastic Flows
which also reads in Itô's form 1
dX t = — v(X t )71(X t )dt v(X)dB. 2 This equation is solved by
X t = g(Bt), where g solves
g' = V o g.
2.2
Two-step nilpotent SDE's
In this section we introduce the free 2-step Carnot group over Rd, study its geometry and show that it is universal in the study of 2-step nilpotent diffusions. Almost all what follows will be generalized in the next section. Nevertheless, we believe that the understanding of the geometry of the free 2-step Carnot group is much more easy to get than the geometry of general Carnot groups and however contains the most important ideas of sub-Riemannian geometry. Let d > 2 and denote ASd the space of d x d skew-symmetric matrices. We consider the group G2 (Rd) defined in the following way
G2 (Rd ) = (Rd x ASd, 0) where ® is the group law defined by (4:11,w1) 0 (a2 1 w2) = (cEi +2 1
1 + w2 + —al A). 2
Here we use the following notation; if ai, a2 E Rd, then ai A CV2 denotes the skew-symmetric matrix (criA — ala0 . Consider now the vector fields
1
0
0
1
2
i
E x.3
1
i>i
defined on Rd x ASd. It is easy to check that:
(1) For x
E Rd X
ASd
,
[D„Di i(x) =
. . , I.< i < j < d;
ex',3
d,
25
SDE's and Carnot Groups
(2) For x E Rd x AS d,
[[D,,
Dk i(s) = 0, 1 < i j , k < d; ,
(3) The vector fields (D„, [D ,
1
are invariant with respect to the left action of G2 (Rd ) on itself and form a basis of the Lie algebra 22 (Rd ) of G2 (Rd ). It follows that G2(Rd) is a d(d2+1) -dimensional step-two nilpotent Lie group whose Lie algebra can be written 132 (R d )
_
ED [Rd , Rd] .
Notice, moreover, that the scaling
(2.2)
c • (a, co) = (ca, c2 w) defines an automorphism of the group G2 (Rd ) . Definition 2.1 group over Rd .
The group G2(Rd ) is called the free two-step Carnot
Example 2.1
(1) The Heisenberg group can be represented as the set of 3 x 3 matrices: (1 s z 0 1 y J, x, y, z 00 1
R.
The Lie algebra of 11E is spanned by the matrices
0 1 0\ 0 0 0\ 001 D1 =f 0 0 0 , D2= 0 0 1 and D3= 0 0 0 , (
0 0 0
0 0 0
for which the following equalities hold [Di , D2 ] = D3 [D1, D3] = [D2 D3] O. Thus
R2 e [R,
0 0 0
26
An Introduction to the Geometry of Stochastic Flows
and, G2(R2 ). (2) Let us mention a pathwise point of view on the law of G2 (R2 ) which has been pointed to us by N. Victoir. If X : [0, -koo) R 2 is an absolutely continuous path, then for 0
(42 —
[t i ,t2] x
-t 411 S[ i,t2]X)
where S[ t1 ,t2 ]x is the area swept out by the vector XtX during the time interval [t1, t 21. Then, it is easily checked that for 0 < t1 < t2 < t3 A[ti ,t3]X —
[ti ,t2]X
A[t2 ,t3 ]x,
where ® is precisely the law of G2(R2 ), i.e. for (x 1 , yi, zi), (x2, Y27 z2) E R3 7
(x i ,
Z1) 10(X21 Y27 Z2) — (aCi + X27 Y1 + Y21 Z1 + Z2
, —
2
_ x21 )).
Let now (Bt ) t>0 be the Brownian motion considered in the equation (2.1). There exists a unique G 2 (Rd )-valued semimartingale (Bnt>0 such that Br 0G2(Rd ) and d
dB; = E
Dipn o dB, O < t < T.
Definition 2.2 The process (Bnt>o is called the lift of the Brownian motion (Bt ) t >0 in the group It is immediate to check that we have
B; = (Bt , (14 Hi, o 2 0
— o dB)
, t > O. 1
It is interesting to note the above Stratonovitch integrals are also Itô integrals, i.e.
t
foJo
o dB,39 — B,39 o dB! = f Bis dB1 .13,39 dFs .
Also observe that we have the following scaling property, for every (Bc*t)t>0 —law
(VI—C B t* L>0
c > 0,
27
SDE's and Carnot Groups
The process B* can be seen as a diffusion process in Rd x .AS d (sometimes called the Gaveau diffusion, see [Gaveau (1977)] and [Malliavin (1997)]) whose generator is given by 1 2
02
1 E (xi 0 -I0(x1 ) 2 2 i
xi
0 ) 0
axi )
+
1
0 .2 .
.
i
Moreover, as shown by [Gaveau (1977)], the law of B* is characterized by the following generalization of Lévy's area formula: Lemma 2.1
Let A be adxd skew-symmetric matrix. Then, for t > 0,
E (ei f(t(AB.,dB.)
I Bt
z) = det
tA (si n tA
2 exp
(I
—
tA cot tA 2t
It is now time to say few words about the natural geometry of G 2 (Rd). First, we note that there is a natural scalar product g associated with the previous operator, precisely we define for (a,4.0), (a',4.0 1 ) E 92 (Rd ), a')Rd,
g ((a, w), (a 1 , w')) -=
where W iRd denotes the usual scalar product on Rd. This scalar product, only defined on 9 2 (Rd), i.e. on the tangent space to G2 (Rd) at 0G2 (Rd , can be extended in a usual manner to a left invariant (0, 2)-tensor, still denoted g, and defined on the whole Lie group G2 (Rd ) . Precisely, for x E G2(Rd), we define gi on the tangent space to G 2 (Rd) at x in such a way that for 1 < j, k , < d,
gx (Di (x), D i (x)) = 0 if i gz (Di (x), D t (x)) = 1, gi (D i (x), [Di , D k](x))
0,
D .11(4 [Dk, Di[(x)) = O. Since g is not definite positive, the associated geometry is not Riemannian but sub-Riemannian. The relevant object for studying this geometry is the so-called horizontal distribution 7-( which is defined as the smoothly varying family of vector spaces = span (D 1 (z),
Dd (x))
,
x E
28
An Introduction to the Geometry of Stochastic Flows
An absolutely continuous curve c: [0, 1] for almost every s E [0,1] we have ci(s) E
G2 (Rd ) is called horizontal if
Proposition 2.1 Let c = (ct,w) be an absolutely continuous curve [0, 1 1 G2 (Rd). It is a horizontal curve if and only if
(s) = c(s) A cl (s). Proof. that
The curve c is horizontal if and only if there exist A1,
Ad such
c/(s) = EAi(s)Di(c(s)). ti=1
Since
a
D•(x) = 8si
1
a
1
-2 E
wx-1,7 2 E
j
j>i
a :7 , 1 < i < d,
we obtain, first, by an identification of the coefficients in front of the Ar 's
Ai(s) = c(s). The identification of the terms in front of the 4,77 's leads exactly to
cu i (s) = —21 a(s) A cx' (s).
LI
The length of an horizontal curve c with respect to g is defined by
1(c) =
\ gc(s) (c' (s), c' (s))ds.
We can now turn to the first version of Chow theorem, whose proof shall be given later (see Theorem 2.4). Theorem 2.2 Given two points x and y E G2(Rd ), there is at least one horizontal curve c: [0, 1] G2(Rd ) such that c(0) = x and c(1) = y.
The Carnot-Carathéodory distance between x and y and denoted dg (x , y) is defined as being the infimum of the lengths of all the horizontal curves joining x and y. It is easily checked that this distance satisfies dg (c.x,c.y) =-cdg (x,y), for any c> 0, x, y E G2 (Rd ), where the multiplication by c in the group corresponds to the scaling defined by the formula (2.2). A horizontal curve with length dg , y) is called a sub-Riemannian geodesic joining x and y.
SDE's and Carnot Groups
29
Proposition 2.2 Let c =-- (û,w) be a smooth horizontal curve [0,1] G2 (Rd ). Then, it is a sub-Riemannian geodesic if and only if there exists a skew-symmetric d x d matrix A such that a" — Act'. Proof. First, we note that the equation given in the proposition is nothing else than the Euler-Lagrange equation
d OL OL Oc' Oc' associated with the Lagrangian with constraints
1 — g e( s) (c' (s), ci (s)) + 2E 2
L (c(s), (s))
where A is an arbitrary skew-symmetric d x d matrix and 9 i = dxi ,i is the one-form on132(Rd ) which vanishes on 92(Rd) excepted on the vector space spanned by [Di , Di ]. Indeed, since c is horizontal we can write
thus 1
d
L(c(s), c i (s)) = — E(cr'i (s))2 2
(cri (s)a31.(s) — ai (s)aii (s)) ,
and
OL Oa: = ii(s)
d
EA
a (s).
j=--1
Now, to conclude the proof we have to check that all the sub-Riemannian geodesics are indeed critical points of the constrained Lagrangian that has been considered. Since the proof of this fact is not obvious and quite technical, we refer the interested reader to [Golé and Karidi (1995)].
Remark 2.1 It can be shown that in G2 (Rd), all the sub-Riemannian geodesics are smooth (see e.g. [Golé and Karidi (1995)1).
30
An Introduction to the Geometry of Stochastic Flows
From the previous proposition, we deduce that the sub-Riemannian geodesics in G2 (Rd) are generalized helices, i.e. curves with constant curvature. Precisely, let c = (ce, co) be a sub-Riemannian geodesic. The curve c is smooth and solves the equation
— Aa',
(2.3)
for some skew-symmetric d x d matrix A. It is now an elementary fact in linear algebra that we can then write a decomposition Rd = Ker(A) eH e • • • Hm where are planes (i.e. two-dimensional spaces) such that AH = H. Therefore, a direct quadrature of the equation (2.3) shows that the projection of c onto any one of these planes is part of a circle, and its projection on the kernel is a line segment. Now, since the geodesics of the geometry of G2(Rd ) are known, it is natural to be interested in the shape of balls. We denote for E > 0, Bg (0, E) the open ball with radius E for the Carnot-Carathéodory metric dg . By direct but tedious computations, it can be shown that the sphere aB g (0, E) is not a smooth sub-manifold of G2 (Rd). Precisely, OBg (0, E) has a singularity in each of the vertical directions [Di , Di ] = 027. For instance in the Heisenberg group, i.e. in G2 (R2 ), the unit sphere looks like an apple (see Fig. 2.1) whose parametric equations are
=
cos u sin v—sin u(1—cos v)
y(u, y) =
sin u sin v+cos u(1—cos v)
X(lt, V)
Z (U, V)
V 1
uSin .2 V
where u E [0, 27r], y E [-27, 271-1. By letting the parameter y describe R, we obtain the wave front of the Heisenberg group, whose structure is quite complicated (see Fig. 2.2 which represents half of the wave front). c_a-ji For the Euclidean metric of G2(Rd ) = Ra x R 2 , the size of the sphere Bg (0, E) in the horizontal directions is approximatively E and approximatively E2 in the vertical directions. Precisely, as it will be seen later, if for E > 0, we denote
Box(E) = {(a,w) E G2(Rd), ai i< E, wi,k l< E 2 1, there exist positive constants c l and e2 and E0 such that for any 0 < BOX(CiE) C
Bg (0, E) C Box(c2E).
E < E0,
SDE's and Carnot Groups
31
Fig. 2.1 The unit Heisenberg sphere.
From this estimation, we deduce immediately that the topology given by the distance dg is compatible with the natural topology of the Lie group G2(Rd ). We can also deduce that the Hausdorff dimension of the metric space (G2(Rd), dg ) is dim V1 ± 2 dim V2 = d2 which is quite striking since the dimension of G2 (Rd) as a topological manifold is only equal to 4 '12+1) . For further details on these properties of the metric space (G 2 (Rd), dg ), we refer to the next section. We now come back to the study of the stochastic differential equation
d
Xr ° --= Xo ±
t
E f vi(x.fo ) 0 dBis , t > 0,
(2.4)
i=1 °
where xo c Rn, V1, ...,Vd are C°° bounded vector fields on Rn and (Bt)t>o is a d-dimensional standard Brownian motion and show the universality of the group G2 (Rd ) in the study of two-step nilpotent diffusions.
32
An Introduction to the Geometry of Stochastic Flows
Fig. 2.2
The wave front in the Heisenberg group.
Theorem 2.3 Assume that the Lie algebra L = nilpotent. There exists a smooth map
F :Rn X G2(Rd )
Vd) is two-step
Rn
such that, for xo E Rn , the solution (XT°)t>o of the stochastic differential equation (2.4) can be written
Xr° = F(xo , Bn, where (Bn t >0 is the lift of (Bt)t>0 in the group G2(R d ).
Proof.
Let us consider the function F :Rn x G2(Rd ) defined for x
E Rn
and g = (a, w) E G 2 (Rd) by (d
F (x, g) = exp
E ai Vi + —2 E [vi, i=1
(x).
1
the Stratonovitch integration satisfies the usual change of variable formula, an iteration of Itc3's formula shows, by using Chen's development Since
SDE'.s and Car-not Groups
33
theorem, that the process (F(xo , Bn)t>o solves the equation (2.4). We conclude by the pathwise uniqueness property. The conclusion of this section is hence the following: The natural geometry associated with a 2-step nilpotent stochastic differential equation is the sub-Riemannian geometry of a quotient of the free 2-step Carnot group G2 (Rd). The next step for us is now to generalize the previous study to any nilpotent stochastic differential equation.
2,3 N step nilpotent SDE's -
We introduce now the notion of Carnot group. Carnot groups are to subRiemannian geometry what Euclidean spaces are to Riemannian geometry. Numerous papers and several books are devoted to the analysis of these groups (see e.g. [Bellaiche (1996)], [Folland and Stein (1982)], [Goodman (1976)1, [Gromov (1996)1). Definition 2.3 A Carnot group of step (or depth) nected Lie group G whose Lie algebra can be written ED
N
is a simply con-
VNI
where [Vi, Vi ] = and
= 0, for s > N. Example 2.2
The group (R d , -I-) is the only commutative Carnot group.
Example 2,3 The group G2 (Rd ) considered in the previous section is a Carnot group of depth 2. Example 2.4 law
R2n X R endowed with the group
Consider the set IHEn
(x, a) * (y, 0) =
y, a 1 (3 -I- —21 w(x, y)) - -
where w is the standard symplectic form on R2n, that is (4)(x, y )
—In
,„ t a'
In
0)
34
An Introduction to the Geometry of Stochastic Flows
Observe that Hi is the Heisenberg group. On f) n the Lie bracket is given by = (CI,w(x, Y)), and it is easily seen that =
ED
v2,
where V1 = R2 x {0} and V2 = {0} x R. Therefore lln is a Carnot group of depth 2. We consider throughout this section a Lie group G which satisfies the hypothesis of the above definition. Notice that the vector space V1, which is called the basis of G, Lie generates g, where g denotes the Lie algebra of G. Since G is step N nilpotent and simply connected, the exponential map is a diffeomorphism and the Baker-Campbell-Hausdorff formula therefore completely characterizes the group law of G because for U,V E g, exp U exp V = exp (P(U,V)) for some universal Lie polynomial P whose first terms are given by
P(U,V) =U +V +1[U,V1+ — 112 HU,V1,V1 — —112 [[U,V],U] 418
gr, [17, gr, viii + • • • .
IV ITT
(see Appendix B for an explicit formula). On g we can consider the family of linear operators ö t : g g, 0 which act by scalar multiplication ti on V. These operators are Lie algebra automorphisms due to the grading. The maps (St induce Lie group automorphisms A t : G G which are called the canonical dilations of G. Let us now take a basis U1, Ud of the vector space V1. The vectors Ui 's can be seen as left invariant vector fields on G so that we can consider the following stochastic differential equation on G:
t>
d
d t =-
t
E f U(E)dB s,t > 0,
(2.5)
i.1
which is easily seen to have a unique (strong) solution (53- t)t>0 associated with the initial condition B0 = O. Definition 2.4 The process (13t)t>o is called the lift of the Brownian motion (Bt)t>0 in the group G with respect to the basis (U1, .•., Ud).
SD.E's and Carnot Groups
Remark 2.2 1
Ed
U.
35
Notice that (Et)t>0 is a Markov process with generator second-order differential operator is, by construction, left-
invariant and hypoelliptic. Proposition 2.3
We have (N
fit = exp
EE
A,(B)tu,
, t > 0,
k=1 I=-- (ii,.•.,ik)
where: Al(B)t
E
(-1) e(a)
aeek k2 k 1) \e(a)
f
od13' 1
' 1 .
Ak [0 ,t]
This is obviously a straightforward consequence of the ChenStrichartz development Theorem 1.1, whose proof is given in Chapter 1. Nevertheless, since the general proof of Theorem 1.1 is purely algebraic we believe that it is interesting to hint a more pathwise oriented proof. We follow closely [Strichartz (1987)1 and proceed in two steps. In the first step we show that if co : R>o Rd is an absolutely continuous path, then the solution of the ordinary differential equation Proof.
d
dx t =
Eui(xt )d.1 i=1
is given by N
xt = exp
EE
Al(W)tUÏ)
k=1
where
(-1) e(a) creek k2
k — 1) e(a)
Lk
• [O ,t]
In the second step, we observe that Stratonovitch differentiation follows the same rules as the differentiation of absolutely continuous paths. Step 1. Let co : R>0 Rd be an absolutely continuous path. Let t > 0 and n E N*. We consider a regular subdivision 0 = to
36
An Introduction to the Geometry of Stochastic Flows
of the time interval At]. The linear continuous interpolation of the path (w.)o<s
(4) - = 444
d. If (is)o<s
cas
= Euk(±-5)dJ4, o < s
it is easily seen that d
it
=
expEptk _ (
k \ W
7 k ' • • eXp
k=1
d Gu tik — E
W tko) Uk •
k=1
Now we use the Baker-Campbell-Hausdorff formula (see Appendix B) to write the previous product of exponentials under the form (N
it
=
exp E
E
Ar(W)tUI
7
k= 1 /={i1,--,ik}
and similar arguments, using furthermore the convergence of Riemann sums, to those given in the proof of the Chen-Strichartz formula show that A7(w)t
Ai(w)t.
Since Xt,
we conclude that (N Xt =
exp
E E
Ai(w)tui
•
k=1 /={i i ,..., ik}
Step 2. Since the It6's formula with Stratonovitch integrals has the same form as the usual change of variable formula, an iteration of It6's formula shows the expected result. Observe that, due to the elementary fact (A/(B)ct)t>o
=law ( c AI (B) t ) t>0
37
SDE's and Carnot Groups
we have the following scaling property, for every c > 0
(Bct)
t>0
= law (Avjijt) t>o
This scaling property leads directly to the following value at OG of the density )-5t of Pt with respect to any Haar measure of G: Pt (Os) = —B-, t > o,
EiN
where C > 0 and D = i dim V. Moreover, from [Alexopoulos and Lohoué (2004)1, for fit, t > 0, the following estimates hold: Proposition 2.4
For (i i ,
ip ) E 11, ..., dr , g E G, q ? 0, t > 0, A
1 u 1 .. Uipall3t(.9) 15_ t D+P'-gF2 exp where
Ap ,q
g , d(0G.9)2
Bp, q t
and Bp,q are non-negative constants.
We now turn to the geometry of G. The Lie algebra )3 can be identified with the set of left-invariant vector fields on G. From this identification and from the decomposition g=
ED ..•
VN,
we deduce a decomposition of the tangent space TG to G at x E G: TG
(x) ED
VN(x),
where Vi (x) is the fiber at x of the left-invariant differential system spanned by V. This decomposition endows naturally G with a left-invariant (0,2)tensor g. Precisely, for x E G, we define gr as being the scalar product on TG such that: form an orthonormal basis; - 0, if i or j is different from 1. (2) gs 11)(x)xV i (x) =
(1) The vectors Ui (X),
...,Ud(X)
An absolutely continuous curve c : [0,11 G is called horizontal if for almost every S G [0,11 we have c'(s) E Vi(c(s)). The length of a horizontal curve c with respect to g is defined by
/(c) = f Vg, (8) (ci(s),e(s))ds. We can now state the basic result on the geometry of Carnot groups: The
Chow's theorem.
38
An Introduction to the Geometry of Stochastic Flows
Theorem 2.4 Given two points x and y E G, there is at least one horizontal absolutely continuous curve c : [0, 1] G such that c(0) = x and c(1) = y. Proof. Let us denote G the subgroup of diffeomorphisms G G generated by the one-parameter subgroups corresponding to U 1 , ..., Ud. The Lie algebra of G can be identified with the Lie algebra generated by U1, • ••, i.e. 9. We deduce that G can be identified with G itself, so that it acts tranG, g g(x) sitively on G. It means that for every x E G, the map G is surjective. Thus, every two points in G can be joined by a piecewise smooth horizontal curve where each piece is a segment of an integral curve of one of the vector fields Remark 2.3 In the above proof, the horizontal curve constructed to join two points is not smooth. Nevertheless, it can be shown that it is always possible to connect two points with a smooth horizontal curve (see [Gromov (1996)] pp. 120. The Carnot-Carathéodory distance between x and y and denoted dg (s, y) is defined as being the infimum of the lengths of all the horizontal curves joining x and y. It is easily checked that this distance satisfies dg (A c x, A c y) = cdg (x, y), for every c > 0, x, y E G. Remark 2.4 The distance dg depends on the choice of a basis for Nevertheless, all the Carnot- Carathéodory distances that can be constructed are bi-Lipschitz equivalent. A horizontal curve with length dg (s, y) is called a sub-Riemannian geodesic joining x and y. The topology of the metric space (G, dg ) is really of interest. The basic tool to investigate this topology is the so-called ball-box theorem. Denote for E> 0, Box(E) = {g E 9, I gi
Ili<
< < N},
where gi denotes the projection of g on Vi and gi j the norm of gi with respect to any norm of V,: (all the norms on Vi are equivalent). We have then the following so-called ball-box theorem. Proposition 2.5 There exist positive constants cl and c2 and E0 such that for any 0 < E < E0 7
exp (Box(c l E)) c B g (0 , E) C exp (Box(c2E)) ,
39
SDE's and Carnot Groups
where B g (0,E) denotes the open ball with radius
E
in the metric space
(G, dg ). Remark 2.5
Let us mention that the previous proposition is actually a consequence of the equivalence of all the homogeneous norms on G. A homogeneous norm on G is a continuous function ii ' 11: G -4 [O , +cc) , smooth away from the origin, such that:
(1) 11 Acx 11= c 11 x 11, e > 0, s E G; (2) 11x -1 11=11 x11, x E G; (3) 11x 1= 0 if and only if x = OG. For analytical questions in Carnot groups (estimations,...) it is often more convenient to work with a given homogeneous norm rather than with the Carnot-Carathiodory distance itself. We have important consequences of the ball-box theorem. First, the topology given by the distance dg is compatible with the natural topology of the Lie group G. Secondly, we can compute the Hausdorff dimension of the metric space (G, dg ). To make what follows clear, let us recall some facts about Hausdorff measure and Hausdorff dimension (see e.g. [Falconer (1986) 1 for a detailed account of this material). Let (M, d) be a metric space, and let 52 be an open subset of M. Consider an open cover /4 = {U,,} of f2 and set for s > 0
ps (S1,11) =
E (diam(EQ) s . a
This quantity (possibly infinite) is called the approximate s-dimensional Hausdorff measure. We will use the notation 111 l< e to mean that each U„ has diameter less than e. The e-approximate s-dimensional measure of SI is the number
inflif(n, 11),1 11 1< e}. Finally, set As
(Q) = y9 (n).
This quantity is called the s-dimensional Hausdorff measure of a Proposition 2.6 There is a unique value D, called the Hausdorff dimension of the open set SI, with the property that iz 8 (f1) = +oo for all s < D and 11,8 (f) = 0 for all s> D.
40
An Introduction to the Geometry of Stochastic Flows
The metric space (M, d) is said to have an Hausdorff dimension equal to D, if any open set of M has an Hausdorff dimension equal to D. Proposition 2.7 is equal to
The Hausdorff dimension of the metric space (G, dg )
D=
E
dim V'.
Proof. We shall show that the unit ball Bg (0, 1) has Hausdorff dimension D = EjN.___ 1 j dim Vi . We know that, due to the dilations, the volume of a ball Bg (0, E) is CE". Consider now a maximal filling of Bg (0, 1) with balls of radius E. An upper bound for the number of balls N(E) in this filling is
1
N(E) < D Now, the set of concentric balls of radius 2E constructed from this filling covers B g (0, 1). Each of these balls has diameter smaller than 4E, thus the Hausdorff s-dimensional measure of Bg (0, 1) is smaller than
lim N(E) 8 , E.o which is 0 if s> D. Therefore the Hausdorff dimension is smaller than D. Conversely, given any covering of Bg (0, 1) by sets of diameter < E, there is an associated covering with balls of the same diameter. The number M(E) of these balls has the lower bound:
M(E)
D
We deduce that for every s > 0,
EE
s
Es ED
s—D
,
,
cover
which shows that if s < D then the Hausdor ff s-dimensional measure of Bg (0, 1) is Therefore the Hausdorff dimension is greater than D
Remark 2.6 Observe, and this is typical in sub-Riemannian geometry, that the Hausdorff dimension of G is therefore strictly greater than the topological dimension.
Carnot groups have their own concept of differentiation. In order to present this concept introduced in [Pansu (1989)], we first have to define a concept of linear maps between two Carnot groups.
41
SDEus and Carnot Groups
Let G1 and G2 be two Carnot groups with Lie algebras 91 and 92 . A Lie group morphism : G1 G2 is said to be a Carnot group morphism if for any t > 0, g E G1,
gg z .9) = where AG' (resp. AG 2 ) denote the canonical dilations on G1 (resp. G2). In the same way, a Lie algebra morphism a : gi 92 is said to be a Carnot algebra morphism if for any t > 0, x E 131) a(8t91 X) = 6ra(X),
where 6G1 (resp. 8 G2 ) denote the canonical dilations on 9 1 (resp. 92 ). Observe that if 0 is a Carnot group morphism, then the derivative dq5 is a Carnot algebra morphism. Let now F : G 1 G2 be a map. When it exists, the Pansu's derivative dpF, is defined by dpF(g)(h) = lim (g2 ) 1 (F(g) -1 11
h))
g, h E Gi.
and dpF(g) is a Carnot group morphism. In this setting, Pansu has proved a deep generalization of the classical Rademacher's theorem which asserts that a Lipschitz map between Euclidean spaces is almost everywhere differentiable. Indeed, any Lipschitz map (with respect to the Carnot-Carathéodory distances of G1 and G2 ) is almost everywhere Pansu differentiable. An important consequence of this is that can be no bi-Lipschitz map between Carnot groups that are not isomorphic as groups. We conclude now our presentation of the Carnot groups with the free Carnot groups. The Carnot group G is said to be free if 9 is isomorphic to the nilpotent free Lie algebra with d generators. In that case, dim VI is the number of Hall words of length j in the free algebra with d generators (see Appendix B). We thus have, according to [Bourbaki (1972) 1 or [Reutenauer (1993) 1 pp.96: dim Vi =
1 7
3
Ept(i)di, j < N, iii
where it is the Möbius function. We easily deduce from this that when N +oo, dim 9
dN
42
An Introduction to the Geometry of Stochastic Flows
An important algebraic point is that there are many algebraically non isomorphic Carnot groups having the same dimension (even uncountably many for n > 6), but up to an isomorphism there is one and only one free Carnot with a given depth and a given dimension for the basis. Actually, as in the theory of vector spaces, we can reduce the study of the free Carnot groups to standard numerical models. Let us denote m = dim G. Choose now a Hall family and consider the Rm-valued semimartingale (B )>o obtained by writing the components of (1n(.13t ))t>0 in the corresponding Hall basis It is easily seen that (Bn t>0 solves a stochastic differential equation of that can be written .
d
B;
=
t
E f D i (B) o dB:,
i.1
where the D's are polynomial vector fields on Rm (for an explicit form of the D's, which depend of the choice of the Hall basis, we refer to [Gershkovich and Vershik (1994)1 pp.27) . With these notations, we have the following proposition which results of our very construction. Proposition 2.8 On Rm , there exists a unique group law ® which makes the vector fields D1, Dd left invariant. This group law is, unimodulari , polynomial of degree N and we have moreover
(Rm,
G.
The group (Rm ,,O) is called the free Carnot group of step N over Rd. It shall be denoted GN(R d ) and its Lie algebra N(Rd). The process (Bnt>0 shall be called the lift of (B)> o in G N (R d) with respect to the basis (Di, •••,Dd)• Remark 2.7 The important point with 9N(Rd) is that this is not an abstract Lie algebra. This is a Lie algebra of vector fields defined on Rm Remark 2.8 By construction of (GN(R d ),C1) the exponential map is simply the identity. The universality of GN (Rd) is the following. Proposition 2.9 Let G be any Carnot group. There exists a Carnot group surjective morphism r : GN(R d ) G, where d is the dimension of the basis of G and N its depth. 1-A group law on Ben is said to be unimodular if the translations let the Lebesgue measure invariant.
43
SDE's and Carnot Groups
Proof.
Let U1, ..., Ud be a basis of the basis of G. Since GN(R d ) is free, there exists a unique Lie algebra surjective morphism dr : gN(R d ) --4 g such that dz(D i) = Ili, i = 1, ..., d. We can now define a surjective Carnot group morphism 7 : GN(R d ) —p by 7r ( e ) = echr(g), g E gN (Rd). Observe that it defines 7r in a unique way because in Carnot groups the exponential map is a diffeomorphism. 0 Notice that G N (R d ) is, by construction, endowed with the basis of vector fields (D 1 , ..., Dd). These vector fields agree at the origin with
( -ga iT „ k). To make our approach essentially frame independent, it is important to relate the horizontal lifts of the same Brownian motion with respect to two different basis. Proposition 2.10 Let cp :Rd --- Rd be a vector space isomorphism. Let us denote b* the horizontal lift of B in the group GN(Rd) with respect to the basis ((p(Di),•••,(P(Dd)). Then there exists a unique Carnot group isomorphism Tp :GN(R d ) --4 GN(R d), such that for every t > 0:
A' = Tv,(B:). Since g(Rd) is free, y can be extended in a unique way in a Lie algebra isomorphism gN(Rd) --4 2N(R d ). Notice that this extension commutes with the automorphisms be , c > O. Now, since GN(Rd) is simply connected and nilpotent, we can define a group automorphism Ty, : GN (Rd) ____> G r (Rd)) It(by the property
Proof.
Tv,(exp x) = exp(v(x)), x E
gN(R d).
It is then easily checked that for every t > 0:
Ê; =-_ Ty, (.13;`). and moreover that Tv, commutes with the dilations A c , e> O.
0
The maps Tv give the formulas for a change of basis in the theory of free Carnot groups. Notice that without the freeness assumption, Tv, may fail to exist.
Remark 2.9 Notice that the map Aut (Rd) is a group morphism.
—
p
(GN(Rd )),
y
Finally, we now come back the study of the stochastic differential equation (2.1) and assume that the Lie algebra Z = Lie(Vi, ..., Vd) is nilpotent of
44
An Introduction to the Geometry of Stochastic Flows
depth N, i.e. every commutator constructed from the Vi 's with length greater than N is O. Theorem 2.5
There exists a smooth map
F : Rn X GN(R( ) Rn such that, for xo C 1Rn , the solution (Xr) t >0 of the stochastic differential equation (2.1) can be written
Xf° = F(x0,B:), where (Bn t >0 is the lift of (Bt )t>0 in the group GN(Rd)
Proof.
This is a straightforward extension of Theorem 2.3. Indeed, as before, we notice that the Stratonovitch integration satisfies the usual change of variable formula, so that an iteration of Itô's formula shows, that the process [exp
(E E
Ai(B)tv,)1 (x0),
k.i /.(i i ,...,ik ) solves the equation (2.1). We deduce hence by pathwise uniqueness property that [ (N = exp
EE
A i (B) tvi
(X0)•
k.1 /.(i i ,.„,ik )
The definition of
GN(Rd )
shows that we can therefore write
XT° = F(xo, Br). Observe that this theorem implies the following inclusion Remark 2.10 of filtrations: for t > 0,
o-(Xf°) C a (AI(B)t,
I I l< N) ,
where cr(Xr) denotes the smallest a-algebra containing Xf° , and (111(B)t, I I l< N) denotes the smallest a-algebra containing all the functionals A1(B) twith I, word of length smaller than N. The above theorem shows the universal property of GN(R d ) in theory of nilpotent stochastic flows. Let us mention its straightforward counterpart in
SDE's and Carnot Groups
45
the theory of second order hypoelliptic operators (this property is implicitly pointed out in the seminal work [Rotschild and Stein (1976)1). Proposition 2.11
Let
L
d
_ E vi2 i=1
be a second order differential operator on Rn . Assume that the Lie algebra Z = Lie (V1 , ..., Vd ) which is generated by the vector fields Vi's admits a stratification Ei e • • • e
EN ,
with E l .-= span(14,...,Vd), 11 i1 == Ei+1 and [El, EN] = O. Then, there exists a submersion map 7r : Rni --4 Rn , with m = dim GN(Rd) such that for every smooth f : Rn -- R,
AGN (liRd) (f a 7r) = (cf) o 7r7 where AG N (Rd)
= Ei-1 Di?,
is the canonical sublaplacian on GN(R d ).
2.4 Pathwise approximation of solutions of SDEs We now come back to the general case where the Lie algebra Z -,-Lie (V1 , ..., Vd) is not nilpotent anymore. In that case it is, of course, not possible in full generality to represent (Xf° )t>0 in a deterministic way from the lift of the driving Brownian motion (B)>o in a finite dimensional Lie group. Nevertheless, the following result which is a consequence of a result due to [Casten (1993) ] (see also [Ben Arous (1989b)1) shows that it is still possible to provide good pathwise approximations with lifts in the free Carnot groups.
Theorem 2.6
Let N > 1. There exists a smooth map
F :111n x GN(Rd ) --4 Rn such that, for xo E Rn, the solution (X0 )>o of the SDE (2.1) can be written
Xj° = F(xo , Bi') + t4IRN(t), where:
An Introduction to the Geometry of Stochastic Flows
46
(1) (.13n to is the lift of (13t)t>0 in the group GN(Rd); (2) The remainder term RN(t) is bounded in probability when t
0.
Proof. Let N 1, and denote ri : Rn R, j = 1,... , n, the canonical projections defined by ri (x)=xj. By using iterations of ItE.'s formula and the scaling property of Brownian motion, we get
[exp
(E
E
Ai(B)tvi) (vi,••.Vik Iri)(xo )
k.i
k)
1 -1- 11:1 (t) odBI+t 4 Lk [0 ,t]
(XT ° ) + t Li2-1 ftiN (t) where the remainder terms t ---> 0.
It is possible to show that more precisely, 3 a, c > 0 such
Remark 2.11 that VA > c,
limp
t—oD
kiN and ft-iN are bounded in probability when
(
sup s Pit2 I RN (s)
Aa . At) < exp —
0<s
For further details, we refer to [Casten (1 993)].
2.5
An introduction to rough paths theory
As already pointed out, in general, the solution of the stochastic differential equation t
d X tx° = Xo
f vi(xfo) 0 dBia , 0 < t < T, i.1 °
is not a continuous function of (Bt ) 0
(f B13 si. d82 ./3 dB. ) o
47
SDE's and Carnot Groups
is continuous with respect to B as soon as the vector fields Vi 's commute. This problem of the continuity of the 116 map
BX, is solved in the setting of the rough paths theory which has recently been developed in [Lyons (1998)1 (see also the survey [Lejay (2004)1 and the book [Lyons and Qian (2002) ]) , Following N. Victoir, we believe that the free Carnot groups are a good framework for the rough paths theory. For p > let us denote 52PGN(Rd) the closure of the set of absolutely continuous horizontal paths x* : [0,7 ] GN(Rd) with respect to the distance in p-variation which is given by n-1 (5)9 (X *
y*) = sup 7r
Ed, (4,(4)- 1, yz+z(xLi)-1)
P
k=1
where the supremum is taken over all the subdivisions
= {0 < ti < • • < tn < T} and where dN denotes the Carnot-Carathéodory distance on the group GN(Rd). Consider now the map I which associates with an absolutely continuous path s: [0, T] Rd the absolutely continuous path y: [0, 7 ] Rd that solves the ordinary differential equation
El d
yt
=
t Vi(y s )dxl.
i.1
It is clear that there exists a unique map I* from the set of absolutely continuous horizontal paths [0, T[ GN(Rd ) onto the set of absolutely continuous horizontal paths [0, 7 ] GN(Rn ) which makes the following diagram commutative
x*
The fundamental theorem of Lyons is the following: Theorem 2.7 If N > [p], then in the topology of p-variation, there exists a continuous extension of 1* from f2PGN(R d ) onto SPGN(Rn)
48
An Introduction to the Geometry of Stochastic Flows
One of the important consequences of this theorem, is that it enables to define the notion of solution for ordinary differential equations which are driven by paths x whose regularity is only 1 -HOlder. From Lyons theorem, P to do so, we just have to find y E SPGN(Rd ) which satisfies 7ry = s, where 7r is the canonical projection GN(Rd ) .-- Rd (finding such a lift, which is not unique in general, is always possible from [Lyons and Victoir (2004)1). In this direction let us mention the work [Coutin and Qian (2002)1 who defined a notion of solution for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter greater than i. Observe that if p < 2, the Lyons fundamental theorem reduces to the classical theory of ordinary differential equations where the integrals are understood in Young's sense. The case p c (2,3) is of particular importance since it covers the theory of stochastic differential equations driven — E Holder conby Brownian paths (a Brownian path is almost surely tinuous, E > 0). This point of view on stochastic differential equations is for instance used in [Friz and Victoir (2003)1 and [Ledoux et al. (2002), 1 to obtain new and simplified proofs of Stroock-Varadhan support theorem and of the Freidlin-Wentzell theory.
Chapter 3
Hypoelliptic Flows
Chapter 1 has shown that, thanks to the Chen-Strichartz development the map
I : C ([0,7],Rd)
C ([0, T],
, B
X,
established by the stochastic differential equation d
XT° = X0 +
Ef vi(x:0)
o dErs ,
i=1
could be formally factorized in the following manner
I=FoH. The map
H : C ([0,7],Rd)
C ([0,71, exp
,
is an horizontal lift in exp (9d,) where gd,,, is the free Lie algebra with d generators. And F is simply a map exp (gd,) R. In Chapter 2 this factorization was made totally rigorous in the case where the vector fields Vi 's generate a nilpotent Lie algebra. In this chapter we now go one step further by studying the case where the Vi 's do not generate a nilpotent Lie algebra anymore but satisfy the so-called strong Hiirmander's condition. In that case, it shall be shown that the Gromov's notion of tangent space can be used to approximate locally the geometry of the It6's map by the geometry of a Carnot group. We start the chapter with the probabilistic proof of H6rmander's theorem. This proof is based on a stochastic calculus of variations that has been developed by Paul Malliavin (cf. iMalliavin (1978)1). The idea of the proof 49
50
An Introduction to the Geometry of Stochastic Flows
is to show that the Itô's map / is differentiable in a weak sense and then to show that, under 1-15rmander's conditions, the derivative is non degenerate. Then, we introduce the basic background of differential geometry that is needed to study locally the stochastic flow associated with a hypoelliptic differential system. This local study of the flow is used to derive the behaviour in small times of a hypoelliptic kernel on the diagonal. After, we provide a large class of examples of hypoelliptic operators which arise naturally in a manifold context. These are the horizontal Laplacians on principal bundles, whose hypoellipticity is related to non vanishing curvature conditions. The case of the horizontal diffusion over a Riemannian manifold is investigated in details. The local behaviour of the hypoelliptic heat kernel on the diagonal is then used to get informations on the spectrum of hypoelliptic operators defined on compact manifolds. Finally, we conclude the chapter by the study of stochastic flows associated with stochastic differential equations driven by loops.
3.1 Hypoelliptic operators and H8rmander's theorem
Consider a stochastic differential equation d
Xf° = so 4-
t
E x70) f vi( 0
t > 0,
(3.1)
j=1
where xo € Rn, VI, Vd are C' bounded vector fields on Rn and (B)>o is a d-dimensional standard Brownian motion. Let us first recall (see Appendix A) that for every so E Rn and every smooth function f : Rn IR which is compactly supported,
E (f(XP)) =
(e''
(X e ),
(3.2)
where
In this section we shall be interested in the existence of smooth densities for the random variables X ts°, t > 0, So E Ir. According to formula (3.2), this question is therefore equivalent to the question of the existence of a smooth transition kernel with respect to the Lebesgue measure for the operators
51
Hypoelliptic Flows
et.c. Let us recall the following definition which comes from functional analysis. Definition 3.1 A differential operator g defined on an open set 0 C is called hypoelliptic if, whenever u is a distribution on 0, u is a smooth function on any open set O' c 0 on which gu is smooth. It is possible to show that the existence of a smooth transition kernel with respect to the Lebesgue measure for et L is equivalent to the hypoellipticity of L. Therefore, our initial question about the existence of smooth densities for the random variables Xtz°, t > 0, so E Rn is reduced to the study of the hypoellipticity of L. the Lie algebra generated by the vector fields Vi 's Let us denote by and for p > 2, by el) the Lie subalgebra inductively defined by LP
{[X, Y], X E LP-1 ,Y E
el.
Moreover if a is a subset of Z, we denote
a(s)
{V(x), V E a}, x E Rn .
The celebrated II6rmander's theorem proved in [Hi5rmander (1967) 1 is the following: Theorem 3.1
Assume that for every so E Rn , Z(X0)
= Rn ,
then the operator L is hypoelliptic.
This is also a necessary condition when the V, 's are with Remark 3.1 analytic coefficients (see [Derridj 09711]). Remark 3.2 It is also possible to obtain hypoellipticity results for second order differential operators which can not be written as a sum of squares (see [Oleinic and Radkevic 09731 ]). The original proof of 116rmander was rather complicated and has been considerably simplified in [Kohn (1973)] using the theory of pseudo-differential operators. The probabilistic counterpart of the theorem, which is the existence of a smooth density for the random variable Xti°, t > 0, has first been pointed out in [Malliavin (1978) 1 where, in order the reprove the theorem under weaker assumptions, the author has developed a stochastic calculus of variations which is now known as the Malliavin calculus (see Appendix A and [Nualart (1995) 1 for a very complete exposition). After Malliavin's
52
An Introduction to the Geometry of Stochastic Flows
work, let us mention the work [Bismut (1981)1 in which the author uses interesting integration by parts formulae to prove the theorem. This is this probabilistic counterpart that we shall now prove. Theorem 3.2
Assume that at some x o C Rn we have
(3.3)
Ax0) = Rn ,
then, for any t > 0, the random variable XT° has a smooth density with respect to the Lebesgue measure of Rn, where (Xtx°)t>o denotes the solution of (3.1). We shall show that XT° admits a density with respect to the Lebesgue measure, by using the Malliavin covariance matrix r (see Appendix A) associated with XT°. Observe, that for notational convenience we take t = 1, but the proof is exactly the same for any t > O. We proceed in several steps. In a first step, we perform the computation of r, in a second step we show that r is almost surely invertible and finally, we shall qualitatively explain why for every p> 1, Proof.
E
(
I
1 1) detr P
<
Step 1. By definition, we have
1 r= (f (DsX1 DsX 3 )1Ra dS) 0
i
but from Theorem (A.7) of Appendix A,
DPC I = Jo aJo±,tVi(Xt), 3 —
where (J o
1, ...,d, 0 < t <1,
)>o is the first variation process defined by Jo—it
OX' t OX
Therefore,
f
1
joT = J0_,1 Ltv(xt)Tv(xoTj_ 0_,1 t dt where V denotes the n x d matrix (Vi—Vd).
53
Hypoelliptic Flows
Step 2. Since J01 is almost surely invertible, in order to show that
r is
invertible with probability one, it is enough to check that with probability one, the matrix
j(71,t v(xt ) Tv ( xt) T j(7_,1 t dt
C —
0
f
is invertible. For this, let us introduce the family of processes
Ct =
f J(7_,18 17(X 8 ) T V(X 8 )T J0-1 8 dt, t > O.
We denote the kernel of Ct by Kt C Rn and get a decreasing sequence of random subspaces of Rn. From Blumenthal zero-one law, the space V Ut>o Kt is deterministic with probability one. Let now y E V, and consider the stopping time
:= inf{s > 0, T yCs y > 0} Then 19 > 0 almost surely. From the expression of Ct , we get therefore that for 0 < s <
y By applying this at
(11 1 .9 14(X 8) = 0
1,
d.
s = 0, we obtain first TY
Vi(x0) = 0, =
Observe now that Jc7Lsvic
xs) =
where I denotes the stochastic flow associated with equation (3.1), and where (1,8*Vi denotes the pull-back action of ID on V. Therefore, according to the Itô's formula given in Proposition A.6 of Appendix A, we obtain that for t < 0, ft
d
E
j
=
that is
t
Ef T d
.1=1 °
y (;[ v,
V1) (SO)
o
dB ,39 = 0, i = 1, ..., d.
54
An Introduction to the Geometry of Stochastic Flows
Therefore, due to the uniqueness of the semimartingale decomposition, for 0 < s
By applying this at s = 0, we obtain then
T
r
117.1,Vii(X0) = 0, i, --= 1, ..., d.
An iteration of the Itô's formula given in Proposition A.6 of Appendix A shows then that, we actually have Ty
U(x0)
=
o,
u c z(xo),
so that y = 0. From this, we conclude that V = 0 and thus that C is invertible with probability one. Therefore, the random variable XT° admits a density with respect to the Lebesgue measure. Step 3. To show that the density of for every p> 1, E
(
XT° is smooth, we have to show that
1 ) I det r 1P
<+00.
Recall now that, with the previous notations
r=
T Jo.i.
By differentiating the stochastic differential equation (3.1) with respect to the initial condition, we obtain (non-autonomous linear equations for the processes (Jo.t)t>o and (.10-1 t ) t >0; see [Malliavin (1997)], pp. 240, for further details. These equations make easy the proof of the fact that for every p> 1,
1 <+00. I det Jo,i 12P)
E
Therefore, it remains to show that for every p> 1, E
( 1 ) <+00. I det C IP
The idea is to introduce again the process
Ct —
8 V(X.9 )TV(X.9 7.1 0-- 1 scit ,
f U
t > 0,
55
Hypoeiliptic Flows
and to control the smallest eigenvalue of Ct by showing that it can not be too small. It can be done by estimating the quantity d
sup P ( Tv Ct v < livii= 1
E) ---=
sup P livii=1
3=1
f
t (Tv Jo-L sVi(X5))
2
ds < E
0
with the help of the so-called Norris lemma. This lemma roughly says that, when the quadratic variation or the bounded variation part of a continuous semimartingale is large, then the semimartingale can only be small with an exponentially small probability. We do not go into details on this part, since it is very technical, and we believe that the most interesting aspect of the probabilistic proof of Hiirmander's theorem is to see how Z(x 0 ) appears in the Malliavin matrix. For further details on the Norris lemma and its use, we refer to [Nualart (1995)1 pp. 116-123. Remark 3.3 Therefore, under the assumption (3.3), there exists a smooth function p on (0, +oo) x Rn such that for any smooth function f :1R 1' IR which is compactly supported (Pt f)(xo) = f p(t, y) f (y)dy , where (Pt ) t>0 denotes the transition function of the strong Markov process (X°) >0 . Moreover, the function p satisfies Kolmogorov's forward equation:
where r* is the adjoint of L.
Remark 3.4 There is a special case of Harmander's theorem, which covers very many of the applications one meets in practice; this is the case where L is elliptic at x o , that is (Vi (x 0 ),...,Vd(x0)) are enough to span Ir. This special case had earlier been obtained by Hermann Weyl. Remark 3.5 Let us mention that it possible to extend Hiirmander's theorem to stochastic differential equations valued in a separable Hilbert space of the type d
dXt = (AX t + a(X t ))dt +
Ei=i ai (X t )d.13;,
56
An Introduction to the Geometry of Stochastic Flows
where A generates a strongly continuous group; for further details we refer to [Baudoin and Teichmann (2003)].
One can extend slightly H6rmander's theorem. Theorem 3.3 Let s 0 E Rn and N E N. If £N+1(x0) •-=-Rn, then for any t > 0, the random variable
(Xr , 14) has a smooth density with respect to the Lebesgue measure of IV x GN(Rd), where (XT°) t >0 is the solution of (3.1) with initial condition x o and (Bn t >0 the lift of (B t ) t>0 in the free Carnot group GN(R d ).
With a slight abuse of notation, we still denote Vi (resp. Di) Proof. the extension of Vi (resp. Di ) to the space Rn X GN (Rd). The process (XP,Bn t >0 is easily seen to be a diffusion process in Rn X GN(Rd) with infinitesimal generator
Thus, to prove the theorem, it is enough to check the H6rmander's condition for this operator at the point (xo, 0). Now, notice that for 1 < i , j < n, [Vi , Di] = 0, so that
Lie(Vi + Di,
Vn
Dn) (X0 , 0)
£N+1(x
e )
liNoa
d ),
N (Rd)) is step N nilpotent. We denoted Lie(Vi D1, •••, Vn -}- Dn) because riN the Lie algebra generated by (V1 + DI, •.., Vn +DO. The conclusion follows 0 readily. Example 3.1 For N = 0, we have Go(Rd ) = {O} and Theorem 3.3 is the classical H6rmander's theorem. Example 3.2 For N = 1, we have G I (Rd) Rd and Theorem 3.3 gives a sufficent condition for the existence of a smooth density for the variable
(X' ° , B e ). Example 3.3 For N = 2, we have G2 (Rd ) Rd x R 1> and Theorem 3.3 gives a sufficent condition for the existence of a smooth density for the variable
(.1q° , Bt, ABO•
Hypoelliptic Flows
57
where AB
3.2
= t (1 f t Ec1B1 — Bis c1B) 2
0
Sub-Riemannian geometry
The goal of this section is to present the basic results of sub-Riemannian geometry, which is the natural geometry associated with hypoelliptic operators. We essentially aim at generalizing the geometry of the Carnot Groups that has been presented in the previous chapter. From now on, we consider d vector fields Vi : Rn which are Coe bounded and shall always assume that the following assumption is satisfied. Strong 118rmander's Condition: For every
x E Rn, we have:
span{V/(x),/ E Uk>i{l, d} k } = We recall that if I = (i1, ik) C {1,-..,d} k is a word, we denote by VI the commutator defined by ----
Let us mention that in sub-Riemannian litterature, the strong Fkirmander's condition is more often called the Chow's condition or bracket generating's condition. Definition 3.2 The set of linear combinations with smooth coefficients of the vector fields VI, Vd is called the differential system (or sheaf) generated by these vector fields. It shall be denoted in the sequel D. With this definition, we can roughly say that the goal of the subRiemannian geometry is to study the intrinsic object of the distribution, such the symmetry groups, the local differential invariants, etc... Observe that D is naturally endowed with a structure of C(Rn, R)module. For x E Rn, we denote
D(x) = {X(x), X
E D }.
If the integer dim D(x) does not depend on x, then D is said to be a distribution. Observe that the Lie bracket of two distributions is not necessarily a distribution, so that we really have to work with differential systems. In the
58
An Introduction to the Geometry of Stochastic Flows
case where dim D(x) is constant and equals the dimension of the ambient space, that is n, the distribution is said to be elliptic. The study of elliptic distributions is exactly the Riemannian geometry. The sub-Riemannian geometry is much more rough than the Riemannian one. Some of the major differences between the two geometries are the following: (1) In general, it does not exist a canonical connection like the Levi-Civita connection of Riemannian geometry; (2) The Hausdorff dimension is greater than the manifold dimension (see Chapter 2, in the case of the Carnot groups); (3) The exponential map is never a local diffeomorphism in a neighborhood of the point at which it is based (see [Rayner (1967)1); (4) The space of paths tangent to the differential system and joining two fixed points can have singularities (see [Montgomery (2002)1). Before we go into the heart of the subject, it is maybe useful to provide some examples. Example 3.4 Let G be a Carnot group and consider for D the left invariant differential system which is generated by the basis of G. Then, D satisfies the strong Iliirmander's condition and the sub-Riemannian geometry associated with D is precisely the geometry that has been studied in Chapter 2. Let M be a manifold of dimension d. Assume that there exists on M a family of vector fields (V1 , ..., Vd) such that for every x E M, (Vi(s),...,Vd(x)) is a basis of the tangent space at s . Let us denote by D the differential system generated by (V1 , ..., Vd) (it is actually a distribution). Then, D satisfies the strong 116rmander's condition and the subRiemannian geometry associated with D is the Riemannian geometry on M which is induced by the moving frame (Vi (x), ...,
Example 3.5
Example 3.6 Let us consider the Lie group SO(3), i.e. the group of 3 x 3, real, orthogonal matrices of determinant 1. Its Lie algebra 50(3) consists of 3 x 3, real, skew-adjoint matrices of trace 0. A basis of 50(3) is formed by
0 VI = (-1 0
1 0 00 ) , V2 0 0
=
/ 00 ( 0 0 0 -1
0 1 0
),
V3
=(
0 0
\-i.
0 1 ) 00 00
59
Hypoelliptic Flows
Observe that the following commutation relations hold [VI , V2] = V3, [V2, V3] =
[V3,141 = V2)
so that the differential system D which is generated by V1 and V2 satisfies the strong 116rmander's condition. The group SO(3) can be seen as the orthonormal frame bundle of the unit sphere V and, via this identification, D is generated by the horizontal lifts of vector fields on V. Therefore, in a way, the sub-Riemannian geometry associated with D is the geometry of the holonomy of S2 . This example shall be widely generalized at the end of this chapter; actually many interesting examples of sub-Riemannian geometries arise from principal bundles. Example 3.7 Let us consider the Lie group SU(2), i.e. the group of 2 x 2, complex, unitary matrices of determinant 1. Its Lie algebra su(2) consists of 2 x 2, complex, skew-adjoint matrices of trace 0. A basis of su(2) is formed by 1( i 0 = -20 )
7
1 ( 0 i\ 10
V2
1(0 '
V3
Note the commutation relations
[171 , v2 ] =
V3, [V2, V31 =
[V3,1/11 = V2,
so that the differential system D which is generated by V1 and V2 satisfies the strong Iiiirmander's condition. Let us mention that there exists an explicit homomorphism SU(2) SO(3) which exhibits SU(2) as a double cover of SO(3), so that this example is actually a consequence of the previous one. Definition 3.3 An absolutely continuous path e: [0,11 Rn is said to be horizontal (with respect to D) if for almost every s E [0,1],
(s) E D(c(s)). We can now give the Chow's theorem in its full generality. Theorem 3.4 Let (x, y) E R x R. There exists at least one absolutely continuous horizontal curve C: [0,1] Rn such that c(0) = x and c(1) = y. Unlike the case of a Carnot group, the proof of this theorem is far to be easy (see the Chapter. 2 of [Montgomery (2002)] for a complete and detailed proof). Thanks to this theorem, it is possible like we did it in the case of
60
An Introduction to the Geometry of Stochastic Flows
the Carnot groups, to define a distance on Rn. From the definition of D, any absolutely continuous and horizontal curve c: [0, 11 11In satisfies d
e(s).Ec:(s)v,(c(s)) i=1
for some absolutely continuous curves ci : [0, 1] defined by
1( c)
= fo
R. The length of c is
d
Ecii (s)2ds.
The Carnot-Carathéodory distance between two points a and b in Rn is now defined as being d(a, b) = inf 1(c), where the infimum is taken over the set of absolutely continuous horizontal curves c: [0, 11 Rn such that
c(0) = a, c(1) = b. It can be shown, by proving a generalization in this framework of the ballbox theorem seen in Chapter 2, that the topology given by the CarnotCarathéodory distance d is the usual Euclidean topology of 1Rn . A horizontal curve with length d(a , b) is called a sub-Riemannian geodesic joining a and b. With Chow's theorem in mind, we address now the problem of existence of geodesics, whose Riemannian analogue is the classical Hopf-Rinow theorem. Theorem 3.5
Any two points of IV can be joined by a geodesic.
There is a major difference between Riemannian geodesics and subRiemannian ones: unlike the two step Carnot group case, it may happen that a geodesic is not smooth.
3.3
The tangent space to a hypoelliptic diffusion
First, we have to introduce some concepts of differential geometry. The set of linear combinations with smooth coefficients of the vector fields VI , Vd is called the differential system (or sheaf) generated by these vector fields.
61
Hypoelliptic Flows
It shall be denoted in the sequel D. Notice that D is naturally endowed with a structure of Cao (Rn, R)-module. For x c Rn, we denote
D(x)
IX (x), X E D}.
If the integer dim D(x) does not depend on x, then D is said to be a distribution. Observe that the Lie bracket of two distributions is not necessarily a distribution, so that we really have to work with differential systems. The Lie brackets of vector fields in V generates a flag of differential systems,
D C D2 c • • • C D k c • • • , where DC is recursively defined by the formula Dk
_ pk-1 [D, V " '].
As a module, Dk is generated by the set of vector fields VI , where I describes the set of words with length k. Moreover, due to Jacobi identity, we have [Di , Di] C Di +.1 . This flag is called the canonical flag associated with the differential system D. 116rmander's strong condition, which we supposed to hold, states that for each x E 11/n , there is a smallest integer r(x) such that Dr(x) = Rn . This integer is called the degree of non holonomy at x. Notice that r is upper continuous function, that is r(y) < r(x) for y near x. For each x E Rn, the canonical flag induces a flag of vector subspaces,
D(x) C D2 (X) C • • C Dr(x) (X) = R. The integer list (dim Dk(x))1
Af(D) =
ED
•••
ED Vk ED
• .
The Lie bracket of vector fields induces a bilinear map on AT (D) which respects the grading: [Vi , Vi] C Vi+j . Actually, .A/(V) inherits the structure of a sheaf of Lie algebras. Moreover, if x is a regular point of D, then this bracket induces a r(x)-step nilpotent graded Lie algebra structure on
62
An Introduction to the Geometry of Stochastic Flows
(D)(x). Observe that the dimension of .Af(D)(x) is equal to n and that from the definition, (VI (x), Vd (X)) Lie generates .111.(D)(x). Definition 3.4 If s is a regular point of D, the v(s) -step nilpotent graded Lie algebra Af(D)(x) is called the nilpotentization of D at x. This Lie algebra is the Lie algebra of a unique Carnot group which shall be denoted Gr(D)(x) and called the tangent space to D at x. Remark 3.6 of Gr (r)(Rd ).
At a regular point x, Gr(D)(x) can be seen as a quotient
Remark 3.7
The notation Gr is for Grornov.
Definition 3.5 If x is a regular point of D, we say that x is a normal point of D if there exists a neighborhood U of x such that: (1) for every y E U, y is a regular point of D; (2) for every y E U, there exists a Carnot algebra isomorphism
Af(D)(x)
Af(D)(Y),
such that 0(Vi (x)) = Vi (y), i = 1,..., d. Remark 3.8 To say that x is a normal point of D is equivalent to say that any relation of the type r(x)
a I VI (x), E E I.(ii,•••,ik)
al E R,
k=1
can be extended in a smooth way to a relation, r(x)
EE
a 1(y)Vi(y),
k=1
More roughly speaking, if x is a normal point of D the isomorphism class of Af(D)(x) is constant in a neighborhood of x. Let us mention that in typical cases there exist regular points which are not normal (see [Gershkovich and Vershik (1988)1). To illustrate these notions, we give now the nilpotentization and the tangent space in the examples already seen. Example 3.8 Let G be a Carnot group with Lie algebra j and consider for D the left invariant differential system which is generated by the basis
63
Hypoelliptic Flows
of G. Then, each point is normal and it is immediate
that
for every x E G,
Af(D)(x) =
Gr(D)(x) = G. Example 3.9 Let M be a manifold of dimension d. Assume that there exists on M a family of vector fields Vd) such that for every s E Vd(x)) is a basis of the tangent space at s and denote by D the differential system generated by (V1 , Vd). Then, each point is normal and for every x E M, Af(D)(x) = R d,
Gr(D)(x) = Rd . Example 3.10 Let us consider on SO(3) the left invariant differential system D generated by (
=
0 10 1 0 0 0 0 0
—
0 0 0 and V2 = 0 0 1 o —i 0
\
In that case, it is easily checked that each point is normal and that for every x E SO(3),
Af(D)(x) = Gr(D)(x) = G2 (R2 ). Example 3.11 Let us consider on SU(2) the left invariant distribution D which is generated by T
V1=
Each
iii
_ i ) and V2 =
1 (0
point is normal and for every x E SU(2), Af(D)(x) =
Gr(D)(x) = G2 (R2) •
64
An Introduction to the Geometry of Stochastic Flows
A really striking fact is that the tangent space Gr(D)(x) at a regular point is not only a differential invariant but also a purely metric invariant. Actually Gromov discovered that it is possible, in a very general way, to define a notion of tangent space to an abstract metric space. This point of view is widely developed in [Gromov (1996) 1 and [Gromov (1999) ] , and is the starting point of the so-called metric geometry. The Gromov-Hausdorff distance between two metric spaces M1 and M2 is defined as follows: SGH(Mi, M2) is the infimum of real numbers p for which there exists isometric embeddings of M1 and M2 in a same metric space M3, say i l : M1 --- M3 and M2 --4 M3, such that the Hausdorff distance of ii(Mi ) and i2 (M2) as subsets of M3 is lower than p. The important property of the GromovHausdorff distance is the following theorem. Suppose M1 and M2 are complete metric spaces with 8GH(M1, M2) = O. Then M1 and M2 are isometric.
Theorem 3.6
Thanks to this distance, we have now a convenient notion of limit of a sequence metric spaces. A sequence of pointed metric spaces (Mn , x n ) is said to Gromov-Hausdorff converge to the pointed metric space (M, x) if for any positive R Definition 3.6
lim 8GH (BMn (x, R) ,BM (x,R)) = 0,
n—).+00
where BK., (x, R) is the open ball centered at x n with radius that case we shall write
R in Mn . In
iim (M n , X n ) = (M, x) . n—)-Foo
If M is a metric space and A > 0, we denote A - M the new metric space obtained by multiplying all distances by A. Definition 3.7 Hausdorff limit
Let M be a metric space and xo
E
M. If the Gromov-
lim (n - M, X0)
n.-i-oo
exists, then this limit is called the tangent space to M at xo. We have the following theorem due to Mitchell [Mitchell (1985)] (see also
[Pansu (1989)1): Proposition 3.1 Let xo E 1Rn be a regular point of D, then the tangent space at xo in Gromov-Hausdorff 's sense exists and is equal to Gr(D)(xo)•
65
Hypoelliptic Flows
Actually, even if so is not a regular point of D, the tangent space in GromovHausdorff sense exists. Therefore, from this theorem, it is possible to define Gr(D)(xo) at any point of D. Nevertheless, if xo is not a regular point, then Gr(D)(x0) is not a Lie group (see [Bella:iche (1996)1). Example 3.12
Let us consider in R2 , the two vector fields
and
V2 -- =
r0y - .
X o
These vector fields span R2 everywhere, except along the line x = 0, where adding (Vi,
V21 =
a
is needed. So, the distribution D generated by V1 and V2 satisfies the strong Heirmander's condition. The sub-Riemannian geometry associated with D is called the geometry of the Grusin plane. In that case, for every (x, y) E R2 ,
Gr(7))(x,y) . R2 , if x
0,
whereas,
Gr(D)(x, y) = G 2 (R2 )/ exp(RV2 ), if x = O. Before we turn to applications in the theory of hypoelliptic diffusions, we still have to generalize the notion of Pansu's derivative (see Chapter 2, Section 2.3.) in our context of general sub-Riemannian manifolds. This generalization is due to [Margulis and Mostow (1995)1, we shall only hint their notion of Pansu's derivative, since the rigorous construction is quite technical but rather clear from an intuitive point of view. Let G be a Carnot group and
F : G --- Rn be a map. Then F is said to be Pansu differentiable at g E G if there exists a Carnot group morphism
dpF(g) : G --- Gr(D)(F(9)))
66
An Introduction to the Geometry of Stochastic Flows
such that for small t
F(gh) g F(g) (AGr(D)(F(g)) dpF(g)(h)) , h c G, where AG (resp. AGr(D)(F(g)) ) are the canonical dilations on the Carnot group G (resp. Gr(D)(F(g))). This approximation formula should be understood in the same way than the usual approximation formula
F(x + h)
F(x) dF(x)(h),
for a map F defined between two manifolds. We now apply all these new notions to study the stochastic differential equation
d
x ° = So i=1
fo
t
(x.fo)
dBis .
(3.4)
Let x E Rn be a regular point of D. Let (XT) t >0 denote the solution of (3.4) with initial condition x. There exists a map
Theorem 3.7
F: Gr(s)(R d)
Rn
such that
= F(.14)
t 7 2 "R(t),
where:
(1) (Bn t >0 is the lift of (Bt)t>0 in the group GN(R d); (2) the remainder term R(t) is bounded in probability when t
O.
Moreover the map F is Pansu differentiable at OGr(x) and the Pansu's derivative
dpF(OG, (x) ) : Gr(i) (Rd)
Gr(D)(x)
is a surjective Carnot group morphism. Proof.
From Theorem 2.6 of Chapter 2, we have
= F(Bn t r. " R(t), t > 0,
Hypoelliptic Flows
67
where the remainder term R is bounded in probability when t --4
0,
and
(r(x)
F(Bn =
[exp
EE
Ai(B)Vi)]
(X).
k=1 1=(ii,..•,ik)
Let now g E G and y: [0,11 --4 Rd an absolutely continuous path whose lift y* : [0,1 ] --4 Gr(s) (Rd) satisfies
A and
YI = g. We have
((z) F (A tG r(x )(ad) g) =
Etk E
Exp
k=1
A i (y ) 1 V1 )]
(X).
I=(ii,.••,ik)
Therefore, in small times
r(.)
t
"F (A G r( r )(Rd ) g) x +
E tk E k=1
A/(01v1(x)",
I=(ii,...,ik)
0
which leads to the expected result. We have a stronger approximation result in the normal case.
Let x E Rn be a normal point of D. Let (Xn t >0 denote the solution of (3.4) with initial condition x. There exist a surjective Carnot group morphism Theorem 3.8
rx : Gr(x)
(Rd )
--4
Gr(D)(x)
and a local diffeomorphism U C Gr(D)(x) --- R n such that
Xf = Os (7rxBn + t r121---" R(t), 0 < t < T where:
(1) U is an open neighborhood of the identity element of Gr(D)(x); (2) B* is the lift of B in the free Carnot group Gr(x)(Rd);
68
An Introduction to the Geometry of Stochastic Flows
(3) T is an almost surely positive stopping time; () R is bounded in probability when t O. From Theorem 2.6 of Chapter 2, we have
Proof.
(r(x) =
EE
[exp
A,(B) t
v,
(s)+tTL'El:ER(t), t > 0,
k=1 I -=(ii,•••,ik)
where the remainder term R is bounded in probability when t since x is a normal point of D, we can write ( r(x)
Exp
E E
A,(B) t
v,)] (x) = ozujo,
O. Now,
t
k=1 I=(ii,...,ik)
where
U C Gr(D)(x) Rn is a local diffeomorphism; (2) .6 is the lift of B in the Carnot group Gr(D)(x), with respect to the family (V1 (X), ...,Vd(X)) (recall that by construction, this family Lie generates Af(D)(s)) ; (3) T is an almost surely positive stopping time. (1)
:
Now, since gr(z)(Rd) is free, there exists a unique Lie algebra surjective homomorphism c : gr (z)(Rd ) .11T(P)( 3 ) such that c(D) = Vi (x). Since Carnot groups are simply connected nilpotent groups for which the exponential map is a diffeomorphism, there exists a unique Carnot group morphism 7r1 : Gr (x) Gr(D)(x) such that dirt = a,. We have 7r,(Bn Êt which concludes the proof.
Remark 3.9 Gr(D)(x).
Observe that 7rx .13; is a lift of B in the Carnot group
Remark 3.10 Observe that the map Rn is : U C Gr(D)(x) bi-Lipschitz with respect to the respective sub-Riemannian distances of
Gr(D)(x) and Rn . Remark 3.11 We stress that theorem (3.8) is not true in general if x is not a normal point of D. Indeed, let us assume that the nilpotentization .11(D)(x) is not constant in a neighborhood of x and that there exists a bi-Lipschitz map qpi C Gr(D)(x) IV. In that case an extension of Pansu-Radernacher's theorem (see previous chapter) due to IMargulis and Mostow (1 995)1 would imply that 01 is almost everywhere Pansu differentiable and the derivatives would provide Carnot group morphisms between
:u
69
Hypoelliptic Flows
groups which are not isomorphic. This argument actually shows that there is no inversion local theorem with respect to the Pansu 's derivative. A case of particular importance which is covered by the previous theorem is the elliptic case which corresponds to the case d = n and (14(x), ..., V,,(x)) is a basis of Rn for any x E Ir. In that case, recall that every point is normal and we obtain
Corollary 3.1 Let s E Rn and let (Xn t>0 denote the solution of (34) with initial condition x. There exists a local diffeomorphism
such that
Xf = 11) x (B t ) + tR(t), 0 < t < T where:
(1) U is an open neighborhood of O; (2) T is an almost surely non negative stopping time; (3) R is bounded in probability when t --- 0. Another immediate corollary of Theorem 3.8 is the behaviour in small times of a hypoelliptic heat kernel on the diagonal (see [Ben Arous (1989a)], [Léandre (1992)1 and [Takanobu (1988)]). Corollary 3.2 Let x be a regular point of D. Let pt, t > 0, denote the density of Xs with respect to the Lebesgue measure. We have, Pt(x) "-'t-00
C(x) t
D(x) 2
1
where C(x) is a non negative constant and D(x) the Hausdorff dimension of the tangent space Gr(D)(x). Proof.
We use Theorem 3.7 to write XT = F(.14) + t r-x4 R ( t ) , :LI
where: (1) (Bnt>0 is the lift of (B t )t>0 in the group (2) R(t) is bounded in probability when t --4 0;
70
An Introduction to the Geometry of Stochastic Flows
(3) the map
F : Rn X Gr (s)(Rd) Rn is Pansu differentiable at the origin with a surjective Pansu's derivative. Now, we have
(F(137 )) t>0
=1"
(
F
G r(
,
x )(Rd)
>
t 0
and
Gr(x) (Ile) n„")
A Gr(D)(x) dp RoG
)( B *) ,
in small times. Since dpF(OG,(x)(Rd) ) is surjective, the random variable
dpRO G
, (x) (liz
admits a density with respect to the proof.
3.4
a ) )(
BI)
Lebesgue measure. This concludes the
Horizontal diffusions
In this section, we provide interesting examples of hypoelliptic diffusions which arise naturally in Riemannian geometry. We shall assume basic knowledge in Riemannian geometry and stochastic processes in this setting as can be found in the Appendices A and B. We shall also assume some very basic knowledge of the theory of principal bundles; for further details, we refer to the Chapter 2 of [Kobayashi and Nomizu (1996)]. Let (M, g) be a d dimensional connected compact smooth Riemannian manifold. That is, the manifold M is endowed with a (0,2) definite positive tensor g. This non-degenerate metric tensor can e.g. stem from an elliptic differential system (V1 , Vd). That is, VI , Vd are smooth vector fields on M such that for every xo E M, (V1 (x0), Vd(x0)) is a basis of the tangent space to M at xo . In that case g is defined at xo by the condition that (V1 (xo), Vd(X0)) is an orthonormal basis. The tangent bundle to M is denoted TM and T,,M is the tangent space at m E M: we have hence TM =
Hypoe/liptic Flows
71
An orthonormal frame at m E M is a linear isomorphism u : Rd --> TTJA such that for every X, y E Rd , gm (u(x), u(y)) --= (x , y) Ri d ,
where (x, y) Rd denotes the usual scalar product in Rd . The set of orthonormal frames at m shall be denoted 0 (M) m . Observe that the Lie group Od (R) of d x d orthogonal matrices acts naturally on 0 (M),,, by u --4 u 0 g, u E 0 (M) m , g E Od (R). Set now
0 ( m) = U 0 (m)„, . mEM
This set is called the orthonormal frame bundle of M. It is easily checked that it can be endowed with a differentiable manifold structure with dimension d(1`- 1-1)-. that makes the canonical projection 7r : 0 (M) --4 M a smooth map. We summarize the above properties by saying that (0 (M) , M, Od (R)) is a principal bundle on M with structure group the group Od (R). Working in this bundle aims at making equivariant the objects we are dealing with. As in Appendix B, it is possible to show the existence of a unique metric, torsion free, connection on M. That is, if we denote by V(M) the set of smooth vector fields on M, there is a unique map
V : V(M) x V(M) --4 V(M) such that for any U, V, WE V(M) and any smooth f,g : M --4 R:
(1) (2) (3) (4) (5)
Vfu+ gvW = fVuW + gVvW; Vu(V + W)-= VV + VuW; Vu(f V) = f VuV --1-U(f)V; VuV — VvU = [U,V]; U(g(V,W)) = g(VuV,W)+ XV, VW).
A vector field X along a smooth curve (ct)t>0 on M is said to be parallel along the curve if we always have
V'e, X = O. A smooth curve (ut ) t>0 in 0(M) is called horizontal if for every x E Rd, the vector field u(x) is parallel along the curve ru. A vector X E Tua 0(M) is called horizontal if u is horizontal. The space of horizontal vectors at
72
An Introduction to the Geometry of Stochastic Flows
uo is denoted 7-1„„0(M); it is a space of dimension d. We have the direct decomposition T u0 0(M) = 7-1.0 0(M) ED Vuo 0 (M) where V 0 0(M) denotes the space of vectors tangent to the fiber 0(M),0 . The canonical projection R- induces an isomorphism nu 0 0(M)
Truo M,
and for each X E Tm M and a frame uo at m, there is a unique horizontal vector X*, the horizontal lift of X from 110 such that 7r,,X* = X. For each s E Rd we can define a horizontal vector field If by the property that at each point u E 0(M), H(u) is the horizontal lift of u(x) from u. Definition 3.8 Let (e i , ed) be the canonical basis of Rd . The fundamental horizontal vector fields are given by
Iii = He .. Observe that the differential system on 0 (M) generated by the horizontal vector fields HI, ..., Hd, is nothing else than the horizontal distribution for the Levi-Civita connection on M, that is 7-10(M). The Bochner's horizontal Laplacian is by definition the sum of squares operator given by d
AO(M) = i=1
The fundamental property of the Bochner's horizontal Laplacian is that it is the lift of the Laplacian operator Am of M. That is, for every smooth R, f MI
O(M)( f o 7r) = (Am f) o As a direct consequence of this, it follows that the solution of the stochastic differential equation d
t
t > 0, x=1
is a Rd
—t ) where U0 E 0 (M) and (B
—
valued standard Brownian motion,
t>0
is such that the M-valued process (r(Bn) t>0 is a Brownian motion. That is,
Hypoelliptic Flows
73
(r(Bt*)) t>0 is a Markov process with generator lAm. The process (Bn t>0 is called a horizontal Brownian motion. Conversely, it can be shown that any Brownian motion (Bt ) t >0 on M can be written (r(BI)) t>0 where (Bn t>0 is a horizontal Brownian motion. The process (Bp) t>0 is -Chen called the lTorizontal lift of (Bt)t>o • The linear Brownian motion (fit )
which drives the stochastic differential equation t>0 driven by (N) t>c, is called the anti-development of (Bt)t>o. This passage through the orthonormal bundle is the classical EelsElworthy-Malliavin's approach to Riemannian Brownian motions (see e.g. [Emery (1989)], [Hsu (2002)1, or Part V of [Malliavin (1997) 1 ). The advantage of this construction of Riemannian Brownian motions is that, firstly it is intrinsic, and secondly it provides a pathwise construction obtained by solving a globally defined stochastic differential equation. The bijective map —
Bt) t>0
—
(B t) t >0
is called the Itô's map of the manifold. Also observe that this construction proves that the natural filtration of a Riemannian Brownian motion is the same than the natural filtration of a linear d-dimensional Brownian motion; a fact, which a priori was not obvious from the original definition of the Brownian motions on a manifold (see Appendix A). Let us now see how the Levi-Civita connection V manifests itself on the bundle (0 (M) , M, Od (R)). Let us denote od(R) the Lie algebra of Od (R), that is the space of d X d skew symmetric matrices. For every M E o d (R), we can define a vertical vector field Vm on 0 (M) by
(Vm F)(u) lim F t,c1
(uet") — F(u)
R. Observe that the map M Vm(u) with u E 0 (M) and F: 0 (M) is an isomorphism from od(R) onto the set V0(M) of vertical vector fields at u. Then, the connection form on 0 (M) (often called the Ehresmann connection), is the unique skew-symmetric matrix w of one forms on 0 (M) such that:
(1) w(X) = 0 if and only if X E 7-t0(M); (2) V(x) = X if and only if X E VO(M). Associated with this Ehresmann connection on 0 (M) , which corresponds to the Levi-Civita connection on M we have two Cartan's structural equations
74
An Introduction to the Geometry of Stochastic Flows
(see Appendix B, Section B.8 for the two Cartan's structural equations on M). The first structural equation reads
de + where
A w = 0,
e is the so-called tautological one form on 0 (M) e(x)(u). u'rX, X E Tu 0 (M).
defined by:
The second structural equation reads cha-l-coAw= and can be taken as a definition of the curvature form O. It can be shown that
52(X, Y)(u) =
rY)u, X, Y E Tu O(M) ,
where R denotes the Riemannian curvature tensor on M. For u E 0 (M), let us now consider the map
Alfu : 0d(R)
od(R)
such that
Tu(m).
E nr(Hi ,B-3 )(u)m,,3 ) 1
. 1
We have the following theorem (compare to Chernyakov's theorem, see [Gershkovich and Vershik (1994)] pp. 22): Theorem 3.9
Assume that for every u E
(M), the application 'l'a is
an isomorphism, then: (1) the horizontal distribution 7-(0(M) satisfies the strong H6rmander's condition; (2) every u E 0 (M) is normal and Gr(?-(0(M))(u) =G 2 (Rd). From the Cartan's formula, we have
Proof.
e([rii ,H3 1) .Hie(Hi) -H3 e(H)-de(Hi , I15 ), e
is the tautological one-form on 0(M). Now, from the first strucwhere tural equation, we have
de=
-enw
75
Hypoelliptic Flows
where co is the Ehresmann connection form. This implies,
eaHi , Hip = 0.
Thus, the commutator of two fundamental vector fields is vertical. Using the second structure equation
dui =
—0)
A co +52,
where SI is the curvature form, we obtain in a similar way with Cartan's formula, cd([112, 113])
= —t2 (11i,
Thus, [Ili
=
VS-2(Hi ,Hj) •
Since the map M Vm (u) is an isomorphism, for every u E 0(M) the family (Hi (u), [Hi , Hd(u)) is a basis of Tu 0 (M). Therefore the horizontal distribution 'NO(M) satisfies the strong H6rmander's condition. The fact every u E (M) is normal and Gr(7-t0(M))(u) = G2(1Rd) stems from the fact the growth vector is maximal and constant on 0 (M). The above computations have indeed shown that it is always (d,
Remark 3.12
In dimension d = 2, the assumption that for every u, the application AP u is non degenerate is equivalent to the fact that the Gauss curvature of M never vanishes.
Remark 3.13
If we assume that (M, g) is an oriented manifold the assumption that at u E 0(M), the application Alf u is an isomorphism is equivalent to the fact that the holonomy group at r(u) is SOd(R).
By applying Theorem 3.8 and using the identification G2 (Rd) d(d— 1), R 2 , we obtain therefore
Corollary 3.3
Rd X
Assume that for every u E 0 (M), the application Alf u is an isomorphism. Let (Bt)t>0 be a Brownian motion on M. Then, there exist a local submersion
:
U
c Rd x R d(d2-1)
0(M)
76
An Introduction to the Geometry of Stochastic Flows
and a standard linear Brownian motion (i3 )>o on R d , such that for any smooth function f: M R, f (B t )
= f (1P (0t, 0;(/ (
ft O
430
)-I-t 53 R(t, f), 0 < t
1
where:
(1) U is an open neighborhood of 10 in Rd x R 2 • (2) T is an almost surely non negative stopping time; (9) R is bounded in probability when t O. Proof. Indeed, let (Bt ) t>0 be a Brownian motion on M. Let us denote (Bnt>0 the horizontal lift of (B)>o to 0(M) and (0t)t>0 the antidevelopment of (Bt)>o. By applying Theorem 3.8 to (Bflt>o, we obtain the expected result.
Under the non-degeneracy assumption, we can moreover give an expression for the horizontal heat kernel in small times. On 0 (M), let us consider the measure p, which reads in local bundle trivializations as the product of the Riemannian measure on M and of the normalized Haar measure of Od(R).
Corollary 3.4
Assume that for every u E 0 (M), the application Wu is an isomorphism. Then, if pi' denotes the density with respect to pc of a horizontal Brownian motion on 0 (M) started at u, we have
Pt(u)
I det
,2 I t -r
where C> 0 is a universal constant which does not depend on M. Proof.
It follows readily from the approximation
(Bn = f
exp ( E HA H1 f+ —21 i.1
t
E [4,, Hk] f f3;d0.39 — 00 1<j
(u)
0
+ t 4 R(t),
which stems from Theorem 3.8 (here f : 0(M) --4 and 0 the anti-development of B).
R is a smooth function D
We can extend without diffi culties the previous discussion to the case of general principal bundles. Let M be a d-dimensional connected smooth
77
Hypoelliptic Flows
manifold. Let now G be a Lie subgroup of the linear group GLd (R) with Lie algebra g. A principal bundle ( M, G) consists first of a submersion
such that each fiber is isomorphic to G. There is also a right action of G on 1: with the property that orbits are exactly the fibers of the bundle. Finally, we must ask that (It, M, G) is locally trivializable in the sense that for every s E M, there is a diagram
0 r-1 (U) --4 U x G
\
/ U
that commutes with the right actions of G on both 7r-1 (U) and U x G, where 0 is a local diffeomorphism and U a neighborhood s. For uo E 1, we denote Vuo the space of vectors tangent to the fiber 1::,0 • As before, for every X E g, we can define a vertical vector field Vx on l:; by
F (uetx ) — F(u) t
(vxF)(u) . lim t—,0
,
F I::
R,
and the map X --4 V x (u) is an isomorphism from g onto the set VuO(M). A connection in (It, M, G) is an assignment of a subspace 7-tu0 g; of Tuo It to each uo E It such that:
(1) T" %= Huo 3, ED vuo 1: 1
(2) nuog It .=- (Rg).74i0 It for every uo E 1
It and g E G, where Rg is the
transformation of 1:: induced by g E G, Rg uo = uog; (3) 7-( u0 It depends differentiably on uo . Given a connection in ( It M, G) we define the connection form w as the g valued one-form on such that:
(1) cd(X) = 0 if and only if X E H (2) V(x) = X if and only if X E V We also define the curvature form Si as the g valued two-form on I:: such that: du.) -I- w A (4) = Si.
78
An Introduction to the Geometry of Stochastic Flows
As in the orthonormal frame bundle over a Riemannian manifold, we can define d fundamental horizontal vector fields H1, H. Indeed, for every uE the canonical projection 7r induces an isomorphism
T,M,
gI
71-4, :
and for each X C TmM and u0 E Ig at m, there is a unique horizontal vector X*, the horizontal lift of X from u0 such that r-,,X* = X. Now, for each x E Rd we can define a horizontal vector field Hx by the property that at each point u C I , i (u) is the horizontal lift of u(x) from u. The fundamental horizontal vector fields are then given by Hi = He,, where (el, ed) is the canonical basis of Rd. Observe that the differential system on It generated by the horizontal vector fields H1, Hd is 7-t We define finally the fundamental horizontal diffusion on the bundle M, G) as the diffusion process on with infinitesimal generator (3.5)
2.1 The hypoellipticity property of AI:. is again related to the non-degenerence of the curvature form Q. Namely, for u E %, consider the map o(R)
g
defined by
=
(M)
E
1/-1)(u).
i
Proposition 3.2 Let us assume that at u C , 'W u is surjective, then the operator AI:, satisfies the H6rmander's condition at u. More precisely, (Hi (u), [Hi, Hk[(u)) i
It directly stems from the identity [
Hi ,
= — VS) ( , Hj ) •
is always surjective then all the points are normal and we can moreover compute explicitly the nilpotentizations and the tangent spaces. Let u Consider on the linear space Rd (1) g the polynomial group law given by
If
(y,Y)=- (x-1-y,X
+
1 2
d
d
(x E i Hi , yi Hi) (u)) i=1 i=i
Hypoe iiiptic Flows
79
Then, it is easily seen that (Rd e 9,*) is a two-step Carnot group that we shall denote GEL. Observe that from the assumption that Ilf„ is always surjective, for every u, v E I, there exists a Carnot group isomorphism GS2u GS2v . In this setting, we easily prove: Proposition 3.3 Let us assume that for every u E Then every point u E I is normal and Gr(7-(1)(u)
wu is surjective.
As an immediate corollary, we get Corollary 3.5 Let us assume that for every u E I, Ilfu is surjective. Let pt , t > 0, denote the heat kernel of (3.5) with respect to any Lebesgue measure. We have, Pt(u, u) rs
'd
C (u) t-0 0- +Chill g
where C(u) is a non negative constant.
3.5
Regular sublaplacians on compact manifolds
Let M be a connected compact smooth manifold. We consider on M a second order differential operator
which satisfies the strong H6rmander's condition. Let D denote the differential system generated by VI , Vd. We assume that there exists a Carnot group G such that for every x E M,
Gr(D)(x) = G. If the previous assumptions are satisfied, then .0 shall be said to be regular. Since the manifold M is assumed to be compact, it is possible to develop a spectral theory for r which is similar to the spectral theory of elliptic operators. Let us hint this theory. Let us denote X the diffusion associated with L. First, we are going to show that X is Harris recurrent. That is, there is a Borel measure m on M such that for every Borel set A C M:
80
An Introduction to the Geometry of Stochastic Flows
(1) m(A) > 0 implies that for any x E M, and t> 0, P (9 s > t, X, E AI X o = x) = 1; (2)
I.
P (Xt E A 1 X o = x) m(dx) = m(A), t > O.
For further details on Harris recurrent processes, we refer to [Revuz and Yor (1999) ] , pp. 425. Since M is compact, this recurrence property is a consequence of the following lemma:
Lemma 3.1 Let 0 C M be a non empty, relatively compact, connected with smooth, non empty boundary, open set, then for any x E M, and t > 0,
P (9 s> t, X, E 01 X0 = x) = 1. Proof.
Define To
= inf { t > 0, Xt E 0 } 1
and ho(x) =
IID (To <+00 I X 0 = x) , x EM.
It is easy to see that .Ch o
=0
on M — 0. From the general theory of Markov processes, there are two possibilities: Either ho =1 on M and for any x E M, and t > 0,
P( s > t, X s E 0 I X o = x) = 1, or 0 < ho < 1 on M — 0. The Bony's strong maximal principle for hypoelliptic operators (see [Bony (1969) ] ), implies actually that the only 0 possibility is h = 1. Therefore, the diffusion X is ergodic, i.e. for any smooth function f : M --4 IR and any x E M,
fim E (f (Xt) I Xo = x) = f f dm, t—,00 -1m
81
Hypoelliptic Flows
where m is the invariant measure for X. Observe that m is solution of the equation
0, so that, from Heirmaxider's theorem it admits a smooth density. An immediate corollary of the ergodicity of X is the following Liouville's type theorem. Corollary 3.6
Let f
'ME
R be a smooth function such that Cf = 0,
then f is constant. Proof.
Indeed, f=0
implies that for every t > 0, x E M,
(x).
(f(Xt) I Xo = x) Thus,
f(x) = lim E (f (X) I Xo = x) = f f dm. t—>+0,0
We shall now assume furthermore than is self-adjoint with respect to m, i.e. for any smooth functions f,g:IM1--4 R f dm. f( fm g(Cf)dm =Cg) In that case, etL is a compact self-adjoint operator in L 2 (M, m). We deduce that has a discrete spectrum tending to —oc. We furthermore have a Minakshisundaram-Pleijel type expansion for the heat kernel pt associated with C:
y) = E e—rikt +00
(Nk
k
2_,(P (s)
k=0
where: (1) {— !Lk} is the set of eigenvalues of .C; (2) Nk = dim V(Ak) = dim{ f I Cf =—Akf};
(Y))
82
An Introduction to the Geometry of Stochastic Flows
(3 ) (cP;?) is an orthonormal basis of V( juk) for the global scalar product (49 10) = si (7017n. mt Let us denote Sp(L) the set of eigenvalues of C repeated according to multiplicity.
Theorem 3.10
For À> 0, let
N(A) = Card (Sp(C) n [—A, 01) . We have N(A) r•sA_,+ 00
where C(C, M) is a non negative constant and D the Hausdorff dimension of G.
The asymptotic development of the heat semigroup et L on the Proof. diagonal leads to K Tr ( et) = f p t (3 e, x)dm(x) ^-t.o D
t7
M
3
where K is a non negative constant. On the other hand, Tr(e tc )
+00 ,
E
Nke —pkt .
k=0
Therefore, +00
E
K Nke - 1kt-t.0
k=0
t
•
D 2
The result follows then from the following Karamata's theorem: if ii is a Borel measure on [0, oc), a E (0, +oo), then +00
fo
c tA dy(A) —t.0
1 — to,'
implies
IX
di()) ---+°°
xa 0
Hypoelliptic Flows
83
We believe that the constant C(, M) is an interesting Remark 3.14 invariant of the sub-Riemannian geometry that L induces on M (recall that in the Riemannian case, it is simply, up to scale, the Riemannian volume of the manifold). For instance, it would be interesting to know if C(.C, M) is the Hausdorff measure of M.
To conclude this section, this is maybe interesting to study carefully an example. Let us consider the Lie group SU(2). As already observed, a basis of zu(2) is formed by
0 1 _ i ) , 217— —
(
0 i\ —1 —0 )
1 V3 = 2
(
0 i\ i 0)'
and the commutation relations hold (3.6)
[V1, V21 = 173, [V2, V31 --= 14, [V3, Vil --= V2. We want to study the regular sub-Laplacian £= v22 + v32 .
Actually, we shall study the following family of operators defined for c E [0, 1],
Ce = EV? + v22 + V32 . Observe that each .C€ is self-adjoint with respect to the normalized Haar measure of SU(2). For e> 0, LC is elliptic so that Card (Sp(C)
n [-.A, 01) ^-+A_,+00 Cdti,
Card (Sp(L)
n [-AO]) ,),,+„„ CV.
whereas
Therefore, it is interesting to understand the spectrum when Proposition 3.4
E --4
0.
Let E E [0, 1). The set of eigenvalues of Lc is the set
{—A n',m , n EN, 0 < m < n}, where
n2 n AnE ,m, = E-+ — — (1 — e) m 2 + (1 — E ) m n . 4 2 Moreover, the multiplicity of Anc ,,,, is equal to n +1.
84
An Introduction to the Geometry of Stochastic Flows
Proof. We use the theory of representations of SU(2); for a detailed account on it, we refer to Taylor [Taylor (1986) ] , Chapter 2. Let E E [0, 1). Thanks to the relations (3.6), note that G commutes with Ll = 1712 1(22 + 1/32 . Therefore L 6 acts on each eigenspace of We can examine the spectrum of L 6 by decomposing L2 (SU(2)) into eigenspaces of which is equivalent to decomposing it into subspaces irreducible for the regular action of SU(2) x SU(2) given by
((g, h) f)(x) = f (9 — 1 xh), f E L 2 (SU(2)) , g, h, x E SU(2). Now, it is known (see for instance Taylor [Taylor (1986)]) that, up to equivalence, for every k E N, there exists one and only one irreducible representation ir : SU (2 ) Ck+ 1 . Thus, by the Peter-Weyl theorem, the irreducible spaces of L 2 (SU(2)) for the regular action are precisely the spaces of the form Vk =
spanfrkij, 1 < j < k
11,
where rit'i denotes the components of the representation rk in a chosen orthonormal basis of Ck+ 1 . Observe now that each space Vk is an eigenspace of .C 1 . The associated eigenvalue is k(k+2) 4 • If we consider now the left regular representation
(g • f)(x) = f(g —l x), f E L2 (SU(2)) , g, x E SU(2), then Vk is a direct sum of k + 1 irreducible representations of SU(2), each equivalent to 'irk: k+1 Vk
IEDIVIc,1) 1=1
where Vkl = 1
11.
Each 14,1 splits into one-
14,1 = EDV1c,1,12,
where
k k
2 2 and
= i1z on Vk,/, 11.
2}
Hypoelliptic
Since LE = .C 1
—
(
Flows
85
1 — E)1712 we have ,
(k(k -I- 2) 4
.CE --=
(1 —E)/2) ,
on Vk,l,p• From this, we deduce immediately:
8
Card (Sp(LY) n [-A, 01) ^, A,+ 00 —
3
-VW '
whereas Card (Sp(.C) n [—A 1 01) i■-, A,+ 00 2A2 .
3.6
Stochastic differential equations driven by loops
On the free Carnot group GN(Rd), consider the fundamental process (B) t >0 defined as the solution of the stochastic differential equation d
t
.13; ----- E f D i (B:) 0 dB, t > O. i=1 o That is, (En t>0 is the lift in GN(Rd ) of the Brownian motion (Bt)t>oMore generally, if (Mt)t>0 is a Rd-valued semimartingale, we shall denote (M t*) t>0 the lift of (M)>o in GN (Rd). Let us denote pt (s, y), t > 0, the smooth transition kernel with respect to the Lebesgue measure of (Bn t>0 . Let us recall (see Appendix A) that it is defined by the property that —
P (B4 8 E dy I B: .=-- x) = Pt (xl Y)dY, for any t, s > 0 and x, y E GN(Rd). From a partial differential equations point of view, it is also the fundamental solution of the equation
Op _ 1
(` ,1-, 1
P. i )
In this first proposition we construct the N-step Brownian loop. This process is the Brownian motion (Bt)>0 conditioned by Bi, --= OG N (Rd ) •
An Introduction to the Geometry of Stochastic Flows
86
Proposition 3.5 Let T> O. There exists a uniqueRd-valued continuous process (PtNT ) 0
=
f Di ln pT_.13 ( sNi 3 OGN (R d ) ) ds, t
d. (3.7)
It enjoys the following properties: almost surely; (2) for any predictable and bounded functional F, (1)
= OG N (Rd),
E (F ((Bt)o
(PtN,T)o
Let us consider the Wiener space of continuous paths:
(C([0, T],R d ),(Xt)o
(1) C([0,7 ] ,Rd) is the space of continuous functions [0,7 ] Rd; (2) (X t ) t>0 is the coordinate process defined by X(f) = f (t), f E C([0,] Rd ); (3) P is the Wiener measure on [0, T], that is the law of a d-dimensional standard Brownian motion indexed by the time interval [0,7 ] ; (4) (Bt)o0 conditioned by Bik, = OGN(litd ) . Of course, Q is not equivalent to the Wiener measure ip on XT, but the following equivalence relations take place
d(I2/ xt
PT—t (X 1*1°GN(Rd))
t
PT (OG N (Rd) OG N (litd)
Therefore from Girsanov's theorem (see Appendix A), under the probability Q, the process
— f Di lnpT_, (X:, 0GN (Rd
ds, t < T,
i
1, ..., d,
is a standard Brownian motion. It means that the law of the process (PtN,T)0
87
Hypo elliptic Flows
Let us now prove the semimartingale property up to time T. In the elliptic case [Bismut (1984a)1 deals with the end point singularity of the Brownian bridge on a Riemannian manifold by proving an estimate of the logarithmic derivatives of the elliptic heat kernel. Unfortunately, such an estimate does not seem to be known in the hypoelliptic case, so that we have to deal by hands with the end point singularity of our equation. To prove that (P*)0
(PsA:4:1' OGN
(Rd
I ds < -4-oo
with probability 1. To prove this, it is actually enough to check that
E (f
lnpT_, (PsN *,OGN (R a)) f ds) <+00.
Since the processes (PTN' *t,T)0
E(
LT
I Di in p-i, _ s (P.97 , OG N (Rd)) Ids
2
=E (ITT IDi ln p s (13
* , OG N (Ra))
[ds .
Using now (3.8), we get LT ID i lnpT_, (PsNj,* , O GN (R d ) ) Ids
E
(
=E
2
f:T
(B 4i; ,OGN (R d ) )
Di lnp, (B: ,OGN (R d-) )
ds PT kOG N (Rd) ,C)GN (Rd ))
Therefore, from the Proposition 2.4 of Chapter 2, it remains then to check that
E T I Di ln (f T
(B:,OGN ori))
ds)
<
+00.
0
We claim now that E (I Di hip.,(B; ,OG N (Rd))
1 E(I His D.
An Introduction to the Geometry of Stochastic Flows
88
Indeed, an integration by parts formula proved in [Driver and Thalmaier (2001) 1 (see also [Elworthy and Li (1994)1) says that for any bounded measurable function f, E (f (B:)(D i lnp,(B:, OG N (Ra)))) = —E (f (B:)B si ) . And the above inequality is proved by taking for the sign function of Di ln p, (x, OGN (Rd ) )• Therefore, E
I
Di lnp8 (B:,0)
1 - E0
ds) <
Dds C
io
-± d < o Nrs
which concludes the proof. Definition 3.9 loop of depth N.
The process (PtNT )o
Example 3.13
The process (PtiT)o
'
Observe that this equation can be explicitly solved and leads to the following expression ft
T ( 11 t)
Jo
dB,
_ s.
Example 3.14
The process (./1,T)0< t
(
1 BT, — 2
Remark 3.15
(I
Bi o 8
0
Notice
- o
that in law,
(PN t ,T)o
nN
(3.9)
Consider now on Rn stochastic differential equations of the type d
Xt = X0
+
t
E f0 pc s ) i=1
t
(3.10)
89
Hypoelliptic Flows
where: (1) So E Rn; (2) VI , Vd are C" bounded vector fields on Rn ; ptd (3) ( )0 0. the Lie algebra generated by the vector As we already did, we denote by fields Vi's and for p> 2, by 2,P the Lie subalgebra defined by
Y E C}.
,CP = {[X, Y], X E Moreover if a is a subset of Z, we denote
a(x) {V(x), V E a}, a. E Rn . Proposition 3.6 For every x o E Rn , there is a unique solution (X tx0 )0
(3.10).
Proof.
We refer to the book [Kunita (1990) ] , where the questions of existence and uniqueness of a smooth flow for stochastic differential equations driven by general continuous semimartingales are treated (cf. Theorem 3.4.1. p. 101 and Theorem 4.6,5. p. 173). We consider now the following family of operators (71q)T>0 defined on the space of compactly supported smooth functions f : Rn R by
(74 f)(x) --= E (f (Xi)) , s E Rn. The operator 74 shall be called the depth N holonomy operator. Remark 3.16 property.
Of course, the operator 7-t does not satisfy a semi-group
Theorem 3.11 tion, In L 2 ,
Let f : Rn R be a smooth, compactly supported func-
lim T f f T.() TN+ 1
.
N f,
where AN is a second order differential operator. It can be written rn
An
90
the
Introduction to
Geometry
of Stochastic Flows
where: (1) 771 =
N+1
E
p(k)d4 1 ;
kiN+1
(2) Qi E zN-1-1 is a universal Lie polynomial in V1,...,Vd which is homogeneous of degree N +1.
Proof. Before we start the proof, let us precise some notations we already used. If I = (ii , ...,ik) E {1, ...,d} k is a word, we denote I / I= k its length and by V/ the commutator defined by V1
= [ 14. 11
[V2:21 "t
[Vik-1 1
Vik
The group of permutations of the index set {1, ..., k} is denoted E ek , we denote e(o) the cardinality of the set { j E { 1,
k — 1} , (j) >
ek.
If
+ 1)}
Finally, we denote AI(P.I,\T r )t =
(-1) e(u)
E crEek k2
f
odPtNi
(PtN,T)0
i )o
(k —1) e(cr)
:T tr- 1 i
o o
N,cr -1 ik dPtk,T
Due to the scaling property
we can closely follow the proof of Theorem 2.6, Chapter 2, to obtain the following asymptotic development of f (X ): ( 2N +2
(exp
EE
Ai(F!/,vT )Tvi) f)
(X)
k=1 2N+3
-1-T 2 R2N+ 3 (T, f,x), where the remainder is bounded in probability when T of (Pt5)o
EE k=1
2N-1-2
A i (p) T v,
=E
k=N+1
E
O. By definition
(13.1:1T)TVI,
91
Hypoelliptic Flows
so that ( 2N+2
f (X1,) =
E
( exp
E
Ai (P.1\, T T )TVI f (x)
k=N-1-1 /=(ii,.•.,ik) 2N+3
+T 2 R2N +3(T, f ,x). Therefore, ( 2N-1-2
E
71, f (x) =E ( ( exp
E
A./(pi,NrT )Tvi) f) (X))
k=N-1-1 I=(ii,...,ik)
+ T T2+-3-- 14.. - ' 2N+3(T, f, x), where
11 2N-1-1(T, f, s) — E (R2N+1(T, f, s)) . Since, by symmetry, we always have E (A/ (P.I,NT)T) r = 01 we have to go at the order 2 in the asymptotic development of the exponential when T —p 0. By neglecting the terms which have order more than T; ±2 , we obtain
71, f (x) =1(x) +
E -&1 E (A./(P.1,vT)TAJ(P1, )T) (VIVJ f)(x) I ,J
4-
T 3142 R;PI-F3(T,
where the remainder term 11;N+3 (T, f, x) is bounded in L 2 when T —p Notice now that, since (P ti'TN )o
71 f (x) =f (x) +
E
].1 IE (A1 (P')) (V/2 f)(x)
/=(z1,.••,iN + 1) + r2142 R;N+3 (T, f, x), which leads to the expected result.
0
92
Example 3.15
An Introduction to the Geometry of Stochastic Flows
We have =
1
2
d
V2 i=1
and,
E 1
The convergence jN+i -
TNI-1
Id
T—C1
also holds in U', p> 1.
Theorem 3.12 Assume that ZN +1 = 0, then for any solution (X7 0 )0
Rn
such that, for xo E R n , the solution (Xtz°)oo of the following stochastic differential equation:
Hypoelliptic Flows
93
where (Be', Bg) t> 0 is a d-dimensional standard Brownian motion. From theorem 3.3, the random variable
(ZT , has a smooth density with respect to any Lebesgue measure on W1 x GN (Rd). It implies the existence of a smooth function p: R such that for every bounded measurable function f R
E(AZT)
I -13;. = 0) = f n f(Y)P(Y)dY.
Now, since in law the process (PtNT)0
Remark 3.19 In the case N =.1, we can give another proof of the existence of a density under the assumption 22 (x0) = R. Indeed, let us recall that t dB,
(T — t) J , t < T, and PT,T — O. 0T—s Using this formula, it is not difficult to prove that if (Xt)0
13 ,9 XT =
JOJ.—
( J0— 18Cr(Xs)
i.
T—s I
T
..10-1 u a(X u )du ,
where (.10_,t)0
JO-4 —
acDt ax
and a the n x d matrix field a = (V1 , ...,V4. From this, we can deduce that the Malliavin matrix of XT must be invertible. Indeed, if not, we could find a non zero vector h E Rd such that D,X T • h = 0 for 0 < s < T, this leads to the conclusion that (J0-1 s o(X,) • h) 0<3
94
An Introduction to the Geometry of Stochastic Flows
Remark 3.20 In general, the only condition £N+1 (50 ) = 0 is not enough to conclude that for the solution (X ti 0 )0
1
) and V2 := ( ° f (x)) '
where f is a smooth function whose Taylor development at 0 is O (e.g. f(s) = e- *1_, >0). Nevertheless, if the vector fields 14's are assumed to be analytic, it is true that £N +1 (x0) = 0 implies that almost surely X Tx° = xo• In the case of the existence of a density for X°, we can moreover give an equivalent of this density when the length of the loop tends to O. To this end, let us precise some notations. We set for s G IRn and k> N,
14(x) = spanfili, N <1 I l< kl. In the case where £N+1(x) = Rn , if k is big enough then Uk(x) = Ir. We denote d(s) the smallest integer k > N + 1 for which this equality holds and define the graded dimension d(x)
d im7.1 zN +1 (x
)
E
:
k (dimUk(x) - dim /4-1 (s)) •
k=N+1 Theorem 3.13 Assume that for any x E Rn, £N+ 1 (x ) = Rn . Let us denote pr(x) the density of XI, with respect to the Lebesgue measure. We have pT(x)
m(x) "4
T—
dims z N-1-1
.)
T
2
where m is a smooth non negative function. Proof. Let us, once time again, consider the solution (Znt>0 of the following stochastic differential equation: d
Zf
,x+EI
t Vi(Z;) o dI31, t > 0,
i_ 1 o
where (Bi, ... , Bcti)t>0 is a d-dimensional standard Brownian motion. From Corollary 3.2 , the density at (x, 0) of the random variable (4, ifq) behaves
Hypoelliptic Flows
95
when T goes to zero like ih(x) dimw GN(Rd)-1-dirriii .CN+ 1 (x) 2
where dim n GN (Rd) = N. dim Vi is the graded dimension of GN (Rd), and fit, a smooth non negative function. Always from Corollary 3.2, the density of the random variable YT behaves when T goes to zero like
T
diani GN (lid)
2
/
where C is a non negative constant. The conclusion follows readily.
Appendix A
Basic Stochastic Calculus
In this Appendix, we review briefly the basic theory of stochastic processes and stochastic differential equations. The stochastic integration is a natural, easy and fruitful integration theory which is due to [Itô (1944)1. For a much more complete exposition of the stochastic calculus we refer to [Dellacherie and Meyer (1976) ], [McKean (1969)], [Protter (2004)1, and [Revuz and Yor (1999)]. For further reading on stochastic differential equations we refer to [Elworthy (1982)1, [Ikeda and Watanabe (1989)], [Kunita (1990)], [Stroock (1982)], and [Stroock and Varadhan (1979)1. We assume some familiarity with basic probability theory.
A.1
Stochastic processes and Brownian motion
Let (51, (Ft)>o, ..T, P) be a filtered probability space which satisfies the usual conditions, that is: (1) (..Ft)t>13 is a filtration, i.e. an increasing family of sub-a-fields of .7 . ; (2) for any t > 0, ..7t is complete with respect to P, i.e. every subset of a set of measure zero is contained in Ft; (3) (Ft ) t >0 is right continuous, i.e. for any t > 0, 'Ft ' A stochastic process (Xt)t>0 on
(t-,w)
n-Fs. 9>t
(n,(Y-t)t>0,-F,P) is an application --4 Xt (w), 97
98
An Introduction to the Geometry of Stochastic Flows
which is measurable with respect to B(R>0) F. The process (X)>.o is said to be adapted (with respect to the filtration (Ft)t>o) if for every t > 0, Xt is .Ft-measurable. It is said to be continuous if for almost every (4) E the function t X t (w) is continuous. A stochastic process (X)>.o is called a modification of (X) >.o if for every t > 0,
The following theorem known as Kolmogorov's continuity criterion is fundamental. Theorem A.1
Let a,
E e > O. If a process (X)>o satisfies for every
s, t > 0,
E (I Xt — X 3 1")o which is a continuous process. One of the most important examples of stochastic process is the Brownian motion. A process (Bt ) t >0 defined on (S-2, (..rt)t>o, P) is said to be a standard Brownian motion with respect to the filtration (71)t>0 if:
(1) B0 = 0 a.s.; (2) B1 =law .N.(0, 1); (3) (Bt)t>0 is F-adapted; (4) For any t> s > 0, Bt —
D
— law
pp
t89
(5) For any t> s > 0, Bt — B, is independent of ..F„. If a process is a Brownian motion with respect to its own filtration, we simply say that it is a standard Brownian motion, without mentioning the underlying filtration. A d-dimensional process (Bt) t>0 = (Bi B`ti)t>0 is said to be a d-dimensional standard Brownian motion if the processes (Btl) t>o, (Btd ) t>0 are independent standard Brownian motions. A standard Brownian motion (Bt)t>0 is a Gaussian process, that is for < ta , the random variable every n E N*, and 0 < t 1 <
(B t„ is Gaussian. The mean function of (Bt ) t>0 is
E(Bt ) = 0,
99
Basic Stochastic Calculus
and its covariance function is E(B8 .13t ) = inf(s, t). It is easy to show, thanks to Kolmogorov's continuity criterion, that it is always possible to find a continuous modification of a standard Brownian motion. Of course, we always work with this continuous modification. The law of a standard Brownian motion, which therefore lives on the space of continuous functions R>0 R is called the Wiener measure. If (B t ) t >0 is a standard Brownian motion, then the following properties hold:
(1) For every c > 0, (B) >. o =law (N/aBt)t>o; (2) (t-13)>o =1" (Bt)t>o; (3) For 'almost every co E Q, the path t Bt (w) is nowhere differentiable and locally Milder continuous of order a for every a < (4) For every subdivision 0 -= to < • • • < t t whose mesh tends to 0, almost surely we have n-1
lim E (B t„, - B 1 ) 2
=----- t.
i=0
A.2 Markov processes Intuitively, a Markov process (Xt)t>o defined on a filtered probability space (11 , (rt)t>o, P) is a process without memory.
Definition A.1 A transition function {Pt ,t > 0 } on R is a family of kernels Pt: R x B(R) [0,1] such that: (1) For t > 0 and x E R, Pt (x .) is a Borel measure on B(R) with Pt (x R) < 1 (2) For t > 0 and A E B(R), the function x Pt (x , A) is measurable with respect to B(R); (3) For s, t > 0, x E R and A € B(R), Pt+,(x , A) ----
Pt (y , A) P, (x , dy);
(A.1)
Equation (A.1) is called the Chapman-Kolmogorov equation. We may equally think of the transition function as being a family ( Pt) t> 0 of positive bounded operators of norm less or equal to 1 on the space of bounded Borel
100
An Introduction to the Geometry of Stochastic Flows
functions f: IR --- JR defined by
(Pt f)(x) = f f(Y)Pt(x,dY). R
Observe that the Chapman-Kolmogorov equation is equivalent to the semigroup property Pt-I-8 — PtPs.
Definition A.2 An adapted continuous process (Xt)t>0 defined on (S2, (1-t ) t >0,..F,P) is said to be a Markov process (with respect to the filtration (■r- 1 ot>o) if there exists a transition function (Pt ) t>0 such that for every bounded Borel functions f : R —f R, 0 < s < t,
E (f (Xs+t) I Fs) = (Pt f )(X .9) . In the theory of stochastic processes, it is often useful to deal with random times. A real valued positive random variable T is said to be a stopping time with respect to the filtration (F)>o if for any t > 0, IT < t} E -Ft .
If T is a stopping time the smallest a-field which contains all the events IT < t}, t > 0, is denoted FT. Definition A.3 An adapted continuous process (Xt)t>0 defined on (n,(Ft)t>0,..F,P) is said to be a strong Markov process (with respect to the filtration (...rt ) t>0) if there exists a transition function (Pt ) t>0 such that for every bounded Borel functions f : R —p t > 0,
E (f (Xs+t) I Fs) = (Ptf )(X s), where S is any stopping time that satisfies almost surely 0 < S < t.
Observe that a strong Markov process is always a Markov process and that a standard Brownian motion is a strong Markov process. For further details on Markov processes, we refer to Chapter III of [Rogers and Williams (2000)].
A.3
Martingales
Consider an adapted and continuous process (Mt ) t>0 defined on a filtered probability space (f2 , (-Ft)t>o, F, IF).
Basic Stochastic Calculus
101
Definition A.4 The process (Mt ) t>0 is said to be a martingale (with respect to the filtration (..rt)t>()) if:
(1) For t > 0, E (I Mt i) < +oc; (2) For 0 < s < t, E (Mt I ..rs) = Ms. For instance a standard Brownian motion is a martingale. For martingales, we have the following proposition, which is known as the stopping theorem.
Proposition A.1
The following properties are equivalent:
(1) The process (Mt)t>0 is a martingale; (2) For any bounded stopping time T, E(MT) = E(M0); (3) For any pair of bounded stopping times S and T, with S < T,
E (MT 1 I's) = Ms. Actually, if T is a bounded stopping time, then the process (MtAT)t>o is also a martingale. A martingale (Mt ) t>0 is said to be square integrable if for t > 0, E (M?) < +oo. In that case, from Jensen's inequality the function t --4 E (M?) is increasing. We also have the so-called Doob's inequality
E
(
sup M?) < 4 sup E (Mt2) , t>0 t>o
which is one of the cornerstone of the stochastic integration. Observe therefore that if supt>0 E (111?) < +oo, the martingale (Mt)t>0 is uniformly integrable and converges in L 2 to a square integrable random variable Moo whicsatfe
E (Moo I -Ft ) = Mt , t > O. If (Mt)t>0 is a square integrable martingale, there exists a unique increasing process denoted ((M))>0 which satisfies;
(1) (M)0 = 0; (2) The process (V — (M)t)t>0 is a martingale. This increasing process (01,\ t ) t >0 is called the quadratic variation of the martingale (Mt)t>0. This terminology comes from the following property. If 0 = to < t 1 < •.. < tn, = t is a subdivision of the time interval [0, t}
102
An Introduction to the Geometry of Stochastic Flows
whose mesh tends to 0, then in probability n-1
lim
E (m, - Mti) 2 = (M)t. ti=0
For technical reasons (localization procedures), we often have to consider a wider class than martingales. Definition A.5 The process (Mt ) t >0 is said to be a local martingale (with respect to the filtration (..rt)t>0) if there exists a sequence (Tn ) n >0 of stopping times such that: (1) The sequence (Tn ) n>0 is increasing and Tn = +00 almost surely; (2) For every n > 1, the process (MtAT,i)t>0 is a uniformly integrable martingale with respect to the filtration (..Ft)t>o• A martingale is always a local martingale but the converse is not true. Nevertheless, a local martingale (Mt ) t >0 such that for every t > 0, E (sup I M, 1 s
<+ 00
,
is a martingale. If (Mt ) t>0 is a local martingale, there still exists a unique increasing process denoted ((M) t ) t>0 which satisfies: (1) (M)0 = 0; (2) The process (M? — (M))>0 is a local martingale. This increasing process ((M)t)t>o is called the quadratic variation of the local martingale (Mt)t>o • By polarization, it is easily seen that, more generally, if (Mt ) t >0 and (Nt ) t>0 are two continuous local martingales, then there exists a unique continuous process denoted ((M, N) t ) t>0 and called the quadratic covariation of (Mt)t>o and (Nt)t>0 which satisfies:
(1) (M, N)0 =0; (2) The process (MNt — (M,N) t ) t>0 is a local martingale. Before we turn to the theory of stochastic integration, we conclude this section with Lévy's characterization of Brownian motion. Proposition A.2 Let (M)>o be a d-dimensional continuous local martingale such that Mo = 0 and
,M 2 )t = t, (M,M) t = 0 if i
j.
103
Basic Stochastic Calculus
Then (Mt ) t>0 is a standard Brownian motion.
A.4
Stochastic integration
Let (Q, (rt)t>o,F,P) be a filtered probability space which satisfies the usual conditions specified before. We aim now at defining an integral fot H8 dM, where (Mt)t>0 is an adapted continuous square integrable martingale such that supt>0 E (Mi) < +00 and (lit ) t>0 an adapted process which shall be in a good class. Observe that such an integral could not be defined trivially since the Young's integration theory does not cover the integration against paths which are less than I-1-181der continuous. First, we define the class of integrands. The predictable a-field P associated with the filtration (Ft)t>0 is the a-field generated on 1R> 0 x by the space of indicator functions lism, where S and T are two stopping times such that S < T. An adapted stochastic process (Ht)t>0 is said to be predictable if the application (t, w) Ht(w) is measurable with respect to P. Observe that a continuous adapted process is predictable. Let us first assume that (Ht) t>0 is a predictable elementary process that can be written n-1
where 1/1 is a ..FT, measurable bounded random variable, and where (Ti)1
H8dM,
H
rm ,__,ATi + , -mtATi).
i=1
Then, we observe that the process (fo lisc/Ms )
t>0
is a bounded martingale
which satisfies furthermore from Doob's inequality 2
E (sup (f li sciA/3 ) 0.0 0
< 4 11 H 11 2°0
We also note that (
ig
rt
\2
0 118C1M8 )
n-1 =E
(E (MtATH-1 — MtAT0 2) .
104
An Introduction to the Geometry of Stochastic Flows
Assume now that (Ht)t>0 is a bounded continuous adapted process. To define fc: HAM„, the idea is of course to approximate (lit ) t> c, with element tary processes ( Hn t>0 and to check the convergence of (fo li3Pc/M3 )
t>0 with respect to a suitable norm. Precisely, let us define for any p E N*, the following stopping times:
T1'; = 0 Tr = inf > 0,
p -1 }
and by iteration
We now define n-1
= ER-77 1 17 , 7+11 (t). i=--1
For this sequence of processes ( Hnt>o, it easy to show that 2 E
(sup (1 (Hr — H.q,)c1Ms)) --4 19,q-)+0,0 O. 0
t>0
From this, we can deduce that there exists a continuous martingale denoted (fo 118 dM3 ) t>0 such that
fo HAM,
= lim I I/PM, 13-40°
J0
uniformly for t on compact sets. We furthermore have 2
E (sup (f HAM 8 )
0.0
0
((f t
lis dA/8 ) 2 )
<4 11 H 11E( 11).
and
E
E (fot liMM),) .
Now, by localization, it is not difficult to extend naturally the definition of fo HAM, in the general case where:
105
Basic Stochastic Calculus
(1) (Mt )t>0 is a semimartingale, that is, (Mt)t>0 can be written
Mt
= At + Nt,
where (At) t >0 is a bounded variation process and (Nt ) t> 0 is a local martingale; (2) (Ht ) t>0 is a locally bounded predictable process. In this setting, we have: ft
t
1-13dM3
J 1-13 dA 3 0
ft
+ J 1-13dN3. Jo
Observe that, since (At ) t>c, is a bounded variation process, the integral ‘g Hs dA s is simply a Riemann-Stieltjes integral. The process
(fct, H3 dN8 )
is a local martingale. t>o The class of semimartingales appears then as a good class of integrators in the theory of stochastic integration. It can be shown that this is actually the widest possible class if we wish to obtain a natural integration theory (see Dellacherie-Meyer [Dellacherie and Meyer (1976) ] or Protter [Protter (2004)1 for a precise statement). The decomposition of a semimartingale (Mt ) t>0 under the form
is essentially unique under the condition No = 0. The process (At ) t>0 is called the bounded variation part of (Mt ) t >0. The process (Nt ) t >0 is called the local martingale part of (Mt)t>0. If (M')>0 and (A/g) t>0 are two semimartingales, then we define the quadratic covariation ( ( A/ 1 , 11/ 2)0 0.o of
(M l )t>0 and (V) t >0 as being ((110 N 2 )t ) t>0 where (Nn t>0 and (N-ht>o are the local martingale parts. Throughout this book we shall only deal with continuous processes so that in the sequel, we shall often omit to precise the continuity of the processes which will be considered. Moreover, we shall preferably use Stratonovitch's integration rather than Itô's integration. If (Nt)o 0, is an ..F-adapted real valued local martingale and if ( 9t)0
T
1
et dNt = I et • dNt + — (e, N)T, 2
106
An . Introduction to the Geometry of Stochastic Flows
where:
(1) Jr oT€) o dNt
is the Stratonovitch integral of (et)o
(Nt)ixt
A.5 Itô's formula The Itô's formula is certainly the most important formula of stochastic calculus. t>0 be a n- dimensional conLet (Xt)t>0 = (X1 - • , tinuous semimartingale. Let now f : Rn R be a C2 function. We have
Theorem A.2
7'1
f (X t ) f(X0)
E
af
• s
, (X s )dXt uxi
n ft 02 f v•, – (X3)d(Xt,Xj), 2 La jo OxiOxj
= f (X 0) -I- Ê —f a –(X 8 ) Ox .10 j= A.6 Girsanov's theorem We give here the most important theorem concerning the impact of a change of probability measure on the class of semimartingales. Let C(R>01R d) denote the space of continuous functions R>0 Rd. Let X 8 , s > 0, denote the coordinate mappings, that is
X 8 (w) = w(s), w E C(R>o, Rd). Set now Bt° = cr(X 8 , s < t) and consider on C(R>o, Rd) the Wiener measure P, that is the law of a d-dimensional standard Brownian motion. Let finally Bt be the usual P-augmentation of .5? and
Boo = V t>oB t • The filtered probability space
(C(R>o Rd) (Bt)t>o, 13 c, , IP)
107
Basic Stochastic Calculus
is called the Wiener space. Observe that, by definition of IP, the process (Xt ) t >0 is under P a d-dimensional standard Brownian motion. We have the following theorem, referred to as the Girsanov's theorem. Theorem A.3 (1) Let Q be a probability measure on (C(R>o,Rd), Boo ) which is equivalent to P. Let us denote by D the density of Q with respect to JP. Then there exists an adapted continuous Rd -valued process such that
(et) >.0
E (D Bt ) = exp
(I esdxs - 1
t
0
I
e, 11 2 ds)20
and under Q,
XXt —
%cis Jo
is a standard Brownian motion. be an adapted continuous Rd-valued process such that the (2) Let process
(oot>o
t
1 t Zt = exp (I es dX, — — f 2 0 0
I
e, 11 2 ds) ,
t > 0,
is a uniformly integrable martingale under P. Define a probability measure on (C(R>o,Rd), Boo ) by: do:2 = Zoo P. Then, under Q, the process
X t — eols o
is a standard Brownian motion.
Observe that the process Zt = exp
(1 esdx, 0
1 —-
2 0
es
1 1 2 ds) , t>o,
satisfies the following equation
zt ,1+ f zesdx,.
108
An Introduction to the Geometry of Stochastic Flows
It can be shown that sufficient conditions which ensure that (Zt ) t>0 is a uniformly integrable martingale are the following: (1) For any t > 0,
E(Z) = 1; (2) For any t > 0,
E (exP ( 1- i t ii es
A.7
1 2 ds)) <+00.
Stochastic differential equations
Let (O, (rt)t>o,P) be a filtered probability space which satisfies the usual
conditions. Let (
Mt t>0
—
(Me',
•
M)>0
denote a continuous d-dimensional semimartingale that is adapted to the filtration (2t)t>1). Consider now d smooth vector fields Vi : Rn Rn , n > 1, i = 1, ..., d. The fundamental theorem for the existence and the uniqueness of solutions for stochastic differential equations is the following: Theorem A.4 Let xo E Rn . On (SI, (.rt)t>o,P), there exists a unique continuous process (X ro) t>0 adapted to the natural filtration of (Mt)t>0 completed with respect to IP and such that for t > 0,
El0 vi ( x:0).dM.:. d
Xf° = xo + The process (X°) equation (A.2).
t
(A.2)
>0 is called the solution of the stochastic differential
In the sequel the stochastic differential equation (A.2) shall be denoted SDE(x0,(1/)10). Observe that SDE (SO, (V) 1<< d, (M)>0) is written in Stratonovitch's form. Thanks to Itô's formula the corresponding Itô's formulation is d
Xr° = xo +
E
t
d
t
f vvyi (xr)d(mis ,* + Ef vi(xr)dm::, 0
0
109
Basic Stochastic Calculus
where for 1 <
j < d, Vv i Vi is the vector field given by n
n
av i )
(X) vvi vi (x)=E Evp(x)-iasi k=1 1=1
a
X E
aXk ,
Rn .
An important point is that the solution (X°) > 0 of (A.2) is a predictable functional of (M)>o. Let us precise what it exactly means. Let C(R>o, Rd) denote the space of continuous functions R>0 Rd . If w(s), s > 0, denote the coordinate mappings, we set Bt = a (w(s) , s < t). A function F : Rn, is said to be a predictable functional if it is predictable C(R>o,R d) as a process defined on C(R>o, Rd) with respect to the filtration (Bt )>o. So, with this terminology, saying that (X°) > 0 is a predictable functional of (Mt ) t >o means that there exists a predictable functional F satisfying for any t > 0,
XT °
P-VMs)0<s
We mention that in general the functional F can not be chosen continuous in the topology of uniform convergence on compact sets (see Section 2.5 of Chapter 2 on the rough paths theory for a discussion about this continuity question). The essential advantage of Stratonovitch's formulation is the following form for Itô's formula. Proposition A.3 Let f : R be a C2 function and let (XT ° )t>o denote the solution of SDE (so, (V)1<o). We have for t > 0, d
f (X f°) = f(x0)
+E
t
(Vif)(X;°) 0 dA//:.
i=1 A.8
Diffusions and partial differential equations
One of the main uses of the theory of stochastic differential equations is to construct and study diffusions. This construction can be performed when the driving semimartingale (M)>o is a standard Brownian motion (Bt)t>0. More precisely, solving the stochastic differential equation SDE(xo, (Vi)io) is a way to canonically associate, on a given probability space, a stochastic process with the second order differential operator
110
An Introduction to the Geometry of Stochastic Flows
Indeed, as a direct consequence of Itô's formula, we get: Proposition A.4 Let xo E Rn and let (X°)>0 denote the solution of SDE (so, (Vi)i0). The process (Xr)t>o enjoys the strong Markov property with respect to the natural filtration of (Bt ) t>0. Furthermore for any smooth function f : Rn IR which is compactly supported, the process
(f (XP) — (,G f)(X .Nds ) t>0
O
is a martingale with respect to the natural filtration of (Bt )>o, where d
is the so-called infinitesimal generator of (XT ° )t>0.
As a consequence of this proposition, the transition function associated with the Markov process (X°)>0 is the semigroup generated by G, that is Pt = eta.
Therefore, for every xo E Rn and every smooth function f : Rn IR which is compactly supported,
E (f (X°)) = (et ' f) (so). It is possible to extend the last formula by adding to G a potential: this is the famous Feynman-Kac formula. Proposition A.5 Let V: Rn R be a bounded and continuous function and let f : IRn R be a smooth function which is compactly supported, then for every xo E Rn,
E (f(Xr)e — f: v(x :°) cis) = (et — ")f) (so), where (X°)>0 is the solution of SDE (so, (17 )10) •
A.9 Stochastic flows As in the theory of ordinary differential equations, in the theory of stochastic differential equations, this is fruitful to look at the solution of SDE (X0 , ( Vi)i0) as a function of the initial condition so.
Basic Stochastic Calculus
111
Theorem A.5 Let (M)>o be a continuous d-dimensional 1-adapted semimartingale. On (SI, (2)>o P), there exists a unique jointly measurable continuous random field (4't(x))t>0,xeRn such that: Rn is a diffeomorphism for any
(1) Almost surely, the map (1) t : t > 0; (2) For t > 0 and x E R 22 , d d't(X)
t
=s+EI V(3 (x)) o i=i 0
We call (t)>o the stochastic flow associated with the above stochastic differential equation. There is a nice Itô's formula for the action of stochastic flows on smooth tensor fields (we refer to Appendix B for the definitions of the tensor fields and of the Lie derivative).
Let K be a smooth tensor field on Rn and let ()>o denote the stochastic flow associated with SDE (so, (V) 1<0)• We have for t > 0 and x E Rn ,
Proposition A.6
d
(4)7 K)(x) --= K(x)
E
ti=1 0
t
(crviK) (s) o dM;.
A.10 Malliavin calculus In this section, we introduce the basic tools of Malliavin calculus which are used in the proof of Heirmander's theorem. For further details, we refer to [Nualart (1995)]. Let us consider the Wiener space of continuous paths: W®d = (C([0,1], Rd ), (B)0<
(1) C([0,1], Rd) is the space of continuous functions [0,1] Rd; (2) (Bt )t>0 is the coordinate process defined by Bt (f) --= f (t), f E
C([0, 1 , Rd); ]
(3) P is the Wiener measure on [0,1], that is the law of a d-dimensional standard Brownian motion indexed by the time interval [0,1]; (4) (.8t)cK t <1 is the (P-completed) natural filtration of (N0
An Introduction to the Geometry of Stochastic Flows
112
A Bi measurable real valued random variable F is said to be cylindric if it can be written
1
F
1 1114133 ,..., j lesidB.)
f (10
0
where hi E L2 ([0, 1], Rd) and f : Rn R is a C' bounded function. The set of cylindric random variables is denoted S. The derivative of F E S is the Rd valued stochastic process (Dtflo
1 1 f his2dB 3 .). E h(t) -axi -- (1 hsl dB, ..., Jo 0 i.1
Dt F
More generally, we can introduce iterated derivatives. If F E S, we set „..,t, F = We can consider DkF as an element of L2 (C([0, 11, Rd), L2 ([0, 11k, Rd))) ; namely, DkF is a random process indexed by [O, l] k . For any p > 1, the operator Dk is closable from S into LP (C([0, 11, Rd), L2 ([ O, 11k, Rd))) . We denote Dk 'P the closure of the class of cylindric random variables with respect to the norm I k
11 FI lk,p =
(
E (FP) + EE (IlDiFilLP2(Iojii,Rd)) j=1
) P
and
p>1 k>1
We have the following key result which makes Malliavin calculus so useful when one want to study the existence of densities for random variables.
Theorem AM
Let F = (F1, • • • I Fri) be a Bi measurable random vector
such that: (1) for every i = 1, ...,n, F E (2) the matrix
r= is invertible.
(f0
(D,Fi,D,Fi)fftdds) 1
Basic Stochastic Calculus
113
Then F has a density with respect to the Lebesgue measure. If moreover, for every p> 1,
E
( 1
)
det
<+00,
then this density is smooth. Remark A.1 The matrix random vector F.
r
is often called the Malliavin matrix of the
Of course, this result is only useful if the class De° is big enough to contain interesting random variables. Theorem A.7 ential equation
Let (XT) t>0 denote the solution of the stochastic differ-
t
d
Xf = x
f V i (Xf°) dB si 0
(A.3)
Then, for every i = 1, , n, XI E D". Moreover,
Dit X 1 = J0,14 1 t Vi(X t ), j
1,
d, O< t < 1,
where (.10—)t>0 is the first variation process defined by -1
0-4 =
OX x Ox
and where DX 1 is the j-th component of Dt Xt.
A.11
Stochastic calculus on manifolds
Let M be a smooth connected manifold. A (continuous) M-valued process (X t )t>0 is said to be a semimartingale if for every smooth function f M R the process (f (Xt))t>0 is a real semimartingale. It is interesting to note that to define the notion of semimartingale on a manifold, no additional structure than the differentiability structure is required. But to define the notion of martingale on a manifold, we have to endow the manifold with an affine connection. On this point we shall not go into details since the notion of martingales on a manifold is not needed in this book, but for further details we refer to [Emery (1989)] and [Emery (2000)].
114
An Introduction to the Geometry of Stochastic Flows
On the manifold M, it is possible to give a sense to stochastic differential equations of the type t
d
XT°
= xo Y f Vi(X:°) 0 dM.:, t >.0
(A.4)
where:
(1) xo E M; (2)
Vd are C°° bounded vector fields on M;
(3) o denotes Stratonovitch integration; (4) (Mil , AV)0
f (XT°) = f(xo) +
t
E f (Vif)(Xr) o i.1 o
t
O.
In this setting, many results given in Sections A7, A8 and A9 remain true. In particular, when the driving semimartingale (Mt ) t>0 is a linear Brownian motion, then the solution of equation (A.4) is a Markov process with infinitesimal generator
1
Let us now assume that the manifold M is moreover endowed with a Riemannian structure (see Appendix B). In that case, there is a natural and canonical second-order elliptic operator defined on M: The LaplaceBeltrami operator M. A continuous M-valued process (Bt)t>0 is said to be a Brownian motion on M if it is a Markov process with generator 1.6,m. It is equivalent to ask that for every C°° bounded function M R, the process
1 (f (B t ) — — f (Am f) (.13,) ds) 20 t>o is a martingale. It may happen that a Brownian motion on a manifold explodes, meaning that the best we can do is to find a process (Bt ) t>0 together with a stopping
115
Basic Stochastic Calculus
time r such that for every C' bounded function M R, the process 1 finf(t,r)
(f
(Amp (B5 ) ds)
(Binf(t,T))
2 jo
t>13
is a martingale. Such phenomenon can not occur if the manifold M is compact. Observe that in general the operator Am can not globally be written under the form d
VO
+ Eyi2,
where Vo, Vd are smooth vector fields on M. Therefore, in full generality, we can not construct Brownian motions on a Riemannian manifold by solving stochastic differential equations on the manifold. An intrinsic way to construct Brownian motions on a manifold by solving a stochastic differential equation can be done by working in the orthonormal frame bundle of the manifold: this the Eels-Elworthy-Malliavin construction. This construction is widely explained in the Section 3.4. of Chapter 3.
Appendix B
Vector Fields, Lie Groups and Lie Algebras
B.1
Vector fields and exponential mapping
Let 0 c 11I1 be a non empty open set. A smooth vector field V on 0 is simply a smooth map
V:
0
It is a basic result in the theory of ordinary differential equations that if K C 0 is compact, there exist E> 0 and a smooth mapping (I) : (—E, E) X K such that for x
O,
K and —E < t < E,
a(I)
— (t, x) x((t,x)), 4)(o,x)= x. at Furthermore, if y: Rn is a Cl path such that for 71
(D(t i, (D(t2, x ) ) = (1) (ti + t2 x). ,
Because of this last property, the solution mapping t the exponential mapping, and we denote (1.(t, x) = e t v (x \) if I t I is sufficiently small. If et" can be defined for any vector field is said to be complete. For instance if 0 = C°'-bounded then the vector field V is complete. 117
4(t, x) is called It always exists t E R, then the Rn and if V is
118
An Introduction to the Geometry of Stochastic Flows
The vector field V defines a differential operator acting on the smooth functions f : R as follows:
(CV f)(x) =
t=0 f (e ti f(x)) XE
O.
We also use the following notation
(V.f)(x),
(Lvf)(x)
that is, we apply V to f as a first-order differential operator. With this notation, observe that
(B.1) We note that V is a derivation, that is a map on C'(0, R), linear over R, satisfying for f, g E C'(0, R),
V( f g) = (V f)g + f(Vg). An interesting result is that, conversely, any derivation on C°° (O, is a vector field, i.e. has the form (B.1). If V' is another smooth vector field on 0, then it is easily seen that the operator VV' — V'V is a derivation. It therefore defines a smooth vector field on 0 which is called the Lie bracket of V and V' and denoted [V, VI. A straightforward computation shows that for x E 0,
n
n
[v, v/](x) i=1
07/
aX
0
3
Observe that the Lie bracket satisfies obviously [V, so-called Jacobi identity, that is:
[V, [V',
aX •
) 49Xi
VI = —[V', V] and the
VI + [V', [V", V]] + [V", [V, VII = O.
Let us now give a geometric meaning of the bracket. If ç : 0' 0 is a diffeomorphism between two open domains in Rn, the pull-back 0*V of the vector field V by the map 0 is the vector field on 0' defined by the chain rule,
0* V(x)= (0 -1 )0(x )(V(0(x))) , x E .
Vector Fields, Lie Groups and Lie Algebras
119
In particular, if V' is a smooth vector field on 0, (etv)*V' is defined on most of 0, for I t I small, and we can define the Lie derivative,
d rvV 1 = (—` etvyv, dt) t=0 as a vector field on O. It actually turns out that we have
LvV 1 = [V, If the vector fields V and V' commute, that is if [V, V' ] = 0 then the flows they generate locally commute; in other words, for any x c 0, there exists a (5 > 0 such that for I ti 1 + It 1< 6 ,
et2v 1 e t,v (x).
e t i v et2 v/ ( x)
Another essential point on the bracket is that one can flow in the direction of the commutator [V, V'] by a succession of flows along V and V'. More precisely, we have
e —ti( e --tV` e tV etV`
B.2
et2fir,V1
0(t3).
Lie derivative of tensor fields along vector fields
A smooth tensor field K of type (r, s), r, s E N, on IV is a smooth map
K : 0 (Rn) ®r (R" )® s , where Rn* denotes the dual space of Rn. By convention a (0,0) tensor is a smooth function f : 0 R. Any (r, s) smooth tensor field K can therefore be written
K
=
VT
al
as,
where the Vi 's are smooth vector fields and the ai 's smooth one-forms, i.e. applications 0 RV*. Let 0 : 0' 0 denote a diffeomorphism between two open domains in Rn , the pull-back of a smooth one-form c: 0 Rn* by the map 0 is the smooth one-form on 0' defined by,
0*a(x) cx 4, (x )(0(x)) , X E 0'. The pull-back by 0 can now be defined on any (r, s) smooth tensor field K in the following way:
An Introduction to the Geometry of Stochastic Flows
120
(1) If K is of type (0, 0), i.e. if K is a smooth function f : 0 —p
then
0* f (x) = (f o 0)(x), x E 01 ; (2) If K =V10 -- -0Vr 0a10-0Ces )
then
Thanks to this action we are now able to define the notion of Lie derivative along a given vector field. Let V denote a smooth vector field on 0 and let K denote a smooth (r, s) tensor field also defined on O. We define the Lie derivative of K along V as the smooth (r, s) tensor field on 0 given by
EvK =__ () t_o ( etv)*K. The Lie derivative along a vector field V enjoys the following properties: (1) If f is a smooth function 0 --4 R, then
rvf =Vf; (2) If K is a smooth (r, s) tensor field and V' a smooth vector field r[v,vii-K = [Cy, Lvdif = (LvGv , — Gv , Ev) K. B.3
Exterior forms and exterior derivative
For k > 0, denote Ak Rn the space of skew symmetric k multilinear maps (Rn) P —i For a E A'R'', 0 E A/R, we define a A 0 E Ak+/Rn in the following way: For any (u1, ..., uk +i) E (nr) k+1 , (a A /61 )(ui, •••, uk+i) = Esig n(a)a(u, (1) ,..., Ucr (0) 13 (v a (k+1) 1 “•7 where the sum is taken over all shuffles; that is, permutations a of {1, ..., k+ l} such that 0 ( 1) < ... < a(k) and cr(k + 1) < ... < o- (k + l). Besides the bilinearity, the basic properties of the wedge product A are the following. For a E Ak Rn , 0 E AiRn , -y E AmIFV: (1) a A 0 = (-1)(3 A a; (2) a A (3 A -y) = (a A /3) A -y .
Vector Fields, Lie Groups and Lie Algebras
121
If (a1 , ..., an) denotes the canonical basis of the dual space of Rn', then it is easily seen that the family fa il A ...A aik,1 < ii < ... < ik < n} is a basis for A'R'. It follows that for k > n, Ak Rn = {0}, while for n 0 < k < n, AkRn is a vector space with dimension ( ., The direct sum
k
of the spaces A'R'', k > 0 together with its structure as a real vector space and multiplication induced by A, is called the exterior algebra of R. It is a vector space with dimension 2n , which is denoted ARn. An exterior differential k-form on IV is a smooth mapping û: R n ---> AkRn. The set of differential k-forms shall be denoted 52k(Rn). The following fact is fundamental. There is a unique family of mappings dk : lik(Rn) _, fe-fi(Rn) (k = 0, ..., n), which we merely denote by d, called the exterior derivative on Rn such that:
(1) d is a A antiderivation. That is, d is R linear and for a E S2 k (Rn), 0 E fe(Rn), d(a A 0) = da A 0 + ( 1) k a A d,3; -
(2) If f: Rn --4 R is a smooth map, then df is the differential of f; (3) d 0 d = 0 (that is, dk+ 1 0 dk = 0). The link between the exterior derivative and the Lie bracket of vector fields is given by the following formula which is due to E. Cartan: For any differential one-form a and any smooth vector fields U and V,
da(U,V) = (Lua)(V) — (Lva)(U) — û([U, V]), where L denotes the Lie derivative.
B.4
Lie groups and Lie algebras
A Lie group G is a group that is also an analytic manifold, such that the group operations G x G --4 G and G —p G given by (g, h) --4 gh and g --4 g -1- are analytic maps. Let e denote the identity element of G. For each g G G we have left and right translations Lg and Rg , diffeomorphisms on G defined by L g (h) = gh and Rg (h) = hg. A smooth vector field V on
122
An Introduction to the Geometry of Stochastic Flows
G is said to be left-invariant if for any g E G,
LV= V. The set of left-invariant vector fields on G is a linear space called the Lie algebra of G and is denoted g. The evaluation map V V(e) provides a linear isomorphism between g and the tangent space to G at the identity. Therefore g is a finite dimensional vector space whose dimension is the dimension of G. An important point is that g is closed under the bracket operation. Namely, for VI., V2 E g, [V1, V2 ] E g. There is a natural map from g to G, the exponential map, defined as follows. It is possible to show that a left-invariant vector field V defines a global flow on G, that is et v is defined for every t E R. In particular, it is defined for t = 1. The map g G, V ev(e), is called the exponential map (by a slight abuse of notation, we shall still denote es" = ev(e)). The exponential map is a diffeomorphism from a neighborhood of 0 in g onto a neighborhood of e in G, and its differential map at 0 is the identity. Nevertheless, let us mention that, in general, the exponential map is neither a global diffeomorphism, nor a local homomorphism, nor even a surjection. It is easily seen from the properties of the exponential of vector fields that for V E g, s,t E R,
e sv e tv = e(s+t)v More generally, if VI, V2 E g commute, that is [Vi., V2 ] = 0, then
e vi ev2 = e v2 e vi It is of course possible to define the notion of Lie algebra independently of the notion of Lie group. Namely, an abstract Lie algebra is a vector space equipped with a bilinear map [., .1 : Z x Z such that:
(1) [X, Y] = —[Y, XI for X, Y EL; (2) [X ,[Y, Z]1+[Y,[Z, X]1+[Z,[X,Y ] 1 = 0 for X, Y, Z E L (Jacobi Identity). If Z is an abstract Lie algebra, the universal envelopping algebra A(Z) is defined as
A(Z) = ZeiJ, where,
(1) Q is the complexification of g, (2) Z2 = eck''10 ,Ctk,
Vector Fields, Lie Groups and Lie Algebras
123
(3) J is the two-sided ideal generated by {X0Y—Y0X—[X,11,X,YEZO. In the case where Z is the Lie algebra of a Lie group G, A(Z) can be identified with the set of all left-invariant operators on G. Besides vector fields there are many interesting left-invariant differential operators on G. For instance, up to a constant multiple, there is a unique left-invariant measure on G which is called a (left) Haar measure. In many but not all cases left Haar measure is also right Haar measure; In that case G is said to be unimodular. For instance all compact Lie groups are unimodular. Besides the Carnot groups, studied extensively in this book, there are some particular Lie groups of matrices that we want to mention. Example B.1 In the following examples we indicate the Lie groups and the corresponding Lie algebras. The exponential map is simply the usual exponential of matrices and the Lie bracket is given by the commutator [A, BI = AB —BA. Let n E N, n > 1, K = R or C, and Mn (K) the set of n x n matrices with entries in K. (1)
GL(1K) -=-- {A E .A4,i(K), det A
0 } , BL ( C) = M(K).
(2)
SL(K) = {A E .Mn (K), det A =---1}, sin (K) = {A E .Mn (K), TrA = 0 } . (3)
O n (R) = {A E GLn (R), AI = A-1 }, on (R) = {A E .A4,(R), A t = —A}. (4) Un (C) = {A E GL,(C), A* = A -1 }, un(C) = {A E Mn (C), A* = —A}. B.5
The Baker-Campbell-Hausdorff formula
The Baker-Campbell-Hausdorff formula shows, and this is a priori a non obvious fact, that the multiplication law in a Lie group is determined uniquely and very explicitly by the Lie algebra structure, at least in a neighborhood of the identity. Let G be a Lie group and let 9 be its Lie algebra. We have seen that if X,Y E g satisfy [X, Y ] = 0 then eX e Y = eY eX . It is
124
An Introduction to the Geometry of Stochastic Flows
actually easy to check that under this commutation assumption, we have e X e Y = eX +Y . This formula is not true for arbitrary X,Y E ft. Nevertheless, the question arises naturally whether one can obtain an explicit formula for eX e l/ . This is the content of the Baker-Campbell-Hausdorff formula which reads
where X and Y are elements of g in a sufficiently small neighborhood U of 0, and where the map P:UxU fi has a universal form which is independent of G. Let us precise the form of P (the formula we give comes from [Dynkin (1947) ] , but referred to as the Specht-Wever theorem in [Jacobson (1962) ]) . For this, we introduce some notations. For X E 43, let adX denote the linear endomorphism g given by (adX)(Y) = [X, 11, Y E 9. We have, +DO
( i) k.1 E (ady)qk(adX)Pk • • • (adY)qi (adX) 171 -
(B.2)
(EL 1(P1 qI)) (re—i Pl!ql!)
ki
where the inner sum is over the set of nonnegative integers (p i , qi ) such that pi + qi >0, (of course either p i = 1 or pl = 0, qi = 1), and we used the convention (adX) 1 = X. We have a simple generalization of (B.2) for more than two
exponentials: e
where p(Xi,
• • • ,
(_1)k_1
xi
ex = ,
X.) is given by (adX)Pk,n
E
dc..)
,
(adX 1 )Pk ,1 • • • ( ad,C)P 1, - • • • adX0P' ,1 (
(nm_,Km!)
the inner sum being taken over all nonnegative integers such that
pi,m > 0, 1. 1, • • . 7 k. Actually, the universality of P actually stems from a purely algebraic idenX,/ ]1 the tity between formal series. Indeed, let us denote by R[[X i ,
125
Vector Fields, Lie Groups and Lie Algebras
non-commutative algebra of formal series with d indeterminates. The exponential of Y E R[[X1,...,Xd]] is defined by eY =
E kzO
irk k!
Now, define the bracket between two elements Y and Z of R[[Xi, -•-, Xci]i by [Y,Z] = YZ — ZY,
and denote by ad the map defined by (adY)Z = [Y, Zj. In this context, the Baker-Campbell-Hausdorff formula reads e Y e Z e P(Y' Z) ,
where
+0°k— (_1) p(y, z) E
E (adZ)qk (adY)Pk • • • (adZ)ql (adY)P ,
1
k
(i Pz
(B.3)
q1) ) (1-111c— ' Pi !q11 )
and, as before, the inner sum is over the set of nonnegative integers such that
(pi ,
qi )
pi + qi >0.
Observe now that since
E
eY eZ =
P+g>0
YPZq p!q!
and ln (e Y e z ) =
+00 t_ 1 )m-1 E k (eYez _
the formula (B.3) can also be written as m YPZq
(- 1 )m -1
m=1
m
p+g>0
p!q
More generally, in the same way, we have for Yi , -•-, Y -1-00 • (-1)m — 1
E
YIP ' • • pi! • • •pn!
m
R[[X1,
= P(Yi, • • - ,Yn),
(B.4)
126
An Introduction to
the
Geometry of Stochastic Flows
where P(Yi , • • - , Yn ) is given by +co r_il k—i
E'
kj
E
(adYn )Pk , n
•••
(adYi )Pk , 1 • • • ( adYn )PI , - • • • ( adYi )P 1 . 1 1
k-1 =--,rrt=1 PlIm) (1-1m-:-_-1 (Ek/
Pi,m!)
the inner sum being, as before, taken over all nonnegative integers such that n
E „,,, m > 0, 1= 1, . - . , k.
rn---1
This is this version of the Baker-Campbell-Hausdorff formula which is used in the proof of the Chen-Strichartz formula.
B.6
Nilpotent Lie groups
An abstract Lie algebra Z is said to be nilpotent if for any X E Z, the map adX : Y —p Y ] is a nilpotent endomorphism of Z. A Lie group G is said to be nilpotent if its Lie algebra g is a nilpotent Lie algebra. If G is a nilpotent Lie group, then the Baker-Campbell-Hausdorff is global. Indeed, in that case, from Dynkyn's formula (B.2) there exists a Lie polynomial 7) : g x g —p g, with rational coefficients, such that for any X, Y E g, ex eY
=
(
)
According to (B.2), the first terms of P are the following:
P(X,Y) = X +Y + -[X, 17] + -1 [[X, 11, 17] — -1 [[X, 17], X] — 7k[Y, [X/ [X , Y]11 — -A [X, [Y, [X, Y111 + - - - • Moreover if G is simply connected then the exponential map g --- G is a diffeomorphism. In that case, the group law of G is thus fully characterized by the Lie algebra structure of g. It can also be shown that a nilpotent Lie group can always be seen as a group of unipotent matrices.
B.7
Free Lie algebras and Hall basis
In this section we describe the construction of a linear basis in the free Lie algebra with d generators. Let Z be the set of the d indeterminates X1 , ..., Xd. Let be an abstract Lie algebra and j : Z --4 Zo a mapping. The Lie algebra Z0 is called free over Z if for any abstract Lie algebra Z
zo
127
Vector Fields, Lie Groups and Lie Algebras
and any mapping f: Z 2, there is a unique Lie algebra homomorphism Z such that f = f o i. It can be shown that there is a free : Lie algebra over Z which is unique up to isomorphism. This Lie algebra is called the free Lie algebra with the d generators X I , Xd. It shall be denoted Z(XI , Xd). Denote now M(Z) the free monoid over Z ,1(h) the length of a word 1 E M(Z) and M i (Z) the set of all length i words. A Hall family over Z is an arbitrary linearly ordered subset H C M(Z) such that: (1) If u, v E H and 1(u) < l(v) then u < 7) ; (2) Z C H;
(3) H n .A42 (Z) =-- {xy, yE Z,x < y}, (4) H (M 2 (Z) U = fa(bc), a, b, c E H, b < a < bc,b
B.8
Basic Riemannian geometry
Let VI ,
Vn be C' bounded vector fields on IV such that for every x E 1Rn, (Vi (x), Vri(x)) is a basis of Rn; Let us denote ( 9', ..., On) E Sti (Rnr the dual basis of (VI , Va ). The first invariant associated with the system (V1, Vn ) is the family of scalar products (.9.).Eliz- on Rn obtained by declaring that for any x E 1Rn', the V, (x)) is orthonormal. The following theorem is the funfamily (VI (x), damental theorem of Riemannian geometry. Theorem B.1 such that:
There is a unique matrix of one-forms w =
('1 ) (10 + (4.) A 9 --= 0, that is, for any 1 < i < n, clO i +
w A 01 = 0;
-4.
(2) w t ---w, that is, for any 1 < j < n, (.4 =
The matrix of one-forms cu is called the connection form, and the equations dB
c.,.) A 9
0, cut =
the first structural equations. Recall now that a connection V on Rn is simply a convention for differentiating a vector field along another vector
128
An Introduction
ta the
Geometry of Stochastic Flows
field. If we denote by V(1119 the set of smooth vector fields on Rn, it is more precisely a map
V : V(Rn) x V(Rfl) V(Ir) such that for any U, V, W E V(Rfl) and any smooth f, g : Rn
R:
(1) V fU-FgVW = fVUW 9VVW;
(2) Vu(V W) = Vul7 VuW; (3) Vu(fV) = fVuV +U(f)V. Since, by assumption, for any x C Rn , the family of vectors (Vi (x), Vn (x)) is a basis for Rn , it is easily seen that the vector fields
Viti Vj
= EWII(ViWk• k=1
The connection V is called the Levi-Civita connection associated with the elliptic system (V1, Vn,) , This connection enjoys the two following additional properties: (1) It is torsion free, that is, for any smooth vector fields X, Y,
VuV — Vv U [U, 17]; (2) It is metric, that is, for any smooth vector fields U, V, W,
U(g(V,W)) = g(V uV, W) + g(V,VuW)• Let us now turn to the second structural equations. The equations
= dco + co A Lo, that is,
3
=--
3
Ewik
A w iF7 1 <
<
n,
k=1
define a skew-symmetric matrix of 2-forms such that for any smooth vector fields X, Y,
En3i.(x, y)vi vxvyvi vyvxvj i=-.1
1 < j < n.
129
Vector Fields, Lie Groups and Lie Algebras
The (1, 3)-tensor R defined by the property that for any smooth vector fields X, Y, Z,
R(X,Y)Z = VxVyZ — VyVxZ — Vix xiZ is called the Riemannian curvature tensor of V while the matrix of twoforms f is called the curvature form. The Ricci curvature Ric is a trace of the Riemannian curvature, it is the (0,2) tensor defined by
Ric(X,Y) =
g (R(X , Vi )Vi ,Y).
Observe that we also have
Ric(X, Y) = The Ricci transform Ric* is defined as the symmetric (1,1) tensor
RiC* (X) =
ER(X,Vi )Vi. -= 1
We also have
Ric*(X) =
E wi (x,vovj .
The last curvature quantity we wish to mention is the scalar curvature s, it is the function defined by
s= At this point, this is probably useful to see an example to understand how all this works in action. Example B.2 Let us consider on the Lie group SO(3) the left invariant frame generated by
( 0 1 0\ VI
= - I. 00 1 1(2 = 0 0 0
0 01 0 0 0
f00 0
0 0 i
, 173 =
\. -1
0-10
Let us recall that the following commutation relations hold
v2J = v3, [v2, v3]
[v3,
- v2.
0 0
130
An Introduction to the Geometry of Stochastic Flows
For the dual frame 9 = (9 1 ,02 ,03), we have
c10 1 = —02 A 93 d92 = -03 AG'
3 = A9269 . Thus, to find the connection form, we have to solve the system 02 A 93 03 A 91 9 1 A 02
= A 02 + C.O1 A 93 = —A A 91 + A 93 A 9 1 +w A192 . =
The previous system admits the unique solution 1 -
( 0 —03 02 03 0 -9 1
292
91
.
0
To find the curvature form, we first compute
dw
1
(9 1 A 0 2 -03 A 9 1 )
(
—' 19 A 02
03 A G'
0 -02 A 03
02 A 93
and then,
w
A
w =--
1
( 0 -9 3 92 ) 03 0 -9 1 A -02 0 1 0
(
0 -03 02 )
03 0 —0' -0 2 8 1 0
0 0 2 A 9 1 93 AG' 2 AO' 0 03 A 02-0 ) . —63 A e, --(93 A 02 o
Therefore,
0 1 A 02 - 03 A 0 1 1 12=ch.,,-1-wAw=— —0 1 A02 02 A03 Ø 4 9 3 A IV —92 A 93 o It follows immediately that for every smooth vector fields X, Y on SO(3)
1
R(X,Vi )Vi = - X, 4
Ric(X, Y) =
3
4
= 1,2,3,
Vector Fields, Lie Groups and Lie Algebras
Ric*(X) =
131
3 4
and
s
9 4
Observe that the same computations could have been performed on SU(2). There is a natural measure associated with the moving frame (V1, ..•, Vd), the so-called Riemannian measure: This is the measure it./ on Rn given by the density
dpc „ dL
1 det (VI (x), Vn (x)) I
=
x ER, '
where L denotes the Lebesgue measure. There is also a natural and fundamental second order elliptic differential operator, the Riemannian Laplacian which is given by A
vz2
Evvyi.
This operator is a Riemannian invariant, that is, it does not depend on the chosen moving frame (Vi, Va ). More precisely, if cp is a smooth map from Rn onto the set of n x n orthogonal matrices, then
E u? _ Evui ui E vi2 _
T1
where Ui = yo(Vi ). Observe that the sum of squares operator
vi2 does not enjoy this property and is therefore not a Riemannian invariant. The term Eni VVYi is actually at the heart of the difference between Riemannian and Euclidean geometry. One can define A in a more intrinsic way. Let us consider the adjoint d* of the exterior derivative d, that is d* is defined by the property that for every smooth and compactly supported one-form a and any smooth and compactly supported function f, we have
f fd*ad,u =f adfdp.
132
An Introduction to the Geometry of Stochastic Flows
Then, it can be shown that we have A --= —d* o d.
A fundamental property of A which stems directly from the previous identity is that it is self-adjoint with respect to itz: For every smooth and compactly supported functions f,g : Rn --4 R,
f
f6,gd itt =-- f gAfclitt.
Bibliography
Alexopoulos G.K., Lohoué N. (2004): On the large time behaviour of heat kernels on Lie groups, Duke Math. Journal, To appear. Azencott R. (1982): Formule de Taylor stochastique et développements asymptotiques d'intégrales de Feynman. In Azema, Yor (Eds.), Séminaire de probabilités XVI, LNM 921, 237-284, Springer. Bakry D. (1994): L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on probability theory, Ecole d'été de probabilités de Saint-Flour 1992, LNM 1581, Springer. Baudoin F. (2002): Conditioned stochastic differential equations: Theory, Examples and Applications to finance. Stochastic Processes and their Applications, Vol. 100, 109-145. Baudoin F., Teichmann J. (2003): Hypoellipticity in infinite dimensions and an application in interest rate theory, preprint. Baudoin F. (2003): Stochastic differential equations driven by loops in Carnot groups, preprint. Baudoin F. (2004a): Équations différentielles stochastiques conduites par des lacets dans les groupes de Carnot, in French, CRAS serie I 338, 719-722. Baudoin F. (2004b): The tangent space to a hypoelliptic diffusion and applications, to appear in Séminaire de Probabilités XXXVIII. Baudoin F., Coutin L. (2004): Etude en temps petit du flot des équations conduites par des mouvements browniens fractionnaires, preprint. Bellaiche A. (1996): The tangent space in sub-Riemannian geometry, In SubRiemannian Geometry, edited by A. Bellakhe and J.J. Risler, Birkhguser. Ben Arous G. (1989a): Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale, Annales de l'institut Fourier, tome 39, p. 73-99. Ben Arous G. (1989b): Flots et séries de Taylor stochastiques, Journal of Probability Theory and Related Fields, 81, pp. 29-77. Berger M., Gauduchon M., Mazet E. (1971): Le spectre d'une variété Riemannienne, LNM Vol. 194. Bismut J.M. (1981): Martingales, the Malliavin calculus and hypoellipticity under general Hbrmander's condition. Z. fiir Wahrscheinlichkeittheorie verw.
133
134
An Introduction to the Geometry of Stochastic Flows
Gebiete, 56, 469-505. Bismut J.M. (1984a): Large Deviations and the Malliavin calculus, Birklauser. Bismut J.M. (198413): The Atiyah-Singer Theorems: A Probabilistic Approach, J. of Func. Anal., Part I: 57 (1984), Part II: 57, 329-348. Bony J.M. (1969): Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Annales de l'institut Fourier, tome 19, ni, p. 277-304. Bourbaki N. (1972): Groupes et Algèbres de Lie, Chap. 1-3, Hermann. Caste11 F. (1993): Asymptotic expansion of stochastic flows, Prob. Rel. Fields, 96, 225-239. Chen K.T. (1957): Integration of paths, Geometric invariants and a Generalized Baker-Hausdorff formula, Annals of Mathematics, 65, n1. Chen K.T. (1961): Formal differential equations, Ann. Math. 73, 110-133. Chow W.L. (1939): Über system von lineare partiellen differentialgleichungen erster ordnung, Math. Ann., 117. Cohen S. (1995): Some Markov properties of Stochastic Differential Equations with jumps, Séminaire de Probabilits, Vol. XXIX, Springer. Coutin L., Qian Z.M. (2002): Stochastic rough path analysis and fractional Brownian motion, Probab. Theory Relat. Fields 122, 108-140. Dellacherie C., Meyer P.A.: Probabilités et potentiel, Hermann, Paris, Vol. I (1976), Vol. II (1980), Vol. III (1983), Vol. IV (1987). Derridj M. (1971): Sur une classe d'opérateurs différentiels hypoelliptiques à coefficients analytiques. Séminaire Goulaouic-Schwartz 1970-1971: Equations aux dérivées partielles et analyse fonctionnelle, Exp. 12, Centre de math., Ecole Polytechnique. Doss H. (1977): Lien entre équations différentielles stochastiques et ordinaires, Ann. Inst. H. Poincaré, Prob. Stat. 13, 99-125. Driver B., Thalmaier A. (2001): Heat Equation Derivative Formulas for Vector Bundles, J. Func. Anal., 1, 42-108. Dynkin E.: Markov processes Vol.I, Springer, Berlin Dynkin E. (1947): Evaluation of the coefficients of the Campbell-Hausdorff formula, Dokl. Akad. Nauk. 57, 323-326. Eckmann J.P. and Hairer M. (2003): Spectral properties of hypoelliptic operators. Comm. Math. Phys. 235, 2, 233-253. Elworthy K.D. (1982): Stochastic Differential Equations on Manifolds. London Math. Soc, Lecture Notes Series 70. Cambridge University Press. Elworthy K.D., Li X.M. (1994): Formulae for the derivatives of heat semigroups, J. Func. Anal. 125, 252-286. Emery M. (1989): Stochastic calculus in manifolds, Springer universitext. Emery M. (2000): Martingales continues dans les variétés différentiables, Lectures on Probability theory and Statistics, Ecole d'Eté de Saint-Flour XXVIII 1998, Lect. Notes in Math. 1738, 1-84. Falconer K.J. (1986): The geometry of fractal sets, Cambridge Univ. Press. Fitzsimmons P., Pitman J.W., Yor M. (1993): Markov bridges, Construction, Palm interpretation and splicing, Seminar on Stochastic Processes, Birkhauser, 101-134.
Bibliography
135
Fliess M., Normand-Cyrot D. (1982): Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T. Chen, in Séminaire de Probabilités, LNM 920, Springer-Verlag. Folland G.B., Stein E.M. (1982): Hardy spaces on homogeneous groups. Princeton University Press. F611mer H. (1981): Calcul d'Itô sans probabilités, Séminaire de probabilités XV, LNM 850, Springer, 143-150. Friz P., Victoir N. (2003): Approximations of the Brownian rough path with applications to stochastic analysis, preprint. Gaveau B. (1977): Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math. 139 (1-2), 95-153. Gershkovich V. Ya., Vershik A.M. (1988): Non holonomic problems and the theory of distributions, Acta App!. Math., 12, 181-209. Gershkovich V. Ya., Vershik A.M. (1994): Non.holonomic Dynamical Systems, Geometry of Distributions and Variational Problems, In Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, Vol. 16, Eds. V.I. Arnold, S.P. Novikov. Go16 C., Karidi R. (1995): A note on Carnot geodesics in nilpotent Lie groups. Jour. Control and Dynam. Systems, 1, 4, 535-549. Goodman N. (1976): Nilpotent Lie groups, Springer Lecture Notes in Mathematics, Vol. 562. Gromov M. (1996): Carnot-Carathéodory spaces seen from within, In SubRiemannian Geometry, edited by A. Bellaiche and J.J. Risler, Birkhauser. Gromov M. (1999): Metric structures for Riemannian and non-Riemannian spaces, with appendices by M. Katz, P. Pansu and S. Semmes. Birkhauser. Gordina M. (2003): Quasi-invariance for the pinned Brownian motion on a Lie group, Stochastic Process. Appl. 104, no. 2, 243-257. H6rmander L. (1967): Hypoelliptic second order differential equations. Acta Math. 119, 147-171. Hsu E.P. (2002): Stochastic Analysis on manifolds, AMS, Graduate Texts in Mathematics, Volume 38. Hunt G.A. (1958): Markov processes and potentials: 1,11,111, Illinois J. Math, 1, 44-93, 316-369 (1957); 2, 151-213. Ikeda N., Watanabe S. (1989): Stochastic Differential Equations and Diffusion Processes. Second Edition. North-Holland Publ. Co., Kodansha Ltd., Tokyo. Itô K. (1944): Stochastic integral, Proc. Imp. Acad. Tokyo, 20, 519-524. Itô K., McKean H.P. (1996): Diffusions Processes and their Sample Paths, Classics in Mathematics, Springer, reprint of the 1974 Edition. Jacobson N. (1962): Lie algebras, New York, Interscience. Kobayashi S., Nomizu K. (1996): Foundations of Differential Geometry, Vol. 1, Wiley Classics Library. Kohn J.J. (1973): Pseudo-differential operators and hypoellipticity. Proc. Symp. Pure Math. 23, 61-69. Kunita H. (1980): On the representation of solutions of stochastic differential
136
An Introduction to the Geometry of Stochastic Flows
equations. In Azema, Yor (Eds.), Séminaire de probabilités XIV, LNM 784, Springer, 282-304. Kunita H. (1981): On the decomposition of solutions of stochastic differential equations. In Williams (Ed.), Stochastic integrals, Proceedings, LMS Duhram Symposium 1980, LNM 851, Springer, 213-255. Kunita H. (1990): Stochastic Flows and Stochastic Differential Equations, Cambridge studies in advanced mathematics, 24. Kusuoka S. (2001): Approximation of Expectation of Diffusion Process and Mathematical Finance, Advanced Studies in Pure Mathematics, Taniguchi Conference on Mathematics Nara, 147-165. Léandre R. (1987): Intégration dans la fibre associée à une diffusion dégénérée, Journal of Probability Theory and Related Fields, 76, 341-358. Léandre R. (1992): Développement asymptotique de la densité d'une diffusion dégénérée, Forum Math. 4, 1, 45-75. Ledoux M., Qian Z.M., Zhang T. (2002): Large deviations and support theorem for diffusions via rough paths. Stochastic Processes and their Applications 102, 265-283. Lejay A. (2004): An introduction to rough paths, to appear in Séminaire de Probabilités XXXVII, Springer. Lyons T.J. (1998): Differential Equations Driven by Rough Signals.Revista Mathemktica Iberio Americana, Vol 14, No 2, 215 - 310. Lyons T., Qian Z.M. (2002): System control and rough paths, Oxford mathematical monographs. Lyons T., Victoir N. (2004): An extension Theorem to Rough Paths, preprint. Malliavin P. (1974): Géomètrie Différentielle Stochastique, Séminaire de Mathématiques Supérieures, Universit é de Montréal. Malliavin P. (1978): Stochastic calculus of variations and hypoelliptic operators. In: Proc. inter, Symp. on Stoch. Duff. Equations, Kyoto 1976. Wiley, 195263. Malliavin P. (1997): Stochastic Analysis, Grundlehren der mathematischen Wissenschaften, Vol. 313, Springer. McKean H. P. (1969): Stochastic Integrals, Academic Press, New York. Margulis G., Mostow G.D. (1995): The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5, 402-433. Menikoff A., Sj6strand J. (1978): On the eigenvalues of a class of hypoelliptic operators, Math. Arm. 235, 55-85. Montgomery R. (2002): A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical surveys and Monographs, Vol. 91, AMS. Mitchell J. (1985): On Carnot-Carathéodory metrics. J. Differential Geom., 21, 35-45, (1985). Nualart D. (1995): The Malliavin calculus and related topics, Springer, Berlin Heidelberg New-York. Nualart D., Rascanu A. (2002): Differential equations driven by fractional Brownian motion, Collect. Math. 53, 1, pp. 55-81. Oleinic 0.A., Radkevic E.V. (1973): Second order equations with non negative characteristic form. A.M.S. and Plenum Press.
Bibliography
137
Pansu P. (1989): Métriques de Carnot-Carathéodory et quasi-isométries des espaces symétriques de rang un, Ann. Math., 129, 1-60. Protter P. (2004): Stochastic integration and differential equations, Vol. 21 of Applications of Mathematics, second edition, Springer. Rayner C.B.: The exponential map for the Lagrange problem on differential manifolds, Phil. Trans. of the Roy. Soc. of London, Ser. A, Math. and Phys., 1127, V. 262, 299-344. Reutenauer C. (1993): Free Lie algebras, London Mathematical Society Monographs, New series 7. Revuz D., Yor M. (1999): Continuous Martingales and Brownian Motion, third edition, Springer-Verlag, Berlin. Rogers L.C.G., Williams D.: Diffusions, Markov processes and Martingales, Vol. 1, second edition, Cambridge university press. Rotschild L.P., Stein E.M. (1976): Hypoelliptic differential operators and Nilpotent Groups, Acta Mathematica, 137, 247-320. Sipilainen E.M. (1993): A pathwise view of solutions of stochastic differential equations, Phd thesis, University of Edinbhurg. Strichartz R.S. (1987): The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, Jour. F'unc. Anal., 72, 320-345. Stroock D.W., Varadhan S.R.S. (1979): Multidimensional Diffusion processes, Springer-Verlag. Stroock D.W. (1982): Topics in stochastic differential equations, Tata Institute, Bombay. Stroock D.W., Taniguchi S. (1994): Diffusions as integrals of vector fields, Stratonovitch diffusion without Itô integration, Vol. in the honour of E. Dynkin, Birkhauser, Prog. Proba. 34, 333-369. Stroock D.W. (2000): An introduction to the analysis of paths on Riemannian manifolds, Mathematical surveys and Monographs, Vol. 74, AMS. Siissmann H. (1978): On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6, 19-41. Takanobu S. (1988): Diagonal short time asymptotics of heat kernels for certain degenerate second order differential operators of Harmander type. Publ. Res. Inst. Math. Sci 24, 169-203. Taylor M.E. (1986): Noncommutative Harmonic Analysis, Mathematical Surveys and Monographs, 22, American Mathematical Society. Taylor M.E. (1996a): Partial Differential Equations, Basic Theory, Springer Verlag, Texts in Applied Mathematics 23. Taylor M.E. (1996b): Partial Differential Equations, Qualitative Studies of Linear Equations, Springer, Applied Mathematical Sciences 116. Thalmaier A. (1997): On the Differentiation of Heat Semigroups and Poisson Integrals, Stochastics and Stochastics Reports, 61, 297-321. Watanabe S. (1984): Lectures on stochastic differential equations and Malliavin calculus, Tata institute of fundamental research, Springer Verlag. Yamato Y. (1979): Stochastic differential equations and nilpotent Lie algebras. Z. Wahrscheinlichkeitstheorie. Verw. Geb., 47, 213-229. Yosida K. (1952): Brownian motion in homogeneous Riemannian space, Pacific
138
An Introduction to the Geometry of Stochastic Flows
J. Math., 2, 263 296. Young L.C. (1936): An inequality of the Holder type connected with Stieltjes integration. Acta Math., 67, 251-282. Zdhle M. (1998): Integration with respect to fractal functions and stochastic calculus I. Prob. Theory Rd. Fields, 111, 333-374. -
Index
Growth vector, 61
Baker-Campbell-Hausdorff formula,
8, 124 Ball-box theorem, 38 Bochner's Laplacian, 72 Brownian motion, 98 Brownian motion on a manifold, 114
Harmander's condition, 57 Hall basis, 127 Hall-Witt theorem, 127 Heisenberg group, 25 Homogeneous norm, 38
Carnot group, 33
Horizontal Brownian motion, 73 Horizontal distribution, 27
Carnot-Carathéodory distance, 28, 38, 60 Cartan formula, 121 Chen's relations, 5 Chen-Strichartz formula, 8 Chow theorem, 28 Chow's theorem, 37, 59 Connection form, 127
Infinitesimal generator, 110 Itô's formula, 106, 109 Itô's integral, 103 Jacobi identity, 118
Kolmogorov's continuity criterion, 98 Differential system, 57, 60 Lie algebra of a Lie group, 122 Lie bracket of vector fields, 118 Lie derivative, 119, 120 Local martingale, 102
Ehresmann connection, 73 Exterior derivative, 121 Exterior forms, 121 Feynman-Kac formula, 110 First structural equation, 127
Malliavin calculus, 111 Malliavin matrix, 113 Markov process, 99 Martingale, 101
Fractional Brownian motion, 17 Fundamental horizontal vector fields,
72, 78 Nilpotentization, 61 Normal point, 62
Gaveau diffusion, 26 Girsanov's theorem, 107 Gromov-Hausdorff distance, 64
Pansu's derivative, 41 139
140
An Introduction to the Geometry of Stochastic Flows
Regular point, 61 Ricci curvature, 129 Ricci transform, 129 Riemannian curvature tensor, 129 Riemannian Laplacian, 131 Rough paths theory, 47 Second structural equations, 128 Semimartingale, 105 Signature of a Brownian motion, 3 Stratonovitch's integral, 105 Sub-Riemannian geodesic, 28, 38, 60 Tensor field, 119 Universal envelopping algebra, 122 Wiener measure, 99 Wiener space, 107
