Masatoshi Fukushima Selecta
Masatoshi Fukushima
Masatoshi Fukushima Selecta Edited by
Niels Jacob Yoichi Oshima Masayoshi Takeda
De Gruyter
Editors Niels Jacob Department of Mathematics Swansea University Singleton Park Swansea SA2 8PP, Wales, United Kingdom E-mail:
[email protected] Yoichi Oshima Faculty of Engineering Kumamoto University 2-39-1 Kurokami Kumamoto 860, Japan E-mail:
[email protected] Masayoshi Takeda Mathematical Institute Tohoku University Aoba, Sendai, 980-8578, Japan E-mail:
[email protected]
ISBN 978-3-11-021524-3 e-ISBN 978-3-11-021525-0 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2010 Walter de Gruyter GmbH & Co. KG, Berlin/New York Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com
Preface
Professor Masatoshi Fukushima is one of the most influential probabilists of our times. His fundamental work on Dirichlet forms and Markov processes made Hilbert space methods a tool in stochastic analysis and by this he opened the way to several new developments. His impact on a new generation of probabilists in his native country as well as in many other countries can hardly be overstated. In publishing a selection of his seminal papers we aim to serve the community, and at the same time we want to express our appreciation of a highly respected, humane scholar. All owners of copyrights of papers being included in the Selecta (see the list at the end of this volume) followed the old and good tradition to grant permission for a reproduction in a Selecta without any charge. Unfortunately this is no longer a policy adopted by all publishers. Therefore we are especially grateful to those who by their generosity continue to support the mathematical community in keeping the tradition of publishing selected or collected works alive. The editors’ thanks go to all who supported us in our enterprise, in particular we want to mention Professor T. Uemura (Kansai University), Dr. K. P. Evans (Swansea University) as well as S. Albroscheit and Dr. R. Plato (Walter de Gruyter). Swansea, Kumanoto and Sendai Fall 2009
Niels Jacob ¯ Y¯oichi Oshima Masayoshi Takeda
V
VI
Curriculum vitae
1935, August 23 1959 1961 1962 1963 1966 1967 1969–1971 1972 1977 1980 1990 1998 1998 2003 2006 2007
Born in Osaka, Japan Graduated from Kyoto University, Faculty of Science Master degree from Kyoto University, Faculty of Science Research Associate of Nagoya University, Faculty of Science Lecturer of Kyoto University, College of General Education Associate Professor of Tokyo University of Education, Faculty of Science Doctor of Science from Osaka University Post Doctral Fellow, University of Illinois, Department of Mathematics, Urbana-Champaign Associate Professor of Osaka University, Faculty of Science Professor of Osaka University, College of General Education Visiting Professor Bielefeld University, Faculty of Physics Professor of Osaka University, Faculty of Engineering Sciences Professor Emeritus, Osaka University Professor of Kansai University, Faculty of Engineering Awarded Analysis Prize of Mathematical Society of Japan Retired from Kansai University Honorary Professor Swansea University
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Contents
Entries in square brackets refer to the bibliography on pages 543–549.
Preface
v
Curriculum vitae
vii
Professor Masatoshi Fukushima – Scholar and Mentor by N. Jacob
1
A construction of reflecting barrier Brownian motions for bounded domains [R5]
3
On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities [R6]
36
Regular representations of Dirichlet spaces [R8]
72
Dirichlet spaces and strong Markov processes [R9]
91
On the generation of Markov processes by symmetric forms [R11]
131
Almost polar sets and ergodic theorem [R12]
165
On an -estimate of resolvents of Markov processes [R19]
181
Dirichlet spaces and additive functionals of finite energy [R22]
189
A note on irreducibility and ergodicity of symmetric Markov processes [R29]
196
Capacitary maximal inequalities and an ergodic theorem [R31]
204
Basic properties of Brownian motion and capacity on the Wiener space [R32]
211
A transformation of a symmetric Markov process and the Donsker-Varadhan theory (with M. Takeda) [R34]
226
-capacities for general Markov semigroups (with H. Kaneko) [R37]
242
IX
Contents
On the continuity of plurisubharmonic functions along conformal diffusions [R38]
249
On Dirichlet forms for plurisubharmonic functions (with M. Okada) [R43]
256
On quasi-supports of smooth measures and closability of pre-Dirichlet forms (with Y. LeJan) [R49]
299
Dirichlet forms, diffusion processes and spectral dimensions for nested fractals [R53]
308
On a spectral analysis for the Sierpinski gasket (with T. Shima) [R55]
319
On a strict decomposition of additive functionals for symmetric diffusion processed [R58]
354
On a decomposition of additive functionals in the strict sense for a symmetric Markov process [R59]
359
Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps (with M. Tomisakai) [R61]
374
On semi-martingale characterizations of functions of symmetric Markov processes [R66]
411
On regular Dirichlet subspaces of 1 and associated linear diffusions (with X. Fang and Y. Ying) [R78]
443
Entrance law, exit system and Lévy system of time changed processes (with Z.-Q. Chen and J. Ying) [R81]
458
Extending Markov processes in weak duality by Poisson point processes of excursions (with Z.-Q. Chen and J. Ying) [R82]
497
Acknowledgments
541
Bibliography
543
X
Professor Masatoshi Fukushima – Scholar and Mentor
Let me start with some remarks on the scientific achievement of Professor Fukushima. First investigations of Professor Fukushima were related to diffusions under boundary conditions and this led him to consider related Hilbert spaces. From here the road was open to Dirichlet forms, Beurling’s and Deny’s axiomatisation of the notion of energy in potential theory. The breakthrough was the 1971 paper in the Transactions of the American Mathematical Society where a Hunt process was constructed associated to a given regular Dirichlet form. This result was immediately recognised by experts as an outstanding one, already 1978 Professor Fukushima was an invited speaker in the session on Probability Theory in the ICM in Helsinki. Adding to this I would like to mention two other honours Professor Fukushima received: The Analysis Prize from the Japanese Mathematical Society and being an Invited Lecturer of the London Mathematical Society. The 1971 paper and the book “Dirichlet forms and Markov Processes” published in 1980 in English changed the landscape of modern probability theory. Of course there are many other contributions of Professor Fukushima’s worth mentioning:
exceptional sets and refinements
plurisubharmonic functions (especially the Acta Mathematica Paper with M. Okada)
stochastic analysis on fractals
boundary behaviour and traces of Markov processes
and many more. Of particular importance was and is the influence of his work to mathematical physics. The construction of diffusion processes on infinite dimensional state spaces highly depend on his seminal contribution. The impact of the “new” book “Dirichlet Forms and Symmetric Markov Processes” written jointly with Y. Oshima and M. Takeda can hardly be overstated. In 1990 when participating in a conference organised by Professor Kunita in Nagoya, I once had a coffee with Professor Shinzo Watanabe. In our discussion Professor Watanabe stated that for him in the 1970’s there had been two major breakthroughs in probability theory: Malliavin calculus and Fukushima’s theory of Hunt processes This is a slightly modified version of a banquet speech given during a meeting to celebrate Professor Fukushima’s 70th birthday.
1
Professor Masatoshi Fukushima – Scholar and Mentor
associated with Dirichlet forms – Professor Watanabe is a very modest person – he should have added his own contributions too. However there is no doubt, Professor Fukushima’s work is of lasting impact. Mathematics is an international subject and Professor Fukushima was and is acting on the international stage. This is natural to all the outstanding Japanese probabilists raised in Professor K. Itô’s school. Professor Fukushima’s work on the 1971 paper was partly done when being in the U.S.A. with the late Professor Doob. Here he also established contacts to Martin Silverstein and he, Professor Fukushima, always emphasised Professor Silverstein’s contributions to our subject. Professor Fukushima was very engaged in the series of Japanese-Russian seminars on probability theory and a frequent visitor to European countries. In addition he was a great help and supporter of many young non-Japanese mathematicians, Y. Lejan, J. Kim, Z.-M. Ma, Z.-Q. Chen, J. Ying, . . . , and of course I have to mention myself. Being a world-open mathematician, a scholar who has visited (partly for longer periods) many countries is one aspect of Professor Fukushima. There is another one: As many cosmopolitans he is deeply rooted in his own culture, i.e. in the traditional Japanese culture. This makes any encounter with him also an encounter with Japan. Through him I myself as well as quite a few of my students and colleagues learnt to appreciate Japan’s great culture. Professor Fukushima belongs to the generation whose childhood was in war-time – I recommend everyone to read Kappa Senoh’s “A Boy Called H” to get a feeling of what this meant to his generation. He as many other Japanese scientists, writers and artists of his generation took on the difficult task to assure his country a respected place in the modern post-war world – and they were rather successful. A final more personal word. Due to his relations to the late Professor Heinz Bauer we met first in Erlangen. I am very happy that I could build on Professor Bauer’s contacts and could even extend them. This refers to the Oberwolfach meetings on Dirichlet forms or a German-Japanese exchange programme supported by DFG and JSPS. Moreover I am grateful that several of my own (former) PhD students could not only visit Japan but could start to build up their own contacts bringing the collaboration to the next generation. You, Professor Fukushima, have made lasting contributions to Mathematics and you have been over the years of great support to many of us. For this we are grateful and we are looking forward to many further years to come with your company. Niels Jacob
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Probab. Theory Relat. Fields 106, 521–557 (1996)
c Springer-Verlag 1996
Construction and decomposition of reflecting diffusions on Lipschitz domains with H¨older cusps Masatoshi Fukushima1 , Matsuyo Tomisaki2 1 Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan (e-mail:
[email protected]) 2 Department of Mathematics, Faculty of Education, Yamaguchi University, Yamaguchi, Japan (e-mail:
[email protected]. yamaguchi-u.ac.jp)
Received: 4 October 1995
Summary. We consider a d -dimensional Euclidean domain D whose boundary is Lipschitz continuous but admits locally finite number of outward or inward H¨older cusp points. Using a method of Stampacchia and Moser for PDE, we first construct a conservative diffusion process on the Euclidean closure of D possessing a strong Feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients aij . The sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy. The second additive functional will be proved to be of bounded variation with a Skorohod type expression whenever aij is weakly differentiable and the H¨older exponent at each outward cusp boundary point is greater than 1 2 regardless the dimension d . Mathematics Subject Classification (1991): 60J60, 60J55, 35J25, 31C25
1 Introduction Let D be a domain in the d -dimensional Euclidean space R d and D = D ! 'D R d be its closure. The d -dimensional Lebesgue measure is denoted by m = m(dx ) or simply by dx . Given measurable functions aij (x ) 1 i j d on D such that aij = aji
1 2
aij (x )i j 2 x D R d
(1.1)
1i j d
for some constant 1, we consider a Dirichlet form on L2 (D) = L2 (D; m) defined by 374
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1
[ ] = H (D) (u ) =
aij (x )'i u(x )'j (x )dx u H 1 (D)
D 1i j d
(1.2) where H 1 (D) = #u L2 (D) : 'i u L2 (D) 1 i d the Sobolev space of order 1. Let #Tt t 0 be the strongly continuous semigroup of Markovian symmetric operators on L2 (D) associated with the Dirichlet form . We denote by C0 (D) [resp. B0 (D)] the space of continuous functions [resp. bounded measurable functions] on D with compact support [resp. vanishing outside a bounded set]. We further denote by C (D) [resp. C0 (D)] the space of bounded continuous functions on D [resp. the restrictions to D of functions in C0 (R d )]. Suppose that the Dirichlet form is regular on L2 (D) rather than on L2 (D) in the sense that H 1 (D) C0 (D) is dense in the space H 1 (D). This is the case for instance when the domain D is of class C in the sense that 'D is locally expressible as a graph of a continuous function of d 1 variables ([16]). According to general theorems ([13]), there exists then a conservative diffusion process M = (Xt Px ) on D associated with the Dirichlet form in the sense that the transition probability pt (x E ) = Px (Xt E ) of M satisfies that pt f is a version of Tt f for any f B0 (D)
(1.3)
However we are now concerned with a highly non-trivial problem of constructing the process M on D with a strong Feller resolvent: G (B0 (D)) C (D)
(1.4)
which particularly implies the absolute continuity of the transition probability: pt (x ) " m
for any t 0 and x D
(1.5)
If both the conditions (1.3) and (1.5) are fulfilled, then we can invoke a general decomposition theorem in [13] of additive functionals (AFʼs in abbreviation) in the strict sense to conclude that the sample path Xt = (Xt1 Xtd ) of M admits the unique decomposition Xti X0i = Mti + Nti
1 i d Px as for any x D
(1.6)
where Mti are martingale additive functionals (MAFʼs in abbreviation) in the strict sense with covariations M i M j t = 2
t
aij (Xs )ds 1 i j d Px as for any x D
(1.7)
0
and Nti are continuous additive functionals (CAFʼs in abbreviation) in the strict sense locally of zero energy. Nti are not necessarily of bounded variation (on each finite time interval) but locally of zero quadratic variation in a certain sense ([13]). Natural questions arise:
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Reflecting diffusions on H¨older domains
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(I) Under what condition on the domain D, there exists a conservative diffusion process M = (Xt Px ) on D satisfying (1.3) and (1.5) ? (II) Under what additional conditions on the domain D and coefficients aij , the second terms Nti of Xti are of bounded variation Px a.s. for any x D ? When D is a general bounded Lipschitz domain and aij = 12 ij 1 i j d , Bass and Hsu gave affirmative answer to the both questions (I) and (II) in [2] and [3] respectively. Actually [2] and its refinement [13; Example 5.2.2] gave an explicit expression of Nti as Nti =
1 2
t
ni (Xs )dLs 1 i d
Px as for any x D
(1.8)
0
where n = (n1 nd ) is the inward unit normal vector at the boundary 'D and Lt is a positive continuous additive functional (a PCAF in abbreviation) in the strict sense associated with the surface measure on 'D; the local time of Xt on the boundary. In this case, the diffusion M is called the (normally) reflecting Brownian motion on D and the decomposition (1.6) with (1.8) is called its Skorohod representation. The first process (Mt1 Mtd ) appearing in (1.6) is the standard d -dimensional Brownian motion starting at the origin in this case. On the other hand, by extending a work of S.R.S.Varadhan and R.J.Williams on an infinite two-dimensional wedge [22], DeBlassie and Toby [7] have formulated under a submartingale problem a normally reflecting Brownian motion on a two-dimensional standard outward cusp domain
C = (x y) R 2 : y x 0 1 and constructed it from the normally reflecting Brownian motion on the upper half plane by means of a conformal map and a random time change. They have also shown in [8] that the constructed process admits the Skorohod representation if 12 but otherwise the process starting at the origin fails to be a semimartingale. By thinking of the direct product of the DeBlassie-Toby reflecting Brownian motion on C with the standard d 2-dimensional Brownian motion, we see that 1 older exponent for the semi-martingale property 2 is still the critical value of the H¨ of the reflecting Brownian motion on the special H¨older domain C R d 2 R d . It is therefore tempting to consider the problem (I) for a general H¨older domain D and further look for a critical value of the H¨older exponent with regard to the question (II). In this paper, we do not deal with a most general H¨older domain. However we assume that D is a general (not necessarily bounded) Lipschitz domain allowing locally finite number of outward or inward cusp boundary points with H¨older exponents uniformly bounded away from zero. Our first aim is to give an affirmative answer to the problem (I) (Theorem 2.1 and Theorem 2.2) by employing the PDE methods of Stampacchia and Moser. We then assume that (1.9) 'j aij L
loc (D) 1 i j d
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M. Fukushima, M. Tomisaki
and give an affirmative answer to the question (II) under the condition that the H¨older exponent at each outward cusp boundary point is greater than 12 regardless the dimension d . Actually an explicit expression of Nti using the boundary local time Lt will be derived in this case by invoking an extended version of a general theorem in [13] to characterize Nti and by combining the Sobolev inequalities obtained in Sect. 3 with the upper bounds of transition functions due to CarlenKusuoka-Stroock [5] (Theorem 2.3). Furthermore, we shall see that the diffusion process constructed in Theorem 2.2 can be, under the condition that 'j aij L (D), related to a submartingale problem (Theorem 2.4), and accordingly, identified in law with VaradhanWilliamsʼs [resp. DeBlassie-Tobyʼs] normally reflecting Brownian motion when aij (x ) = 12 ij and D is a wedge [resp. a cusp C ] in R 2 . The present paper is an essential improvement of the previous one [14] where we gave affirmative answers to questions (I) and (II) only under the restriction that the H¨older exponents at cusps are uniformly greater than d 1 d , which was technically required in getting a modified Sobolev inequality of Moserʼs type - a key inequality in our construction of a strong Feller resolvent. This requirement now turns out to be unnecessary thanks to a specific transformation of a standard cusp domain onto a rectangular set exhibited in the last section. In the next section, we shall formulate a precise condition on the domain D and state main theorems answering the questions (I) and (II). Their proof will be carried out in the subsequent sections.
2 Statement of main theorems Let F be a real valued function defined on a set E R k including the origin such that F (x ) = x +f (x ), where 0 1 R, and f is a k -dimensional Lipschitz continuous function vanishing at the origin. Here denotes the Euclidean norm. In this paper we call such F a H¨older function and we denote its H¨older exponent, H¨older constant and Lipschitz constant respectively by Exp(F ) =
H¨ol(F ) =
Lip(F ) = Lip(f ) = min #K 0 : f (x ) f (y) K x y x y E For x = (x1 xd ) R d , we let x = (x1 xd 1 ) so that x = (x xd ). Let us now consider the following condition (H) on a domain D R d with d 2: (H) There are four constants (0 1) 0 A 1 M 0 and a locally finite open covering #Uj j J of 'D satisfying the following properties : (i)
For each j J , there are a H¨older function Fj of d 1 variables and a constant rj such that Fj is defined on the d 1-dimensional ball centered at the origin with radius rj , Exp(Fj ) ,
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Reflecting diffusions on H¨older domains
525
H¨ol(Fj ) = 0, or 1 A H¨ol(Fj ) A, or A H¨ol(Fj ) 1 A, Lip(Fj ) M , Uj D = # = ( d ) : rj Fj ( ) d , for some Cartesian coordinate system = ( d ). U (ii) 'D j , where Uj = #x Uj : dist(x 'Uj ) . j J
When D is bounded, condition (H) reduces to a simple one that every point x of 'D has a neighbourhood Ux such that 'D Ux is the graph of a H¨older function of d 1 variables. For later convenience, we let J+ = #j J : H¨ol(Fj ) 0 J0 = #j J : H¨ol(Fj ) = 0 J = #j J : H¨ol(Fj ) 0 For j J , denote by aj ( 'D) the origin of Uj with respect to the coordinate system . aj is called an outward [resp. inward ] cusp boundary point of D if j J+ [resp. j J ]. In what follows, we work with the Dirichlet form H 1 (D) on L2 (D) given by (1.1) and (1.2). Let #G 0 be the associated resolvent on L2 (D). It is then Markovian in the sense that 0 G f 1 whenever 0 f 1 and p it is well defined as a bounded linear operator on L (D) for any p [1 ]. Denote by C D the space of those functions in C D vanishing at infinity. Theorem 2.1 Assume that a domain D R d satisfies condition (H). Then G enjoys the following properties : p L (D) C D p 1+ (d (i) G L2 (D) 1) . (ii) G C D is a dense subspace of C D . (iii) There is a function G (x y) continuous on D D off diagonal such that G f (x ) = G (x y)f (y) dy x D f C D (2.1) D
As will be seen in Sect. 4, Theorem 2.1 is still valid under condition (A) stated in Sect. 3. Condition (A) is weaker but less concrete than (H) so that we employ (H) in formulating main theorems. Theorem 2.1 (i) means that G has a strong Feller property. By virtue of Theorem 2.1 (ii) and the Hille-Yosida theorem, there exists a strongly t
0 on C D such that G f = continuous Markovian semigroup #T t
t e T f dt f C D . We have then a Feller transition function by t
0 Tt f (x ) = D pt (x dy) f (y), which gives rise to a Hunt process (cf. [13; Theorem A.2.2]) M = (Xt Px ) on D such that Px (Xt A) = pt (x A) t 0 x D A D M is associated withthe Dirichlet form H 1 (D) of (1.2) since the re solvent G is. Since G C0 D is dense in the Dirichlet space, Theorem 2.1
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(ii) implies that the Dirichlet form is regular. Therefore we can apply general theorems in [13] to the associated pair and M. In particular, pt (x ) is absolutely continuous because G (x ) is ([13; Theorem 4.2.4]). Since has the strong local property and aij are uniformly bounded, we can invoke [13; Theorem 4.5.4]) and [13; Theorem 5.7.2, Example 5.7.1] to conclude that M is a conservative diffusion process on D. Summing up what has been mentioned, we get Theorem 2.2 Under condition (H), there exists a conservative diffusion process M = (Xt Px ) on D with resolvent G of Theorem 2.1. M is associated with the Dirichlet form given by (1.1) and (1.2), and the transition function pt (x ) of M satisfies (1.3) and (1.5). We next formulate a decomposition of the sample path of M and its Skorohod representation. Theorem 2.3 Consider a domain D R d satisfying condition (H). (i) The sample path Xt = (Xt1 Xtd ) of the conservative diffusion process M on D constructed in Theorem 2.2 admits a unique decomposition (1.6) with MAFʼs Mti in the strict sense satisfying (1.7) and CAFʼs Nti in the strict sense locally of zero energy. (ii) Assume condition (1.9) for aij . We also require the condition that Exp(Fj )
1 2
j J+
(2.2)
for the domain D. Then Nti has the following representation : Nti
=
d t
'j aij (Xs ) ds +
j =1
0
d j =1
t
aij (Xs ) nj (Xs ) dLs
0
(2.3)
1 i d t 0 Px a.s. for any x D where Lt is a unique PCAF in the strict sense with Revuz measure being the surface measure on 'D. Note that (2.3) reduces to (1.8) when aij = 12 ij . The above three theorems extend those results of R.F.Bass and P.Hsu in [2] and [3] formulated for a general bounded Lipschitz domain D and for aij = 12 ij . Let us denote by + the set of all outward cusp boundary points. Cb2 D will stand for the set of twice continuously differentiable functions on R d that are together with their first and second partial derivatives bounded on D. Theorem 2.4 Under condition (H) for the domain and the assumption that 'i aij L (D)
1 i j d
(2.4)
the conservative diffusion process M = (Xt Px ) of Theorem 2.2 enjoys the following properties : for each x D,
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Reflecting diffusions on H¨older domains
1. 2.
527
Px (X0 = x ) = 1, t d # f (Xt ) 'i aij 'j f (Xs ) ds
is a Px -submartingale,
0 i j =1
whenever f Cb2 D , f is constant in a neighbourhood of + and d
'i f (x ) aij (x ) nj (x ) 0
# -a.e. on 'D
(2.5)
i j =1
, 3. Ex
I + (Xs ) ds = 0.
0
When d = 2, aij (x ) = 12 ij and D = C the standard outward cusp domain, R.D. DeBlassie and E.H. Toby [7] have shown the existence and the uniqueness of the corresponding submartingale problem for a probability measure Px on = # : is a continuous function from [0 ) into C : for each fixed x C, 1 Px ((0) = x ) = 1 1 t 2 f ((t)) f (s) ds is a Px -submartingale 2 0 whenever f Cb2 (C ) f is constant in a neighbourhood of the origin and, f (x ) n(x ) 0 on 'C
I0 (s) ds = 0 3 Ex 0
Hence, by virtue of Theorem 2.4, the diffusion process of Theorem 2.2 coincides in law with DeBlassie-Tobyʼs one in [7] in this special case. In exactly the same way, we see that, when d = 2, aij (x ) = 12 ij and D is a wedge # : 0 R 2 for a fixed (0 2), the diffusion process of Theorem 2.2 is identical in law with Varadhan-R.Williamsʼs normally reflecting Brownian motion [22]. 3 Lp -estimate, local estimate and Harnack inequality In this and the next sections, we shall work under another condition (A) on a domain D R d which will be seen to be more general than (H) (Proposition 4.1). In this section, we derive some estimates for harmonic solutions of equations associated with H 1 (D) under condition (A). In order to state condition (A), we employ the following notations: B (a ) = #x R d : x a B () = B (0 ) B+ () = #(x xd ) B () : xd 0
C () = #(x xd ) B () : x xd
Q () = #(x xd ) B () : x xd
380
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for a R d 0 (0 1). For a Lipschitz mapping % from a set E R k into R l such that %(x ) %(y) K x y x y E , for some constant K 0, we also denote by Lip(%) the smallest constant K of this property. We now state condition (A) on a domain D R d : (A) The following properties hold for an at most countable index set I , a constant (0 1) and positive constants r M : There are a point ak 'D and its neighbourhood Vk associated with each k I such that
l; (i-1) D Vk Vl = k l I k = (i-2) there are a constant k [ 1) and a one to one mapping %k from B ( ) onto Vk with %k (0) = ak , Vk D equals either %k Ck ( ) or %k Qk ( ) , Lip(%k ) M , Lip(% 1 k ) M . (ii) For any a 'D k I Vk , there are its neighbourhood Wa and a one to one mapping "a from B (r ) onto Wa such that "a (0) = a "a (B+ (r )) = Wa D Lip("a ) M Lip("a 1 ) M .
(i)
In our previous paper [14], we considered the same condition as above, but we assumed (d 1) d . Further we did not consider the case Vk D = %k Qk ( ) , namely, we assumed in [14] that every ak is an outward cusp boundary point but not an inward one. Under those assumptions, we got the same estimates as in this section following the PDE argument due to Stampacchia [18] and Moser [17]. The PDE argument is based on a Sobolev inequality of Moserʼs type formulated in Proposition 3.1 below. As will be proved in the last section, we need not the previous assumption (d 1) d for the validity of Proposition 3.1. Once it is established, we can follow the PDE argument developed in [14] without any change so that we shall state the results of this section omitting the proof and only referring to the corresponding results in [14]. In the rest of this section, we assume (A). First of all, we note the following easily verifiable observation : if is a d d oneto one mapping from an open set U R onto an open set in R with 1 M and if B (a r) U and (a ) = a, then Lip B (a r M ) B (a r)
(3.1)
This observation particularly leads us to the following property of the domain D (see [14; Lemma 3.1]). We set k I : Vk D = %k Ck ( )
= k I : Vk D = %k Qk ( )
IC = IQ
ak for k IC [resp. IQ ] may be called an outward [resp. inward] cusp boundary point. A collection of open sets is said to have a finite intersection property if there exists an integer M such that any subcollection of cardinality greater than M has an empty intersection.
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Lemma 3.1 For any (0 ], there exist positive constantsr and m for which D satisfies the following : For every a 'D # 'D k I %k (B ()), there are a neighbourhood V a of a and a one to one mapping a from B r = V a D Lip( a ) onto V a such that a (0) = a a B+ r 1 ) m . m Lip( a Furthermore, for any r (0 r ] we have a subset A 'D # and a positive constant = (r) such that # a (B+ (r))aA has a finite intersection property, and for every b 'D, the set B(b ) D is contained in one of the following sets: %k Ck () for k IC , %k Qk () for k IQ , a (B+ (r)) for a A. In the following we set Ck () = %k Ck () Qk () = %k Qk () Ba (r) = a (B+ (r))
k IC k IQ a 'D #
for 0 0 r r . Since a Sobolev inequality of Moserʼs type formulated in Lemma 2 in [17] is valid for u H 1 (B+ (r)), we immediately obtain by means of the map a in Lemma 3.1 that for any (0 ] ! (0 1] q [2 2d (d 2)] ( q [2 ) if d = 2 ) there is a positive constant C1 = C1 ( ! q) such that 1 q
uq dx
C1 r
d
1
1 q 2
% u2 dx + r 2
Ba (r)
N
1 2
d i =1
'i u2 dx
Ba (r)
(3.2) for u H 1 Ba (r) N Ba (r) with N !Ba (r) 0 r r , and a 'D # . Here E denotes the Lebesgue measure for measurable sets E . (C1 and the other constants C2 C3 etc. below also depend on d r M and them.) in some cases on and . However we omit indicating Actually (3.2) also holds for u H 1 Ck () and u H 1 Qk () : Proposition 3.1 (i) For any ! (0 1] there is a positive constant C2 = C2 (!) such that
1 q q
u dx Ck ( )
C2
d 1+k k
1 q
12
% 2
2
u dx +
N
d i =1
Ck ( )
1 2 2
'i u dx
(3.3)
for u H 1 Ck () N Ck () with N !Ck (), 0 2 q 2(d 1 + k ) (d 1 k ), and k IC . (ii) Let 2 q in case d = 2 or 2 q 2d (d 2) in case d 3. Then for any ! (0 1] there is a positive constant C3 = C3 (! q) such that
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1 q
uq dx
C3
Qk ( )
d
1
1 q 2
% u2 dx + 2
d
N
i =1
Qk ( )
1 2 'i u2 dx
(3.4) for u H 1 Qk () N Qk () with N !Qk (), 0 and k IQ . The proof of Proposition 3.1 will be carried out in the last section by employing a specific transformation of a standard cusp domain onto a rectangular set. The following Sobolev inequality in an ordinary sense follows from (3.2), (3.3), (3.4) and Lemma 3.1 ([14; Proposition 3.2 (ii)]). (i) There is a positive constant C4 such that 1 2 % ( 1 q d uq dx C4 u2 dx + 'i u2 dx
Proposition 3.2
D
D
i =1
(3.5)
D
for u H 1 (D) 2 q 2(d 1 + ) (d 1 ). (ii) Assume the absence of outward cusp boundary point : IC = . Then the above statement is valid for 2 q 2d (d 2) in case d 3 and for 2 q in case d = 2. We denote the norm of the Sobolev space H 1 (E ) by H 1 (E ) . For an open set E D, let us consider the following spaces :
& (E ) = u C 1 (E ) : u H 1 (E ) u = 0 on 'E D C (3.6) & (E ) = the completion of C & (E ) with respect to the norm H 1 (E ) (3.7) H & (E ) coincides with H 1 (E ) if E D. When E = C () Q () or Note that H 0 k k Ba (r), we can derive the following Sobolev inequalities from Proposition 3.1 ([14; Proposition 3.3]). Proposition 3.3 (i) For any (0 1), there is a positive constant C5 = C5 () such that d 1 q 1 2 q 2 u dx C5 'i u dx (3.8) Ck ( )
i =1
Ck ( )
& (C ()) 0 2 q 2(d 1 + k ) (d 1 k ) k IC . for u H k (ii) For any (0 1) and for any 2 q 2d (d 2) (2 q if d = 2), there is a positive constant C6 = C6 ( q) such that d 1 q 1 2 q 2 u dx C6 'i u dx (3.9) Qk ( )
i =1
Qk ( )
& (Q ()) 0 k IQ . for u H k (iii) Let 0 0 1 and 2 q 2d (d 2) (2 q if d = 2). Then there is a positive constant C7 = C7 ( q) satisfying
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1 q
q
u dx
C7
1 2
d
Ba (r)
2
'i u dx
(3.10)
Ba (r)
i =1
& (Ba (r)) 0 r r a 'D # . for u H We now turn to the Dirichlet form H 1 (D) given by (1.1) and (1.2). We
& (E ) for an open set E D. also consider the following form E H E (u ) =
d
'i u(x )'j (x )aij (x ) dx
& (E ) u H
(3.11)
E
i j =1
& (E ) is a Dirichlet form on with aij 1 i j d satisfying (1.1). Since E H L2 (E ), we have the associated Markovian resolvent #GE 0 on L2 (E ). Let T be a functional defined by f0 $ dx +
T $ = D
d i =1
fi 'i $ dx
$ H 1 (D)
(3.12)
D
for fi L2 (D) i = 0 1 d . Since T is a continuous linear functional on H 1 (D), there is for each 0 a unique element u H 1 (D) such that $ H 1 (D)
(u $) = T $
(3.13)
Here ( ) = ( ) + ( )L2 (D) . We denote this function u by G T . If T is & (E ) respectively and defined by (3.12) with D and H 1 (D) replaced by E and H & (E ) if every fi belongs to L2 (E ), then we have for each 0 a unique u H denoted by GE T such that & (E ) $H
E (u $) = T $
(3.14)
where E ( ) = E ( ) + ( )L2 (E ) . Obviously G T [resp. GE T ] coincides with G f [resp. GE f ] in the case where T $ = (f $) for f L2 (D) [resp. f L2 (E )]. If E = Ck () Qk () or Ba (r), then the norm H 1 (E ) is equivalent to H& (E ) E ( )1 2 in view & (E ) satisfying (3.14) with of Proposition 3.3. Therefore there exists GE 0 T H = 0. Using Sobolev inequalities (3.5), (3.8), (3.9), (3.10) and following a standard argument as in [18; Theorem 4.1] (see also [10]), we can get the following Lp estimates. Theorem 3.1 (i) Let p (d 1) + 1 and 0. Then it holds that, for some C8 = C8 (p ) 0, G T
L (D)
C8
d
fi L2 (D) + fi Lp (D)
i =0
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(3.15)
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where T is given by (3.12) with fi L2 (D) Lp (D) i = 0 1 d . (ii) Let k IC p (d 1) k + 1 and 0 1. Then there is a positive constant C9 = C9 (p ) such that
1
GE T L (E ) C9 p
p 1 d1
d
k
fi Lp (E )
(3.16)
i =0
where E = Ck () 0 0, and T is a continuous linear functional & (E ). given by (3.12) with fi Lp (E ) and $ H (iii) Let p d and 0 1. Then there is a positive constant C10 = C10 (p ) such that d (p d ) p fi Lp (E ) (3.17) GE T L (E ) C10 i =0
Qk ()
k IQ 0 0, and T is a continuous linear where E = & (E ). functional given by (3.12) with fi Lp (E ) and $ H (iv) Let 0 p d and 0 1. Then there is a positive constant C11 = C11 (p ) such that GE T L (E ) C11 r (p d ) p
d
fi Lp (E )
(3.18)
i =0
for 0 E = Ba (r) 0 r r a 'D # , and for T defined by (3.12) with & (E ) fi Lp (E ) instead of D H 1 (D) fi Lp (D) respectively. E H We are next concerned with local estimates for subsolutions of the equations associated with E . A function u H 1 (E ) is called a subsolution if & (E ) $ 0 $ H
E (u $) 0
(3.19)
In the same way as in [18; Theorem 5.1] or in [17; Theorem 1], we obtain the following local estimates from Proposition 3.3 or (3.2) ([14; Theorem 3.2]). Theorem 3.2 (i) Let 0 for some (0 1) and E = Ck () with k IC . Then every nonnegative subsolution u H 1 (E ) of (3.19) satisfies ess sup u C12 ( s)
d 1+k 2k
1 2 2
u dx
Ck (s)
0 s
(3.20)
Ck ( )
for some C12 = C12 () 0. (ii) Let 0 for some (0 1) and E = Qk () with k IQ . Then every nonnegative subsolution u H 1 (E ) of (3.19) satisfies ess sup u C13 ( s) d 2 Qk (s)
385
1 2 u 2 dx
Qk ( )
0 s
(3.21)
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for some C13 = C13 () 0. (iii) Let 0 0 1 0 r r a 'D # and E = Ba (r). Then every nonnegative subsolution u H 1 (E ) of (3.19) satisfies 1 2
d 2
2
ess sup u C14 (r s)
u dx
Ba (s)
0 s r
(3.22)
Ba (r)
for some C14 = C14 ( ) 0. If u H 1 (E ) satisfies E (u $) = 0
& (E ) $H
(3.23)
for some 0, then u 0 and (u) 0 are both nonnegative subsolutions of (3.19). Therefore as an immediate consequence of Theorem 3.2 we get the following result. Corollary Let 0 3.1 (i) # for some (0 1) and E 1= Ck () with k IC resp. Qk () with k IQ . Then every solution u H (E ) of (3.23) satisfies (3.20) [ resp. (3.21) ] with u being replaced by u. (ii) Let 0 0 1 0 r r a 'D # and E = Ba (r). Then every solution u H 1 (E ) of (3.23) satisfies (3.22) with u being replaced by u.
Finally, by means of Proposition 3.1 and Theorem 3.2, we can get the following Harnack inequality for solutions u H 1 (E ) of the equation (3.23) with = 0 ([14; Theorem 3.3]). # Theorem 3.3 (i) Let k IC # resp k IQ 0 0 ! 1 and E = Ck () resp. Qk () . If u H 1 (E ) is a nonnegative solution of (3.23) with = 0 and satisfies #x : u(x ) 1# Ck ( 2) ! Ck ( 2) resp. #x : u(x ) 1 Qk ( 2) ! Qk ( 2) , then there is a positive constant C15 = C15 (!) such that , ess inf u C15
Ck ( 4)
resp.
ess inf u C15
Qk ( 4)
(3.24)
(ii) Let 0 0 r r a 'D # 0 ! 1 and set E = Ba (r). If u H 1 (E ) is a nonnegative solution of (3.23) with = 0 and satisfies #x : u(x ) 1 Ba (r 2) ! Ba (r), then there is a positive constant C16 = C16 ( !) such that ess inf u C16 Ba (r 4)
386
(3.25)
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M. Fukushima, M. Tomisaki
4 Strong Feller resolvent In thissection, we will show that the resolvent #G associated with the Dirichlet form H 1 (D) has the same properties as those of Theorem 2.1 under condition (A). At the end of this section, we will show that condition (H) reduces to (A) and hence Theorem 2.1 follows. Theorem 4.1 Under condition (A), G satisfies the same properties as in Theorem 2.1. Namely, p p 1+ (d (i) G L2 (D) L (D) C D 1) . (ii) G C D is a dense subspace of C D . (iii) There is a function G (x y) continuous on D D off diagonal such that G (x y)f (y) dy x D f C D (4.1) G f (x ) = D
Theorem 4.1 is obtained essentially by the same argument as in [14; Sect. 4] but we give the proof here for completeness. Theorem 4.1 (i) is an immediate consequence of the following theorem. Theorem 4.2 Assume condition (A). Let p (d 1) + 1 T be a functional given by (3.12) with fi L2 (D) Lp (D) i = 0 1 2 d , and 0. Then G T is uniformly continuous in D and accordingly G T can be extended to a continuous function on D. Proof Put u = G T . Fix a k IC and an s (0 2] arbitrarily. Set E = Ck (s). & (E ) be the solution of the equation (3.14) with = 0 Let GE 0 (T u) H and T = T u. We see by means of Theorem 3.1 (i), (ii), % d d 1 1 p 1 k fi Lp (E ) f0 u Lp (E ) + L (E ) C9 (p 1 2)s p i =1
c1 s
1 p
p 1 d1
d
fi L2 (D) + fi Lp (D)
(4.2)
i =0
for some positive c1 independent of s and k . Since u belongs to H 1 (E ) and satisfies (3.23) with = 0, following the same argument as in [17] we get by means of Theorem 3.3 (i) ( 1 Osc ; Ck (s 4) 1 C15 (1 2) Osc ; Ck (s) 2 c2 Osc ; Ck (s) for c2 (0 1) independent of s and k . Here Osc(; F ) denotes the oscillation of a function over a set F : Osc(; F ) = ess supF ess infF . Hence Osc u; Ck (s 4) Osc ; Ck (s 4) + Osc ; Ck (s 4) 2 L (E ) + c2 Osc ; Ck (s) 4 L (E ) + c2 Osc u; Ck (s)
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Combining this with (4.2) and using [18; Lemma 7.3], we get 0 s 4 k IC Osc u; Ck (s) c3 s 1
(4.3)
for some constants c3 0 and 1 (0 1). In the same way we also get 0 s 4 k IQ Osc u; Qk (s) c4 s 2
(4.4)
for some constants c4 0 and 2 (0 1). Recall r and 'D # appearing in Lemma 3.1. Similarly, for any (0 2], there then exist a c5 0 and a 3 (0 1) such that Osc u; Ba (s) c5 s 3 0 s r 4 a 'D # (4.5) The estimate for oscillations on open balls with closures contained in D, which is due to Stampacchia [18], asserts that Osc (u; B (a s)) c6 s 4
0 s 4 a D D
(4.6)
Here is a positive number fixed arbitrarily, D = #x D : dist(x 'D) , and constants c6 0 and 4 (0 1) depend on but are independent of a D D . For an 0 fixed arbitrarily, we see by virtue of (4.3) and (4.4) that there exists an s1 = s1 () (0 4] such that Osc u; Ck (s1 ) k IC (4.7) Osc u; Qk (s1 ) k IQ (4.8) By means of (4.5), we further find an s2 = s2 ( s1 ) (0 rs1 4] such that Osc u; Ba (s2 ) a 'Ds#1
(4.9)
In view of Lemma 3.1, we can find an o 0 such that every pair x y Do with x y o is simultaneously contained in one of sets Ck (s1 ) with some k IC Qk (s1 ) with some k IQ Ba (s2 ) with some a 'Ds#1
(4.10)
(4.6) with this o leads us to Osc (u; B (a s3 ))
a D Do 2
(4.11)
for some s3 = s3 ( o ) (0 o 8]. We now set = ( o 2) s3 . Let x y D with x y . If x or y belongs to Do 2 , then u(x ) u(y) by (4.10), (4.7), (4.8), (4.9). Otherwise, u(x ) u(y) by (4.11). Employing Corollary 3.1 in place of Theorem 3.1 (i) in getting (4.2), we obtain the following in the same way as above :
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Theorem 4.3 Let W be an open set of R d and E = W D. Every solution u H 1 (E ) of (3.23) for some 0 is uniformly continuous in W1 D for every open set W1 satisfying W1 W . We next give Proof of Theorem 4.1 (ii) We first follow an argument in [20; Proposition 5.1] to show G C D C D (4.12) Since C0 D is dense in C D , it suffices to show that G C0 D C D (4.13) in the case where D is unbounded. Let C0 D and 0. Choose an R1 0 such that Supp[] B (R1 ) D
c1 G L2 (DB (R1 ))
(4.14)
c1 being a positive constant specified later. We next take an R2 R1 satisfying the following : Ck ( ) D B (R1 ) Qk ( ) D B (R1 ) Ba (r ) D B (R1 )
for k IC with ak 'D B (R2 ) for k IQ with ak 'D B (R2 ) for a 'D # B (R2 )
We set JC = #k IC : ak 'D B (R2 ) JQ = #k IQ : ak 'D B (R2 ), and A = 'D # 2 B (R2 ). Then, on account of (3.1), there is a constant (0 R2 R1 ) depending on but not on R1 R2 such that ( (
r 2 D2 B (R2 ) Ck Qk Ba ! ! 2 2 2 aA k JC
k JQ
# #a ! A ! D D2 B (R2 ) and, for We consider the set K = k k JC JQ each a K we define a constant s and a set Ea (s) as follows : s = if a = ak k JC Ck (s) Q (s) s= if a = ak k JQ k Ea (s) = s = r 2 if a A Ba (s) B (a s) s = if a D D2 B (R2 ) *
Note that D B (R2 )
Ea (s 2)
aK
Ea (s) D B (R1 )
aK
and Ea (s) (G $) = ( $) = 0 By virtue of Corollary 3.1, we then have
389
& (Ea (s)) $H
(4.15)
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537
G L (Ea (s 2)) c1 G L2 (Ea (s))
(4.16)
where we used a local estimate due to Stampacchia [18] or Moser [17] in the case that a D D2 B (R2 ). It should be noted that c1 is a positive constant independent of a R1 and R2 . By (4.14), (4.15) and (4.16), we find that G L (DB (R2 )) c1 G L2 (DB (R1 ))
(4.17)
which along with Theorem 4.2 proves (4.13). We next adopt Kunitaʼs argument [15]. Let denote by C0 D the space of differentiable functions on R d the restrictions to D of all infinitely continuously
with compact support. For each u C0 D , we define a functional Lu by d d Lu $ = aij 'i u 'j $ dx $ H 1 (D) j =1
D
i =1
Then T = u Lu satisfies the condition of Theorem 3.1 (i) and u = G T for each 0. By virtue of Theorem 3.1 (i), there is for any 0 a C0 D such that u G L (D)
Since C 0 D is dense in C D , we thus obtain the denseness of G C D in C D . Proof of Theorem 4.1 (iii) Since G is Markovian, there exists a function G (x y) satisfying (4.1) by virtue of Theorem 3.1 (i) and Theorem 4.2. Hence it is enough to show that G (x ) belongs to H 1 (U ) and is continuous on U
(4.18)
for any open set U with U D #x , where x D and 0. Let us denote the dual space of H 1 (E ) by H 1 (E ) . There exists for each 1 0 and T H (D) a unique element u H 1 (D) such that (u $) = T $
$ H 1 (D)
We denote this function u by G T . (We already usedthis notation for T given by (3.12) which is actually a general expression of T H 1 (D) (cf. [16; 1.1.14]).) For a while we fix an x D arbitrarily. We define a set Ex (s) according as three different cases. (Case 1) x is a cusp point, that is, x = ak for some k I . In this case we take an s (0 ]. (Case 2) x is a boundary point but not a cusp point, that is, x 'D k I #ak . Choose (0 ] such that x 'D k I %k (B ()). Then for an r given in Lemma 3.1 we take an s (0 r ]. (Case 3) x is an interior point of D. In this case we take an s (0 dx 2], where dx = dist(x 'D).
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Let us put Ex (s) =
Ck (s) Q (s) k
Bx (s) B (x s)
if k IC in Case 1 if k IQ in Case 1 in Case 2 in Case 3
Then there exists a unique element sx H 1 (D) such that 1 $(y) dy $ H 1 (D) sx $ = Ex (s) Ex (s)
(4.19)
Here we note the following lemma which is obtained by the same method as in [14; Lemma 4.3]. Lemma 4.1 Let U be an open set such that U D #x . Then G (x )U H 1 (U ) and sx U converges to G (x )U weakly in H 1 (U ) as s 0. Take an open set V of R d such that U V and x V and set E = V D. On account of Lemma 4.1, G (x )E H 1 (E ) and sx E G (x )E weakly & (E ) and extend it to D by putting $ = 0 on in H 1 (E ) as s 0. Take any $ C D E . Then E (G (x )E $) = lim E sx E $ s0 1 = lim sx $ = lim $(y) dy = 0 s0 s0 Ex (s) E (s) x This implies that G (x )E H 1 (E ) is a solution of (3.23) and hence, in view of Theorem 4.3, G (x ) is continuous in U . We finally note the following proposition which along with Theorem 4.1 implies Theorem 2.1. Proposition 4.1
Condition (H) reduces to condition (A).
Proof For each j J , a H¨older function Fj in (H) (i) is given by Fj (x ) = j x j + fj (x ) where j 1 j = 0 or 1 A j A or A j 1 A according to j J0 or j J+ or j J , and fj is function defined
a Lipschitz continuous on the d 1-dimensional closed ball x R d 1 : x rj with fj (0) = 0 and Lip(fj ) M . Then !
+ (j ) d(j ) B (rj ) : Fj (j ) d(j ) Uj D =
for some Cartesian coordinate system (j ) = (j ) d(j ) = 1(j ) 2(j ) d(j ) . Let us put I = J+ ! J . For k I , ak is the point of 'D corresponding to the origin in (k ) -coordinate system. #ak : k I is then the totality of cusp
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boundary points. For each k I , the neighbourhood Uk of ak contains no cusp
l k l I . boundary point other than ak , and hence ak al k = For each k I , we define a mapping %k from Ek #(x xd ) : x rk xd R into (k ) -space by
%k (x xd ) = (k ) d(k )
(k ) = x d(k ) = k xd + fk (x ) 1 + A + AM . Put = We then have Lip %k 1 + A + M and Lip % 1 k 2(1 + A + M ) Vk = %k (B ( )) and M1 = (1 + A)(1 + M ). Then we see from (A) (i) with = and M = M1 . In particular, (3.1) that #Vk k I satisfies Vk D = %k Ck ( ) if k 0 = %k Qk ( ) if k 0. We next show (A) (ii). Let o be the positive solution of the equation 2 + 2 = 2 o for o = 1. We then take a constant R 1 satisfying % 1 ( o R1 o 1+M +A R R will play the role of r in (A) (ii). and put r = o R. This r Let us fix a p 'D k I Vk arbitrarily. By means of (H) (ii), there is a (j ) (j ) pd(j ) ). We shall j J such that p U j . Denote the -coordinate of p by (p define a mapping "p and a neighbourhood Wp in two cases j I and j I separately.
p p-d ) = In the case that j I , (p (j ) pd(j ) ) belongs to %j (Ej ). Putting (
p (j ) pd(j ) , we have that (% 1 p p-d ) Ej B ( ) and p-d = p j in j accordance to the sign of j . We then define a mapping "p from the set Gp #(x xd ) : x + p- rj xd R into (j ) -space as follows:
"p (x xd ) = (j ) d(j )
(j ) = x + p-
d(j ) = j x + p- j + xd + fj x + p-
Notice that, on the region #x Gp : x o S for S 1, Lip "p 1 + M + j 1 A S 1 . Since the distance of p = "p (0) from ' "p (Gp ) is greater than S o , we can conclude from (3.1) that B (r ) Gp for the above chosen r = o R. In the case that j I , we define a mapping "p from the set Gp #(x xd ) :
x + p (j ) rj xd R into (j ) -space by
"p (x xd ) = (j ) d(j )
(j ) = x + p (j ) d(j ) = xd + fj x + p (j ) In both cases, B (r ) Gp . Accordingly we put Wp = "p (B (r )). It is easy to see that "p is one-to-one, "p (0) = p, "p (B+ (r )) = Wp D, and Lip("p ) 1 . M2 Lip("p 1 ) M2 where M2 = 1 + M + A R 1 R o
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Thus (H) reduces to (A) with I r as above and M = M1 M2 .
5 Decomposition of the sample path and additive functionals This section is devoted to the proof of Theorem 2.3 and Theorem 2.4. To this end, we first prepare an extended version of a general theorem [13; Theorem 5.5.5] to characterize the second term in the decomposition (1.6). Let X be a locally compact separable metric space, m be a positive Radon measure on X with full support and ( ) be a strongly local regular Dirichlet form on L2 (X ; m). We assume that there exists a conservative diffusion process M = (Xt Px ) on X associated with the form whose transition function pt (x ) is absolutely continuous with respect to m for any t 0 and x X . Then the resolvent of M admits a symmetric density G (x y) with respect to m which is -excessive in two variables x y. The potential of a measure is denoted by G (x ) = X G (x y) (dy). The integral of a function f against a measure is denoted by f or f . A positive Radon measure on X is said to be of finite energy integral if there exists a constant C17 such that (x )(dx ) C17 1 ( ) (5.1) X
for some special standard core of . The totality of such measures is denoted by S0 . It is known that S0 if and only if G is finite and that, in this case, G is a -excessive and quasi-continuous version of the potential U considered in [13; Sect. 2.2]. We further introduce two classes of positive Radon measures on X by S00 = # : (X ) sup G (x ) x X
S01 = # : S0 G (x ) x X Obviously S00 S01 S0 . In our later application, the family S01 turns out to be more useful than S00 . An increasing sequence #E of finely open sets is said to be an exhaustive sequence if =1 E = X A positive Borel measure on X is called smooth in the strict sense if there exists an exhaustive sequence #E of finely open sets such that IE S00 = 1 2 Let S1 be the totality of smooth measures in the strict sense. S1 is known to be in one to one correspondence with the (equivalence classes of) positive continuous additive functionals (PCAFʼs in abbreviation) in the strict sense of M under the Revuz correspondence ([13; Theorem 5.1.7]). Lemma 5.1 S1 if and only if there exists an exhaustive sequence #E of finely open sets such that IE S01 = 1 2
393
Reflecting diffusions on H¨older domains
541
Proof It suffices to show that any S01 admits an exhaustive sequence #E of finely open sets such that IE S00 = 1 2 We may choose E as follows: E = #x O : G (x )
= 1 2
where #O is an exhaustive sequence of relatively compact open sets. Then, (IE )(X ) = (E ) is finite, and further G (IE )(x ) for m-a.e. x X by the maximum principle ([13; Lemma 2.2.4]) and hence for every x X by the absolute continuity of the transition function. We denote by u the energy measure of u loc . The Dirichlet form is expressible as 1 (u ) = u (X ) u 2 by using the co-energy measure u . The second assertion of the next proposition replaces S00 in [13; Theorem 5.5.5] by S01 . Proposition 5.1 (i) Suppose that a function u satisfies the following conditions: 1. u is finite valued, finely continuous and u loc . 2. IG u S00 for any relatively compact open set G. Then we have the unique decomposition u(Xt ) u(X0 ) = Mt[u] + Nt[u] t 0 Px ae x X
(5.2)
where M [u] is a CAF in the strict sense such that, for any relatively compact open set G,
[u] 2 [u] Ex Mt M = Ex BtG x G = 0 E (5.3) x tG G B being the PCAF in the strict sense with Revuz measure u and (G being the first leaving time from G. Nt[u] is a CAF in the strict sense locally of zero energy. (ii) Assume further the following property of u:
= (1) (2) with IG (1) IG (2) S01 for any relatively compact open set G and (u ) =
(5.4)
for some special standard core of . Then N [u] = A(1) + A(2) Px as x X where A(1) and A(2) are PCAFʼs in the strict sense with Revuz measures (2) respectively.
(5.5) (1)
and
Proof The first assertion is a consequence of [11; Theorem 2]. Since (1) (2) in (ii) are in the class S1 by the preceding lemma, we can see the validity of identity (5.5) on account of [12; Theorem 3.3, Corollary 3.1].
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We are in a position to prove Theorem 2.3. We fix a domain D R d possessing the property (H) and a data aij satisfying (1.1). We now apply the general theory prepared above to the specific Dirichlet form (1.2) on L2 (D; m), the resolvent G of Theorem 2.1 and the conservative diffusion M of Theorem 2.2. Here m denotes the d dimensional Lebesgue measure. M will be called the reflecting diffusion on D (associated with aij ). Accordingly G will be called the resolvent of the reflecting diffusion on D. Notice that, under the condition (H) for the domain D, the surface measure # on 'D is well defined with a local expression " 1 + Fj ( )2 d E Uj 'D (5.6) #(E ) = E
where E = # : ( Fj ( )) E . Further, with the unit inward normal vector n( ) = (n1 ( ) nd ( )) making sense #-a.e. on 'D according as " 1 + Fj ( )2 Uj 'D n( ) = ( Fj ( ) 1) we have the divergence theorem ' dm = ni d # D 'xi D
1 i d C0 (D)
This formula extends to a wider class of functions and in particular the condition (1.9) for aij guarantees the identity 'j (aij ) dm = aij nj d # C0 (D) (5.7) D
D
Denote by i the coordinate functions: i (x ) = xi 1 i d Then, i 1 (D), and the co-energy measures i j with respect to the Dirichlet form Hloc (1.2) are given by (cf. [13; Example 5.2.1]) i j = 2aij m
1 i j d
(5.8)
Let us denote by B an arbitrary ball in R d . Since aij are bounded, we have IB D i S00 The PCAF in the strict sense with Revuz measure m is just a constant functional t. Therefore Proposition 5.1 (i) implies the decomposition (1.6) where M i 1 i d are CAFʼs in the strict sense satisfying (1.7) with t being replaced by t (B D . Owing to the boundedness of aij however, we can let B R d to get (1.7), proving Theorem 2.3 (i). Turning to the proof of Theorem 2.3 (ii), we have under the condition (1.9) (i ) =
(x ) (dx )
with
D
=
d d ('j aij )m aij nj # j =1
395
j =1
(5.9)
Reflecting diffusions on H¨older domains
543
holding for any C0 (D) because (i ) =
d j =1
=
aij (x )'j (x ) m(dx ) D
d j =1
'j aij dm +
D
d j =1
'j (aij ) dm
D
which equals to the right hand side of (5.9) by virtue of (5.7). Suppose that the surface measure # satisfies IB D # S01
for any ball B R d
(5.10)
We then have from the boundedness of aij and condition (1.9) IB D S01 Hence (5.9) and Proposition 5.1 (ii) lead us to the expression (2.3) in terms of a PCAF L in the strict sense with Revuz measure being the surface measure #, proving Theorem 2.3 (ii). It only remains to show (5.10) for the proof of Theorem 2.3. Theorem 5.1 If (2.2) holds, namely, each outward cusp boundary point is of H¨older exponent greater than 12 , then the surface measure # on 'D satisfies condition (5.10). For any ball B , the compact set B 'D can be covered by finite number of open sets U j appearing in the condition (H) (ii) for the domain D. Besides G (x y) is jointly continuous off diagonal by Theorem 2.1 (iii). For the proof of (5.10), it is therefore sufficient to show I # S0
and G I #(x ) x
(5.11)
where = # = ( d ) : d = Fj ( ) Uj 'D for each fixed j J and rj . Exp(Fj ) will be denoted by j . c1 c2 will denote some positive constants. We further let = # : ( Fj ( )) ( # : ) Lemma 5.2 Let j J+ ! J . (i) I # S0 whenever d 3. When d = 2, this is true if j 12 . (ii) I # S0 for any 0 where = # : .
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Proof (i) In view of (1.1) and (5.1), it suffices to prove the inequality u( Fj ( ))#(d ) c1 D(u u) + (u u)L2 (D) u C0 (D) (5.12)
where D(u u) denotes the Dirichlet integral of u on D. On account of (5.6), the surface measure # has a density #( ) with respect to d satisfying #( ) c2 j 1 Hence the square of the left hand side of (5.12) is dominated by u( Fj ( ))2 d 2j 2 d c22
(5.13)
(5.14)
The second factor equals 0 r 2j +d 4 dr, which is finite under the stated condition. Consider a function C0 (U ) taking value 1 on the set . Then from the expression r 2 2 '
# ( d )u( d )d d u( Fj ( )) = ' d Fj ( ) we see that the first factor of (5.14) is dominated by (u 2 + u2 )d c3 U D
arriving at (5.12). (ii) Since #( ) is bounded on = # : ( Fj ( )) , (5.12) with being replaced by holds for any 0. In order to complete the proof of (5.11), we prepare a lemma on a comparison of resolvent densities. Lemma 5.3 Let K be a compact subset of D and U be a bounded domain containing K such that the domain D1 = D U possesses the property (H). Denote by G1 (x y) x y D 1 , the resolvent density of the reflecting diffusion on D 1 . Then, x y K x =
y (5.15) G (x y) G1 (x y) + C18 for some positive constant C18 depending on the set K . Proof Consider the set F = D U and the resolvent density G0 (x y) x y F , of the part MF of M on the set F . MF is obtained from M by killing the sample paths upon leaving the set F . Then by Dynkinʼs formula G (x y) = G0 (x y) + Ex e G (X y) x y F where ( denotes the leaving time from the set F . Take an open set W such that K W W U . The second term of the right side of the above identity with y being restricted to D W is dominated by
397
Reflecting diffusions on H¨older domains
C18 =
545
G (x y)
sup x D U yDW
which is finite owing to the off diagonal continuity Theorem 2.1 (iii). Let M1 be the reflecting diffusion on D 1 and M1F be its part on the set F ( D 1 ). On account of [13; Theorem 4.4.3], MF and M1F share a common Dirichlet form F on L2 (F ) given by F (u ) = (u ) & (D1 ) [F ] = H
u [F ]
& (D1 ) is defined by (3.7). Therefore we have the inequality where H G0 (x y) G1 (x y) holding for m m-a.e. (x y) F F . In view of the continuity of G and G1 , we get (5.15) for every x F and every y D W . We return to the set Uj 'D specified before Lemma 5.2. Lemma 5.4 Following inequalities hold for x y x =
y and a positive constant C19 depending on the set Uj D : (i) If j J+ and d 2, then
G (x y) C19 x y
d 1j j
(5.16)
(ii) If j J0 ! J and d 3, then G (x y) C19 x y d +2
(5.17)
(iii) If j J0 ! J and d = 2, then G (x y) C19 x y
for any 0
(5.18)
Proof (i) Notice that the Sobolev inequality in the statement of Proposition 3.2 (i) holds with D and being replaced by Dj = Uj D and j respectively. Since the domain Dj is bounded, we can invoke Carlen-Kusuoka-Stroock [5] to conclude in the same way as in [3; Sect. 2] that the resolvent density G1 (x y) of the reflecting diffusion on Dj admits the estimate G1 (x y) c1 x y
x y Dj
(5.19)
for & = 4 (q 2). In particular, by taking q = 2(d 1 + j ) (d 1 j ) we see that (5.19) is valid for & = (d 1 j ) j . We can then use Lemma 5.3 to get (5.16). (ii), (iii) In these cases, Proposition 3.2 (ii) is applicable to the domain Dj = Uj D and we see the validity of (5.19) for & = d 2 [resp. & = 0 ] by taking q = 2d (d 2) [resp. q = 4 + 2 ]. We again use Lemma 5.3 to get (5.17) [resp. (5.18)].
398
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M. Fukushima, M. Tomisaki
Lemma 5.5 Let j J+ ! J . Assume that j #( ) .
1 2
in case j J+ . Then G I
Proof Keeping the expression G ( ( F ( ))#( )d
G I #( ) =
and the bound (5.13) of # in mind, we first prove the finiteness of the potential for = 0 in case that j J+ and j 12 . From (5.16), we have the bound
2
2
+ F ( )
G I #(0) c1
j d 1 2 j
. 1
1 j
d
(5.20)
Since
d 1j 2 j + F ( )2 2j 2
d 1j j
d 1 j
for some 0, where j = H¨ol(Fj ), we obtain
G I #(0) c2
r d +j 3 (d 1 j ) dr + c3
d 1j j
r d +j 3 dr
0
In the case that j J , we get the finiteness of G I #(0) from (5.17) and (5.18) in a similar manner to the above. Next take a =
0 We can choose a neighbourhood V of such that 0
V V Uj and D1 = V D is a Lipschitz domain. Let ˜ = V Then the same reasoning as the proof of Lemma 5.3 works to see that G ( ) ˜ is dominated by c4 d +2 in case that d 3 and by c5 0, in case that d = 2. Since #( ) is bounded on ˜ , we see the finiteness of G I˜ #( ) and hence of G I #( ). Proof of Theorem 5.1
We divide the situation into four cases :
(I) j J+ j 1 2 (III) j J d = 2
(II) j J d 3 (IV) j J0
In view of Lemma 5.2 and Lemma 5.5, we see that (5.11) holds in cases (I) and (II). Hence it remains to prove (5.11) in cases (III) and (IV). We can instead prove a stronger property sup G I #(x )
(5.21)
x
in these cases. Indeed, when j J and d = 2, we have the bound (5.18) of G (x y) for any 0, and we can proceed in a similar manner to the proof of Theorem 6.1 in our preceding paper [14] in getting (5.21) by choosing smaller than j . When
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Reflecting diffusions on H¨older domains
547
j J0 , then we have the bound (5.17) or (5.18) of G (x y) which, together with the uniform boundedness on of the density function # of the surface measure, readily leads us to (5.21). Proof of Theorem 2.4 Take any function f as is stated in the theorem and denote by W a neighbourhood of + on which f is constant. Then IB ( DW ) # S01
for any ball B R d
(5.22)
To see this, it suffices to show I W # S0
and G I W #(x ) x
(5.23)
for the set Uj 'D appearing in (5.11) and exclusively for j J+ . Since W for some 0, the first assertion in (5.23) follows from Lemma 5.2 (ii). The second one for x = 0 [resp. for x =
0] is immediate from the continuity of G (0 y) [resp. from Lemma 5.5]. Now just as computations made in (5.8) and (5.9), we have ' d aij 'i f 'j f m (5.24) f f = 2 i j =1
and
(f ) =
C0 D
(x ) (dx ) D
with =
d
d 'i aij 'j f m 'i f aij nj I DW #
i j =1
(5.25)
i j =1
I DW can be inserted in the last expression because 'i f vanishes on W . In view of (2.4), (5.22), (5.24) and (5.25), Proposition 5.1 applies and we get f (Xt ) f (X0 ) = Mt[f ] + Nt[f ]
Px -a.s. x D
where M [f ] is a MAF in the strict sense with ' t d M [f ] t = 2 aij 'i f 'j f (Xs ) ds 0
and Nt[f ] = 0
t
d
i j =1
'i aij 'j f
'
(5.26)
(5.27)
i j =1
t
(Xs ) ds +
0
d
' 'i f aij nj
(Xs ) d Ls (5.28)
i j =1
Here Lt is a PCAF in the strict sense with Revuz measure I DW #. (5.27) and (5.28) are valid Px -a.e. for every x D. Under the condition (2.5) for f , the second functional in the right hand side of (5.28) is a PCAF in the strict sense. Therefore the desired conclusion follows from (5.26) and (5.28).
400
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M. Fukushima, M. Tomisaki
6 Sobolev inequality of Moser type In this section we will show Proposition 3.1. Throughout this section we assume condition (A). Proposition 3.1 is immediate from the following two propositions. Proposition 6.1 For any ! (0 1] there is a positive constant C20 = C20 (! , M d ) such that % d 2 2 2 2 u dx C20 u dx + 'i u dx (6.1) Ek ( )
N
i =1
Ek ( )
1
#
for Ek () = Ck () resp. Qk () , u H Ek () N being a Borel subset of # Ek () with N !Ek (), 0 and k IC resp. k IQ . Proposition 6.2 (i) Let k IC and 1 p (d 1 + k ) k . Take a q satisfying p q p d 1+ k (d 1 + k k p) if p (d 1 + k ) k , or p q if p = (d 1 + k ) k . Then there is a positive constant C21 = C21 (p q M d ) such that 1 q uq dx Ck ( )
C21
d 1+k k
1 1 q p
% p
u dx +
p
Ck ( )
1 p
d i =1
p
Ck ( )
'i u dx
for u H 1 Ck () 0 . (ii) Let 1 p d and take a q satisfying p q pd (d p) if p d , or p q if p = d . Then there is a positive constant C22 = C22 (p q M d ) such that 1 q uq dx Qk ( )
C22
d
1 1 q p
% p
u dx +
Qk ( )
p
1 p
d i =1
p
Qk ( )
'i u dx
for u H 1 Qk () 0 k IQ . The part (i) of Proposition 6.2 is obtained by the same method as in [1; pp.128– 135] if we employ a transformation " (x ) defined below in place of the transformation rk (x ) in [1; p.130]. The part (ii) is also obtained following argument in [1; pp.103–104]. So we omit the proof of Proposition 6.2. In order to show Proposition 6.1 with Ek () = Ck (), we make use of a mapping " (x ) from a cusp C () onto a direct product set (). We begin with the definition of " . Let R+d = #(x xd ) R d : xd 0 and d be the following product space:
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Reflecting diffusions on H¨older domains
549
#(r t) : 0 r t if d = 2 if d 3 #(r t ) : 0 r 0 t d 2 where d 2 = # 1 2 d 2 : 0 j (j = 1 d 3) 0 d 2 2. Given (0 1), we define the mapping " : R+d d as follows : When d = 2, we set for x = (x1 x2 ) R+2 d =
" (x ) = (r t)
1
r = x ( 0)
t = x1 x2
( R);
When d 3, we set for x = (x xd ) R+d " (x ) = (r t ) 1
( 0) r = x ( 0) t = x xd if x = 0 (0 0) ( d 2 ) =
if x 0 ( d 2 ) satisfying 1 () d 1 () = x x Here 1 () d 1 () is the spherical polar coordinate of S d 2 , that is, for = 1 d 2 d 2 , 1 () = cos 1 2 () = sin 1 cos 2 .. . d 2 () = sin 1 sin 2 sin d 3 cos d 2 d 1 () = sin 1 sin 2 sin d 3 sin d 2 For each r 0, we identify the points (r 0 ) d 2 by regarding them as a same point. Under this identification, " is one to one from R+d onto d and the inverse mapping " 1 is as follows : For (r t) 2 , let = (r t) be the (unique) positive solution of the equation t 2 2 + 2 = r 2
(6.2)
Then " 1 (r t) = (x1 x2 )
x1 = t 1 ( R)
x2 = ( 0)
(6.3)
For (r t ) d with d 3, let = (r t) be the positive solution of the equation (6.2). Then " 1 (r t ) = (x1 x2 xd ) xi = t 1 i () ( R) i = 1 2 d 1
xd = ( 0)
(6.4)
We next note that " : C () ()
one to one, onto
where () is a subset of d given by #(r t) : 0 r 1 t 1 () = #(r t ) : 0 r 0 t 1 d 2
402
if d = 2 if d 3
(6.5)
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M. Fukushima, M. Tomisaki
For the sake of convenience, we write (r t ) () in case d = 2 too, and use (6.4) with the convention that 1 () 1. Since = (r t) is the solution of the equation (6.2), (x1 xd ) = " 1 (r t ) satisfies the following relations : r 'xd = 'r + (1 )t 2 2 1
(6.6)
t 2 'xd = 't + (1 )t 2 2 1 'xd = 0 (j = 1 2 d 2) 'j and for i = 1 2 d 1,
(6.7) (6.8)
1 'xi 'xd = t 1 1 i () 'r 'r ( 1 'xi 'xd = 1 + t 1 1 i () 't 't 'xi 'i () = t 1 (j = 1 2 d 2) 'j 'j
(6.9) (6.10) (6.11)
In the following, A1 A2 denote positive constants depending only on and d . Let 1 0 and (x1 x2 xd ) = " 1 (r t ) (r t ) (). Then we get the following estimates by means of (6.2), (6.3), (6.4), (6.6), (6.7) : (6.12) A1 r xd r 'xd A3 'r 'xd 't A4 r
A2
(6.13) (6.14)
Further we see that the Jacobian determinant is given by J (r t ) =
'(x1 xd ) (d 1) 'xd d 2 Sd () = (1)d +1 xd t '(r t 1 d 2 ) 'r
where
Sd () =
1 sind 3 1 sind 4 2 sin d 3
if d = 2 if d 3
(6.15)
(6.16)
for = 1 d 2 d 2 . By means of (6.12) – (6.16), we readily get J (r t ) A3 r (d 1)
(6.17)
A5 (d 1) +1 C () =
J (r t ) dr dt d A6 (d 1) +1 ( )
We next note the following fact:
403
(6.18)
Reflecting diffusions on H¨older domains
551
(i ) (0) Lemma 6.1 Let
(i1) 0 (i ) (i ) and x (i ) C () i = 0 1 x = (1) (i ) (i ) (i ) = " x = 1 d 2 i = 0 1. For 0 s x . Set r t 1, put r (s) = r (0) + s r (1) r (0) t (s) = t (0) + s t (1) t (0) (6.19) (s) = 1(s) d(s) 2 j(s) = j(0) + s j(1) j(0) j = 1 2 d 2 Then r (s) t (s) (s) belongs to () for 0 s 1. Moreover the following estimate holds : J r (0) t (0) (0) J r (1) t (1) (1) (6.20) A7 (d 1) J r (s) t (s) (s)
for 0 s 1 if d = 2 and for 0 s 1 if d = 3 and
3 (0) (1) d) sin j(0) + sin j(1) =
0 t +t j =1
Proof In view of (6.5) it is obvious that r (s) t (s) (s) () 0 s 1.We d 3 (0) (1) now assume that t (0) + t (1) =
0 in case d 3. Let 0 j =1 sin j + sin j s 1 in case d = 2, and 0 s 1 in case d 3. Then J r (s) t (s) (s) 0 by virtue of (6.12), (6.13), (6.15) and (6.16). Moreover J r (0) t (0) (0) J r (1) t (1) (1) J r (s) t (s) (s) ( (0) (1) (d 1) 2 ( (0) (1) d 2 (0) (1) Sd Sd A3 t t r r (s) (s) A1 r A2 t Sd (s) Note that r (s) r (0) r (1)
and hence r (0) r (1) r (s)
and if d 3, then t (s) t (0) t (1) and for j = 1 2 d 3, (0) sin j (s) sin j sin (1) j consequently
and hence t (0) t (1) t (s) 1 if j(0) 2 j(1) 2 if j(0) 2 j(1) 2
Sd (0) Sd (1) 1 0 Sd (s)
We thus get (6.20).
404
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For an open set E R d , denote by C 1 E the restrictions to E of all continuously differentiable functions on R d . c1 c2 , etc. appearing in what follows denote positive constants depending only on and d . Lemma 6.2 For any ! (0 1], there is a positive constant C23 = C23 (! d ) such that % d 2 2 2 2 u dx C23 u dx + 'i u dx (6.21) C ( )
N
i =1
C ( )
for u C 1 C () , a Borel subset N C () with N !C () 1 and 0 . Proof Let 1 and 0 . For u C 1 C () , we set x = " 1 (r t ) C () u-(r t ) = u " 1 (r t ) = u(x ) d 'u 1 'u(r t ) = 'xi " (r t ) i =1
By virtue of (6.8) – (6.11), % d 1 'u 'xd 1 1 1 'u 'u (x ) i () + (x ) (r t ) = tx 'r d 'xi 'xd 'r i =1
d 1 'u 'u 1 (x ) i () (r t ) = xd 't 'xi i =1 % d 1 'u 'xd 1 1 1 'u (x ) i () + (x ) + t xd 'xi 'xd 't i =1
d 1 'i 'u 'u 1 (r t ) = t xd (x ) () 'j 'x 'j i i =j
j = 1 2 d 2
Combining these with (6.12), (6.13), (6.14), we find that ' t ) u (r t ) A8 'u(r 'r ' t ) u (r t ) A9 r 'u(r 't ' t ) j = 1 2 d 2 u (r t ) A10 rt 'u(r 'j
(6.22) (6.23) (6.24)
(0) (1) Let 0 ! 1 N C () with N !C () x C () and x N . (i ) (i ) (i ) (i ) (i ) = " x i = 0 1. Then Put # = r t
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Reflecting diffusions on H¨older domains
553
u x (0) u x (1) = u- # (0) u- # (1) =
1
0
' (s) ds u- # 's
(s)
where # (s) = r (s) t (s) , and r (s) t (s) (s) are those given by (6.19). Integrating over x (1) N , we find that N u x (0)
(1) J # d # (1)
u dx + (1) ( )
N
1 0
' (s) u- # ds 's
Here d # (i ) denotes the product measure dr (i ) dt (i ) d (i ) for each i = 1 2. From this 2 u2 dx N C ( )
(
2 u dx
2C () N
(0) (0) J # d #
+2 (0) ( )
(1) ( ) 0 s 1
' (s) (1) u- # J # d # (1) ds 's
2(I + II )
'2
(6.25)
Obviously,
I C ()2
u2 dx
(6.26)
N
On account of Lemma 6.1 and (6.17),
d#
II
(0) (1) ( ) 0 s 1
(0) ( )
(1) ( ) 0 s 1
J # (0) J # (1) d # (1) ds J # (s)
2(d 1)
A3 A7
(0) ( ) (1) ( ) 0 s 1
(1) ( ) 0 s 1
' (s) 2 (s) u- # J # d # (0) d # (1) ds 's
d # (1) ds
= c1 2(d 1) +1
' (s) 2 (s) (1) u- # J # J # d # (1) ds 's
(0) ( ) (1) ( ) 0 s 1
' (s) 2 (s) u- # J # d # (0) d # (1) ds 's
On the other hand, we get from (6.22)–(6.24)
406
(6.27)
554
M. Fukushima, M. Tomisaki
' (s) u (s) (0) u (s) (0) (1) ' u- # ' + # t t (1) 's 'r # r r 't d 2 ' u # (s) (0) (1) + j j 'j j =1
d 'u (s) 1 c2 'xi " #
(6.28)
i =1
Denoting
'u " 1 by i and substituting (6.28) into (6.27), we arrive at 'xi
II c3
2(d 1) +3
d i =1
c3 2(d 1) +3
d
(0) ( ) (1)
2 i # (s) J # (s) d # (0) d # (1) ds
( ) 0 s 1
IIi ()
(6.29)
i =1
By fixing # (0) = r (0) t (0) (0) () 1, we make use and 0(s) s (s) (1) (1) (1) (1) # of the transformation # = r t = r t (s) (s) . Putting (s) (s) (s) is given by # = (r t ) = r t , we find that the Jacobian determinant '# (1) '# = s d . Moreover # = (r t s) exhausts a set # (0) s specified by (1 s)r (0) r s + (1 s)r (0) ad + (1 s)t (0) t s + (1 s)t (0) (1 s)j(0) j j s + (1 s)j(0)
j = 1 2 d 2
where ad = 1 if d = 2, = 0 if d 3, and j = (j = 1 2 d 3) d 2 = 2. So we get, for each i = 1 2 d , 2 (0) d # ds i # (s) J # (s) d # (1) IIi () = (0) =
( ) 0 s 1
(0) ( ) 0 s 1
(1) ( )
d # (0) ds
( (0) s )
i (#)2 J (#) s d d #
By exchanging the order of integration, ( r s r 2 d i (#) J (#)s 0 IIi () = ( ) 1s 1s 0 s 1
d) 2 ( j j j s t s t 1 ad j 0 d #ds 1s 1s 1s 1s j =1 2d +1 d 2 i (#)2 J (#) d # (
( )
407
Reflecting diffusions on H¨older domains
555
Combining this with (6.29) and (6.18), we have II c4 2(d 1) +4
d i =1
c5 2 C ()2
i (#)2 J (#) d # ( )
d
2
'i u dx
(6.30)
C ( )
i =1
Since N !C (), (6.25), (6.26) and (6.30) lead us to (6.21).
Lemma 6.3 Let 0 and E () be the following subset of R d with d 2. E () = #(x xd ) B () : xd (x ) where is a continuous function on #x R d 1 : x such that (0) = 0 and (x ) 0 x . Then the statement of Lemma 6.2 with C () replaced by E () above holds. Proof Let u C 1 E () and N ( E ()) be a Borel subset satisfying N !E (). For x = (x xd ) E (), we set x = (x xd ). We also use the polar coordinate with center x : y = x + r r = y x = (y x ) r S d 1 . The following inequality is obvious : u(y) dy + N u(x ) u x N u(x ) N u(y) u y dy + u x u y dy + N
N
from which we obtain N 2 u(x )2 ( 2 2 c1 u dy + N
+
'd u(x s) ds
c1 E ()
'd u(y s) ds yd
d ( i =1
2
yd
dy yE ( )yd 0
xd
N
+
2
xd
r
dy
yE ( )y=x +r
'i u x + s ds
0
2
2 .
2
u dy + 2E () N 2 'd u dy +2E ()2
xd
2
'd u(x s) ds
xd
E ( )
+4E ()d +1
d i =1
'i u(z )2
z E ( )zd 0
408
d 1
x z
. dz
556
M. Fukushima, M. Tomisaki
where we used the following estimate for the last term. ( 2 r dy 'i u x + s ds 0
yE ( )y=x +r
4E ()
r
dy
yE ( )y=x +r
'i u x + s 2 ds
0
r d 1 drd
= 4E () yE ( )y=x +r
4E ()d +1 z E ( )zd 0
r
'i u x + s 2 ds
0
'i u(z )2 dz x z d 1
Therefore we have N 2 u(x )2 dx E ( )
c2 E ()2
u2 dy + E ()2 2
'd u2 dx
N
E ( )
+E ()d +1
d i =1
c3 E ()2
'i u(z )2 dz
z E ( )zd 0
E ( )
u2 dy + E ()2 2 N
+E ()d +2
dx x z d 1
.
'd u2 dx E ( )
. 'i u2 dx
d i =1
E ( )
Noting that c4 d E () c5 d , we get the conclusion.
Proposition 6.1 now follows from Lemmas 6.3 and 6.4. Proof of Proposition 6.1 Let k IC 0 , and N be a Borel subset N !Ck (). In view of [16; Theorem 1.1.7], u %k of C k () satisfying H 1 Ck () provided u H 1 Ck () . By means of [1 : Theorem 3.18], u %k is approximated by functions belonging to C 1 Ck () in H 1 -norm. Noting that 1
% 1 k (N ) is a Borel subset of Ck () and satisfies %k (N ) ! Ck () for some
! (0 1], we obtain (6.1) with Ek () = Ck () from Lemma 6.3. Similarly (6.1) with Ek () = Qk () follows from Lemma 6.4. Acknowledgements. The second named author is very grateful to Professor Y. Ogura for valuable discussions on the transformation in Sect. 6.
References 1. R. A. Adams: Sobolev space. Academic Press, New York San Francisco London, 1975 2. R. F. Bass, P. Hsu: The semimartingale structure of reflecting Brownian motion. Proc. A.M.S. 108 (1990) 1007–1010
409
Reflecting diffusions on H¨older domains
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3. R. F. Bass, P. Hsu: Some potential theory for reflecting Brownian motion in H¨older and Lipschitz domains. Ann. Probab. 19 (1991) 486–506 4. M. Biroli, U. Mosco: A Saint-Venant principle for Dirichlet forms on discontinuous media. Preprint Series Univ. Bonn SFB 224, 1992 5. E. A. Carlen, S. Kusuoka, D. W. Stroock: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincar´e Probab. Statist. 23 (1987) 245–287 6. Z. Q. Chen, P. J. Fitzsimmons, R. J. Williams: Quasimartingales and strong Caccioppoli set. Potential Analysis. 2 (1993) 219–243 7. R. D. DeBlassie, E. H. Toby: Reflecting Brownian motion in a cusp. Trans. Am. Math. Soc. 339 (1993) 297–321 8. R. D. DeBlassie, E. H. Toby: On the semimartingale representation of reflecting Brownian motion in a cusp. Probab. Theory Relat. Fields 94 (1993) 505–524 9. M. Fukushima: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4 (1967) 183–215 10. M. Fukushima: On an Lp -estimate of resolvents of Markov processes. Publ. RIMS, Kyoto Univ. 13 (1977) 277–284 11. M. Fukushima: On a decomposition of additive functionals in the strict sense for a symmetric Markov processes. Proc. International Conference on ʻDirichlet Forms and Stochastic Processesʼ, Beijing, 1993, eds. Z. Ma, M. R¨ockner, J, Yan, Walter de Gruyter, Berlin 1995 12. M. Fukushima: On a strict decomposition of additive functionals for symmetric diffusion processes. Proc. Japan Acad. 70 Ser. A (1994) 277–281 13. M. Fukushima, Y.Oshima, M. Takeda: Dirichlet forms and symmetric Markov processes. Walter de Gruyter, Berlin 1994 14. M. Fukushima, M. Tomisaki: Reflecting diffusions on Lipshitz domains with cusps-analytic construction and Skorohod representation. Potential Analysis 4 (1995) 377–408 15. H. Kunita: General boundary conditions for multi-dimensional diffusion process. J. Math. Kyoto Univ. 10 (1970) 273–335 16. V. G. Mazʼja: Sobolev spaces. Springer-Verlag, Berlin Heidelberg New York 1985 17. J. Moser: A new proof of de Giorgiʼs theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13 (1960) 457–468 18. G. Stampacchia: Equations elliptiques du second ordre a` coefficients discontinus. S´eminar sur les equations aux de´riv´ees partielles, Coll`ege de France, 1963 19. H. Tanaka: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163–177 20. M. Tomisaki: Superposition of diffusion processes. J. Math. Soc. Japan 32 (1980) 671–696 21. M. Tomisaki: A construction of diffusion processed with singular product measures. Z. Wahrscheinlichkeitstheorie verw. Gebiete 52 (1980) 51–70 22. S. R. S. Varadhan, R. J. Williams: Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985) 405–443
This article was processed by the author using the LTEX style file pljour1m from Springer-Verlag.
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418
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540
Acknowledgments
The editors and the publisher wish to thank the following for granting permission to reprint in this volume the papers listed below. Abbreviations in brackets refer to the Bibliography of Masatoshi Fukushima. Academiæ Scientarium Fennicæ [R22] American Mathematical Society [R8], [R9] Cambridge University Press [R53] Kyoto University. Research Institute for Mathematical Sciences [R19] Mathematical Society of Japan [R6], [R12], [R32] Mittag-Leffler Institute [R43] Osaka University (Grad. School of Sciences, Dept. of Mathematics) [R5], [R34], [R38], [R49], [R78] Springer [R11], [R29], [R31], [R55], [R61], [R82] The Japan Academy [R58] University of Illinois (Dept. Mathematics) [R81] We would also like to add that the copyright for [R37] and [R66] is with M. Fukushima, and that for [R59] with the publisher.
541
542
Bibliography
Research Papers (English) [R1]
On Feller’s kernel and the Dirichlet norm, Nagoya Math. J. 24 (1964) 167–175.
[R2]
Resolvent kernels on a Martin space, Proc. Japan Acad. 41 (1965), 260–263.
[R3]
On spectral functions related to birth and death processes, J. Math. Kyoto Univ. 5 (1966), 151–161.
[R4]
On a class of Markov processes taking values on lines and the central limit theorem, (with M. Hitsuda), Nagoya Math. J. 30 (1967), 47–56.
[R5]
A construction of reflecting barrier Brownian motions for bounded domains, Osaka J. Math. 4 (1967), 183–215.
[R6]
On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan 21 (1969), 58–93.
[R7]
On Dirichlet spaces and Dirichlet rings, Proc. Japan Acad. 45 (1969), 433–436.
[R8]
Regular representations of Dirichlet spaces, Trans. Amer. Math. Soc. 155 (1971), 455– 473.
[R9]
Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc. 162 (1971), 185–224.
[R10]
On transition probabilities of symmetric strong Markov processes, J. Math. Kyoto Univ. 12 (1972), 431–450.
[R11]
On the generation of Markov processes by symmetric forms, in Proceedings of the Second Japan-USSR Symposium on Probability Theory, (Kyoto, 1972), 46–79, Lecture Notes in Math. 330, Springer, Berlin, 1973.
[R12]
Almost polar sets and an ergodic theorem, J. Math. Soc. Japan 26 (1974), 17–32.
[R13]
On the spectral distribution of a disordered system and the range of a random walk, Osaka J. Math. 11 (1974), 73–85.
[R14]
Local property of Dirichlet forms and continuity of sample paths, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 29 (1974), 1–6.
[R15]
On an asymptotic property of spectra of a random difference operator (with H. Nagai and S. Nakao), Proc. Japan Acad. 51 (1975), 100–102.
543
Bibliography [R16]
Asymptotic properties of the spectral distributions of disordered systems, in International Symposium on Mathematical Problems in Theoretical Physics, (Kyoto Univ., Kyoto, 1975), 224–227, Lecture Notes in Phys. 39, Springer, Berlin, 1975.
[R17]
On spectra of the Schrödinger operator with a white Gaussian noise potential (with S. Nakao), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), 267–274.
[R18]
Potential theory of symmetric Markov processes and its applications, in Proceedings of the Third Japan-USSR Symposium on Probability Theory, (Tashkent, 1975), 119– 133. Lecture Notes in Math. 550, Springer, Berlin, 1976.
[R19]
On an -estimate of resolvents of Markov processes, Publ. Res. Inst. Math. Sci. 13 (1977/78), 277–284.
[R20]
A decomposition of additive functionals of finite energy, Nagoya Math. J. 74 (1979), 137–168.
[R21]
On additive functionals admitting exceptional sets, J. Math. Kyoto Univ. 19 (1979), 191–202.
[R22]
Dirichlet spaces and additive functionals of finite energy, in Proceedings of the International Congress of Mathematicians, (Helsinki, 1978), 741–747, Acad. Sci. Fennicæ, Helsinki, 1980.
[R23]
A generalized stochastic calculus in homogenization, in Quantum fields–algebras, processes, (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978), 41–51, Springer, Vienna, 1980.
[R24]
On a stochastic calculus related to Dirichlet forms and distorted Brownian motions, in New stochasitic methods in physics, Phys. Rep. 77 (1981), 255–262.
[R25]
On a representation of local martingale additive functionals of symmetric diffusions, in Stochastic integrals, (Proc. Sympos. Durham, 1980), 110–118, Lecture Notes in Math. 851, Springer, Berlin, 1981.
[R26]
Capacity and quantum mechanical tunneling (with S. Albeverio, W. Karwowski and L. Streit), Comm. Math. Phys. 81 (1981), 501–513.
[R27]
On asymptotics of spectra of Schrödinger operators, in Statistical and physical aspects of Gaussian processes, (Saint-Flour, 1980), 335–347, Colloq. Internat. CNRS 307, CNRS, Paris, 1981.
[R28]
On absolute continuity of multidimensional symmetrizable diffusions, in Functional analysis in Markov processes, (Katata/Kyoto, 1981), 146–176, Lecture Notes in Math. 923, Springer, Berlin-New York, 1982.
[R29]
A note on irreducibility and ergodicity of symmetric Markov processes, in Stochastic processes in quantum theory and statistical physics, (Marseille, 1981), 200–207, Lecture Notes in Phys. 173, Springer, Berlin, 1982.
[R30]
Markov processes and functional analysis, in Proc. International Math. Conf. Singapore, 187–202, Eds. L.H. Chen, Y.B. Ng, M.J. Wicks, North Holland, 1982.
544
Bibliography [R31]
Capacitary maximal inequalities and an ergodic theorem, in Probability theory and mathematical statistics, (Tbilisi, 1982), 130–136, Lecture Notes in Math. 1021, Springer, Berlin, 1983.
[R32]
Basic properties of Brownian motion and a capacity on the Wiener space, J. Math. Soc. Japan 36(1984), 161–176.
[R33]
On conformal martingale diffusions and pluripolar sets (with M. Okada), J. Funct. Anal. 55 (1984), 377–388.
[R34]
A transformation of a symmetric Markov process and the Donsker-Varadhan theory (with M. Takeda), Osaka J. Math. 21 (1984), 311–326.
[R35]
A Dirichlet form on the Wiener space and properties on Brownian motion, in Théorie du potentiel, (Orsay, 1983), 290–300, Lecture Notes in Math. 1096, Springer, Berlin, 1984.
[R36]
Energy forms and diffusion processes, in Mathematics + Physics. Vol. 1, 65–97, World Sci. Publishing, Singapore, 1985.
[R37]
-capacities for general Markovian semigroups (with H. Kaneko), in Infinitedimensional analysis and stochastic processes, (Bielefeld, 1983), 41–47, Res. Notes in Math. 124, Pitman, Boston, MA, 1985.
[R38]
On the continuity of plurisubharmonic functions along conformal diffusions, Osaka J. Math. 23 (1986), 69–75.
[R39]
A stochastic approach to the minimum principle for the complex Monge-Ampère operator, in Stochastic processes and their applications, (Nagoya, 1985), 38–50, Lecture Notes in Math. 1203, Springer, Berlin, 1986.
[R40]
Reversibility of solutions to martingale problems (with D. W. Stroock), in Probability, statistical mechanics, and number theory, 107–123, Adv. Math. Suppl. Stud. 9, Academic Press, Orlando, FL, 1986.
[R41]
On recurrence criteria in the Dirichlet space theory, in From local times to global geometry, control and physics, (Coventry, 1984/85), 100–110, Pitman Res. Notes Math. Ser. 150, Longman Sci. Tech., Harlow, 1986.
[R42]
On Dirichlet forms with random data—recurrence and homogenization (with S. Nakao and M. Takeda), in Stochastic processes—mathematics and physics, II, (Bielefeld, 1985), 87–97, Lecture Notes in Math. 1250, Springer, Berlin, 1987.
[R43]
On Dirichlet forms for plurisubharmonic functions (with M. Okada), Acta Math. 159 (1987), 171–213.
[R44]
A note on capacities in infinite dimensions, in Probability theory and mathematical statistics, (Kyoto, 1986), 80–85, Lecture Notes in Math. 1299, Springer, Berlin, 1988.
[R45]
On holomorphic diffusions and plurisubharmonic functions, in Geometry of random motion, (Ithaca, N.Y., 1987), 65–78, Contemp. Math. 73, Amer. Math. Soc., Providence, RI, 1988.
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Bibliography [R46]
On two classes of smooth measures for symmetric Markov processes, in Stochastic analysis, (Paris, 1987), 17–27, Lecture Notes in Math. 1322, Springer, Berlin, 1988.
[R47]
On the skew product of symmetric diffusion processes (with Y. Oshima), Forum Math. 1 (1989), 103–142.
[R48]
Capacities on Wiener space: tightness and invariance (with S. Albeverio, W. Hansen, Z.-M. Ma and M. Röckner), C. R. Acad. Sci. Paris Se’r. I Math. 312 (1991), 931–935.
[R49]
On quasi-supports of smooth measures and closability of pre-Dirichlet forms (with Y. LeJan), Osaka J. Math. 28 (1991), no. 4, 837–845.
[R50]
On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures (with K. Sato and S. Taniguchi), Osaka J. Math. 28 (1991), 517–535.
[R51]
An invariance result for capacities on Wiener space (with S. Albeverio, W. Hansen, Z.-M. Ma and M. Röckner), J. Functional Analysis 106 (1992), 35–49.
[R52]
On 2-capacities for a class of elliptic pseudo differential operators (with N. Jacob and H. Kaneko), Math. Ann. 293 (1992), 343–348.
[R53]
Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, in Ideas and methods in mathematical analysis, stochastics, and applications, (Oslo, 1988), 151–161, Cambridge Univ. Press, Cambridge, 1992.
[R54]
-capacities and Hunt processes in infinite dimensions, in Probability theory and mathematical statistics, (Kiev, 1991), 96–103, World Sci. Publishing, River Edge, NJ, 1992.
[R55]
On a spectral analysis for the Sierpi´nski gasket (with T. Shima), Potential Analysis 1 (1992), 1–35.
[R56]
Two topics related to Dirichlet forms: quasi-everywhere convergences and additive functionals, in Dirichlet forms (Varenna, 1992), 21–53, G. Dell’Antonio and U. Mosco (Eds), Lecture Notes in Math. 1563, Springer, Berlin, 1993.
[R57]
On discontinuity and tail behaviours of the integrated density of states for nested prefractals (with T. Shima), Comm. Math. Phys. 163 (1994), 461–471.
[R58]
On a strict decomposition of additive functionals for symmetric diffusion processes, Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 277–281.
[R59]
On a decomposition of additive functionals in the strict sense for a symmetric Markov process, in Dirichlet forms and stochastic processes, Z. Ma, M. Röckner, J. Yan (Eds), Walter de Gruyter, (1995), 155–169.
[R60]
Reflecting diffusions on Lipschitz domains with cusps—analytic construction and Skorohod representation (with M. Tomisaki), in Potential theory and degenerate partial differential operators (Parma), Potential Analysis 4 (1995), 377–408.
[R61]
Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps (with M. Tomisaki), Probab. Theory Related Fields 106 (1996), 521– 557.
546
Bibliography [R62]
On decomposition of additive functionals of reflecting Brownian motions (with M. Tomisaki), in Ito’s stochastic calculus and probability theory, N. Ikeda, S. Watanabe, M. Fukushima, H. Kunita (Eds), Springer, Tokyo, 1996, 51–61.
[R63]
Dirichlet forms, Caccioppoli sets and the Skorohod equation, in Stochastic Differential and Difference equations, Csiszar Michaletzky (Eds.), Birkhäuser 1997, 56–66.
[R64]
Distorted Brownian motions and BV functions, in Trends in probability and related analysis, (Taipei, 1996), 143–150, N-R.Shieh (Ed), World Sci. Publishing, River Edge, NJ, 1997.
[R65]
Large deviations and related LIL’s for Brownian motions on nested fractals (with T. Shima and M. Takeda), Osaka J. Math. 36 (1999), 497–537.
[R66]
On semi-martingale characterizations of functionals of symmetric Markov processes, Electron. J. Probab. 4 (1999), Paper 18, 1–32, www.math.washington.edu/ ~ejpecp.
[R67]
functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space, J. Funct. Analysis. 174 (2000), 227–249.
[R68]
On limit theorems for Brownian motions on unbounded fractal sets, in Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), 227–237, Progr. Probab., S. Graf (Ed) 46, Birkhäuser, Basel, 2000.
[R69]
On Ito’s formulae for additive functionals of symmetric diffusion processes, in Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999), 201– 211, F. Getsztesy (Ed), CMS Conf. Proc. 28, Amer. Math. Soc., Providence, RI, 2000.
[R70]
On the space of BV functions and a related stochastic calculus in infinite dimensions (with M. Hino), J. Funct. Analysis 183 (2001), 245–268.
[R71]
Dynkin games via Dirichlet forms and singular control of one-dimensional diffusions (with M. Taksar), SIAM J. Control Optim. 41 (2002), 682–699.
[R72]
On Sobolev and capacitary inequalities for contractive Besov spaces over -sets (with T. Uemura), Potential Analysis 18 (2003), 59–77.
[R73]
Capacitary bounds of measures and ultracontractivity of time changed processes (with T. Uemura), J. Math. Pures Appl. 82 (2003), 553–572.
[R74]
A note on regular Dirichlet subspaces (with J. Ying), Proc. Am. Math. Soc. 131 (2003) 1607-1610 (2003); erratum ibid. 132 (2004), 1559.
[R75]
On spectral synthesis for contractive -norms and Besov spaces (with T. Uemura), Potential Analysis 20 (2004), 195–206.
[R76]
Function spaces and symmetric Markov processes, in Stochastic analysis and related topics in Kyoto, Advanced Studies in Pure Mathematics 41 (2004), 75–89, Math. Soc. Japan.
[R77]
Time changes of symmetric diffusions and Feller measures (with P. He and J. Ying), Ann. Probab. 32 (2004) 3138–3166.
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Bibliography [R78]
On regular Dirichlet subspaces of 1 and associated linear diffusions (with X. Fang and J. Ying), Osaka J. Math. 42 (2005), 27–41.
[R79]
Poisson point processes attached to symmetric diffusions (with H. Tanaka), Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005), 419–459.
[R80]
Traces of symmetric Markov processes and their characterizations (with Z.-Q. Chen and J. Ying), Ann. Probab. 34 (2006), 1052–1102.
[R81]
Entrance law, exit system and Lévy system of time changed processes (with Z.-Q. Chen and J. Ying), Illinois J. Math. 50 (2006), 269–312.
[R82]
Extending Markov processes in weak duality by Poisson point processes of excursions, (with Z.-Q. Chen and J. Ying), in The Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Itô (Oslo), F.E. Benth, G.Di Nunno, T. Lindstrom, B. Oksendal, T. Zhang (Eds), 153–196, Springer, 2007.
[R83]
On Feller’s boundary problem for Markov processes in weak duality (with Z.-Q. Chen), J. Funct. Anal. 252 (2007), 710–733.
[R84]
One-point extensions of Markov processes by darning (with Z.-Q. Chen), Probab. Th. Rel. Fields 141 (2008), 61–112.
[R85]
Flux and lateral conditions for symmetric Markov processes (with Z.-Q. Chen), Potential Anal. 29 (2008), 241–269.
[R86]
On unique extension of time changed reflecting Brownian motions (with Z.-Q. Chen), Ann. Inst. Henri Poincare, Probab. Statist. 45 (2009), 864–875.
[R87]
On extended Dirichlet spaces and the space of BL functions, Potential Theory and Stochastics in Albac, Aurel Cornea Memorial Volume, Eds. D. Bakry, L.Beznea, N. Boboc, G. Bucur, M. Röckner, Theta Foundation, Bucharest, AMS distribution, 2009, 101–110.
Expository Writing [E1]
Boundary problems of Brownian motions and the Dirichlet spaces, S¯ugaku 20 (1968), 211–221 (in Japanese).
[E2]
On the Theory of Markov Processes, Butsuri (The Physical Society of Japan) 25 (1970), 37–41 (in Japanese).
[E3]
On Random Spectra, Butsuri (The Physical Society of Japan) 34 (1979), 153–159 (in Japanese).
[E4]
On Spectral Analysis on Fractal, Manufacturing and Technology 48 no. 1 (1996), 55–59 (in Japanese).
[E5]
Fractal and Random walk – Toward the function and the shape of nature, in: Shizennoshikumi to Ningennochie, Osaka University Press, 1997, 211–222 (in Japanese).
548
Bibliography [E6]
Decompositions of symmetric diffusion processes and related topics in analysis, S¯ugaku Expositions 14 (2001), 1–13, AMS.
[E7]
Stochastic Control Problem and Dynkin’s Game Theory, Gien (Organization for Research and Development of Innovative Science and Technology (ORDIST)), Kansai University 111 (2001), 35–39 (in Japanese).
[E8]
Refined solutions of optimal stopping games for symmetric Markov processes, Technology Reports of Kansai University (with K. Menda), 48 (2006), 101–110.
[E9]
On the works of Kiyosi Itô and stochastic analysis, Japanese J. Math. 2 (2007), 45– 53.
Seminar on Probability (in Japanese) [S1]
(with Kiyosi Itô and Shinzo Watanabe) On Diffusion processes, Seminar on Probability 3, 1960.
[S2]
(with Ken-ichi Sato and Masao Nagasawa) Transformation of Markov processes and boundary problems, Seminar on Probability 16, 1960.
[S3]
Dirichlet space and its representations, Seminar on Probability 31, 1969.
[S4]
(with Hiroshi Kunita) Studies on Markov processes, Seminar on Probability 40, 1969.
[S5]
(with Shintaro Nakao and Sin-ichi Kotani) On Random spectra, Seminar on Probability 45, 1977.
Monographs and Textbooks [MT1] Dirichlet Forms and Markov Processes (Japanese), Kinokuniya Co. Ltd., 1975. [MT2] (with Kazunari Ishii) Natural Phenomena and Stochastic Processes (Japanese), Nippon-Hyoron-Sha Co., Ltd., 1980, (enlarged ed.) 1996. [MT3] Dirichlet Forms and Markov Processes, North-Holland Mathematical Library 23, North-Holland, Amsterdam-New York/ Kodansha, Tokyo, 1980. ¯ [MT4] (with Y¯oichi Oshima and Masayoshi Takeda) Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics 19, de Gruyter, Berlin, 1994. [MT5] Probability Theory (Japanese), Shokabo Pub. Co., Ltd., 1998. [MT6] (with Masayoshi Takeda) Markov Processes (Japanese), Baifukan Pub. Co., Ltd, 2008.
549
