Chapter 1. The mathematical work of Masatoshi Fukushima - An Essay

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1 Chapter 1 The mathematical work of Masatoshi Fukushima - An Essay Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura Over more than five decades Professor Fukushima has made remarkable contributions to mathematics, especially to stochastic analysis and its applications to science, in particular to mathematical physics. For many of his colleagues, students and friends it becomes an excellent opportunity to express their appreciation of himself and his work by celebrating his Sanju, the 80th birthday in the Japanese tradition, with an international conference and by publishing a Festschrift in his honour. Naturally such a Festschrift should contain a scientific account of Professor Fukushima s oeuvre. When making here an attempt, i.e. writing un essai in the classical meaning, to present such an account, we must be aware that we are not discussing a completed oeuvre. Professor Fukushima is continuing to make highly original research contributions with deep insights. Obviously it is pre-matured to describe or guess the lasting impact these latest contributions will have. However it is fair to say that some of his earlier works are meanwhile considered as to belong to the classical canon of stochastic analysis, and hence we will concentrate more on those contributions. But we will see how some of his latest research indeed connect with his earlier work in a natural way. Professor Fukushima started his graduate studies in 1959 as a student of Kiyoshi Itô at Kyoto University. For anyone who is interested in the history of probability theory it is a temptation to write about the amazing changes in the field during the three decades between the publication of the works of P. Lévy [24], J. L. Doob [8], K. Itô [19] and [20], and S. Kakutani [22], and the publication of the monograph of K. Itô and H. P. McKean [21]. This period also includes the contributions of the E. B. Dynkin school, and we refer to the monographs [10] - [11], and for us quite important the work of W. Feller, in particular [12] and [13], as well as the contributions of G. A. Hunt [16] - [18]. (Clearly, this list is not exhausting.) However we have already some good, although brief, account of this period, especially when having in mind the developments in Japan, see D. W. Stroock and S. R. S. Varadhan [29], H. Föllmer [15], and in particular to M. Fukushima [F39] as well as [F49]. We should also mention the work of Kôsaku Yosida combining one-parameter operator semigroup theory with probability theory and potential theory on which S. Watanabe 3

2 4 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura [30] gave a readable account. For Professor Fukushima, early in his career, the problem to understand Feller s boundary theory for one-dimensional diffusion processes in higher dimensions, say Brownian motion in bounded domains, took an important place, see [F1] and [F2], and in some sense this is a topic that still occupies him after 50 years, see for example [F44]. Already in these early papers the idea was to use Dirichlet spaces as introduced by A. Beurling and J. Deny [2] and [3]. Feller s work in one dimension could use the fact that (in a rough formulation) integrable functions with an integrable distributional derivative are absolutely continuous. Clearly this does not apply in higher dimensions, but the results of the paper by J. Deny and J.-L. Lions [7], see also J. L. Doob [9], provided a type of substitute, i.e. allowing to pass to quasi everywhere fine continuous versions. In [7] many technical tools originated from classical potential theory, by which we mean really the theory of harmonic functions and the Newton potential, and thus the method had a natural limitation. In the paper [F2] Professor Fukushima introduced already a significant change into the theory of Dirichlet spaces as initiated by A. Beurling and J. Deny: the concept of Dirichlet spaces related to L 2 (X; m) was introduced, where X is a certain Hausdorff space with some finite measure m. Although this looks at first glance as technical or minor, today this is the standard definition of (symmetric) Dirichlet spaces (with some more general space X and measure m), once we add regularity, see the comments to [F3] below. The papers [F3] and [F4] are Professor Fukushima s early seminal contribution in which he laid the foundation of what is nowadays understood as the modern theory of Dirichlet forms. Most of this work was done while he was a post doctoral fellow at the University of Illinois, Urbana - Champaign, working in the group of J. L. Doob. In [F3] parts of the quadratic form based potential theory of symmetric Dirichlet forms are studied, and in particular regular representations are discussed. The basic problem was to find for a given L 2 - related Dirichlet space an isometric version on a locally compact metric space that contains sufficiently many regular (continuous) elements. Now the ground was prepared for [F4], the construction of a strong Markov process associated to a regular Dirichlet space. The ideas picked up from the Deny - Lions paper [7], and employed already in [F2], were combined with the regular representation result. Using the natural capacity in a regular Dirichlet space it is possible to obtain quasi-continuous refinements of elements and this makes it eventually possible to construct the corresponding Hunt process (up to an invariant negligible set). This result was immediately recognised as a breakthrough and Professor Fukushima was invited to give a talk at the ICM in Helsinki The construction of the Hunt process associated with a Dirichlet form we find now in the monograph [F43] and its predecessors [F10] and [F23] follows the modification by M. L. Silverstein which appeared in [27], not the paper [F4]. Professor Fukushima

3 The mathematical work of Masatoshi Fukushima - An Essay 5 always points to the important contributions of Martin Silverstein to the theory of Dirichlet forms [27] and the boundary theory of symmetric Markov processes [28], as did Martin Silverstein appreciated his work. Martin Silverstein was among the first to recognize the importance of the regular representation of Dirichlet forms considered by Professor Fukushima, see [27], Section 2. Maybe in times without citation index, impact factors, etc., it was much easier to add to academic standards also civilized standards in human relations, respect for the work of colleagues, and a certain intellectual honesty. The construction of a Hunt process being associated with a regular Dirichlet form was already a big step forward, but immediately several question arose, for example: 1. Itô s stochastic calculus is an analysis on path level. How can we derive a corresponding theory for symmetric Markov processes which may not be semimartingales? 2. The processes are constructed up to a set of zero capacity. Is it possible to remove such exceptional set? 3. Can this approach to stochastic processes help to treat problems in science, especially mathematical physics? 4. Is it eventually possible to extend Feller s boundary theory to higher dimensions? Itô calculus changed its face under the influence of P. A. Meyer and his school. The basic objects for a reasonable stochastic calculus were identified as semimartingales, but the Hunt processes constructed with the help of Dirichlet forms are in general not semi-martingales. The breakthrough was the paper [F8] where the Fukushima decomposition was introduced. Using versions of elements in the Dirichlet space, additive functionals were introduced with the help of the associated Hunt process and it was proved that every such additive functional admits a decomposition into a martingale and an additive functional of zero energy. Then a stochastic calculus for additive functionals was established. Meanwhile the Fukushima decomposition and the related stochastic calculus is a core element of stochastic analysis with many applications in the studies of diffusions in infinite dimensions or on fractals, and in the study of Markov processes in random environment, just to mention some areas. One of the hardest problem is to construct nice versions of the process associated with a Dirichlet form. The best solution would be to get rid of any exceptional set. Several papers of Professor Fukushima s, including [F13, F16, F21], are devoted to this problem. Arguably the most influential one is that jointly with H. Kaneko [F16]. Here (r, p)-capacities, originally introduced by P. Malliavin [26], were taken up with the aim to get version up to a set of (r, p)-capacity zero. These capacities are monotone in r and in p, and it may happen that for r or p large the only set of

4 6 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura (r, p)-capacity zero is the empty set. H. Kaneko [23] has used this to study diffusion processes in an L p -setting. Many of the non-japanese colleagues of Professor Fukushima may not know about the book [F9] in Japanese co-authored with the physicist K. Ishii titled Natural Phenomena and Stochastic Processes first published in Influenced by the work of K. Ishii, Professor Fukushima turned to the study of the spectra of random Schrödinger operators and related topics, see [F6] written with H. Nagai and S. Nakao, [F7] with S. Nakao, as well as [F18] with S. Nakao and M. Takeda. Subsequently he continued to publish on mathematical physics and was in closed contact with colleagues in Bielefeld, including S. Albeverio, W. Karwowski and L. Streit, see for example [F11]. His interest and work had much impact on the later work of S. Kotani. Within our essay we can not discuss all of Professor Fukushima s contributions, so we mention only briefly his work on ergodic theorems [F5, F12], on problems related to large deviation, for example [F15] with M. Takeda, on stochastic analysis on fractals [F20, F22, F27] in collaboration with T. Shima and M. Takeda, on Dirichlet forms related to complex analysis of several variables [F17] and [F14, F19] with M. Okada, or on BV functions, capacitary inequalities and the geometry of sets [F28] - [F32], including joint works with M. Hino or T. Uemura, or finally, already more related to the boundary theory of Markov processes, the study of reflecting diffusions with M. Tomisaki [F24] - [F26]. However we want and we have to return to the beginning of his career and the attempt to extend Feller s theory. Time changes of Markov processes are closely related to their boundary theory. Motivated by the Douglas integral, which was the main tool of J. Douglas to give a solution to the Plateau problem and that also characterizes the trace of reflecting Brownian motion in the unit disk on its boundary, Professor Fukushima with P. He and J. Ying in [F33] studied time changes of symmetric diffusions and the role played by the Feller measure. They showed that the jumping and killing measures for the time-changed process are bounded below by the Feller measure and the supplementary Feller measure, respectively. A complete solution to this problem is obtained in a joint work of Professor Fukushima with Z.-Q. Chen and J. Ying [F35], where not only the jumping and killing measures of the time-changed process are identified, but also the complete characterization of the Beurling - Deny decomposition, see also the extension in [F36]. The boundary theory for one-dimensional diffusions is well understood thanks to the fundamental works of W. Feller, K. Itô, and H. P. McKean. Much less is known for the boundary theory of multi-dimensional diffusions and of Markov processes with discontinuous sample paths. In his work with H. Tanaka [F34] Professor Fukushima studied the one-point extension of absorbing symmetric diffusions using excursion

5 The mathematical work of Masatoshi Fukushima - An Essay 7 theory. In a series of joint works with Z.-Q. Chen [F38, F40, F41] and with Z.-Q. Chen and J. Ying [F37], he has further studied one-point extensions of general symmetric Markov processes, as well as strong Markov processes in weak duality, and their characterization. Moreover, many-point extensions of strong Markov processes were also studied. These works provide us with a satisfactory picture of the boundary theory for symmetric Markov processes with finitely many boundary points, and more generally, for strong Markov processes having weak duals, with finitely many boundary points. An important tool used in this study is the reflected Dirichlet space first introduced by M. L. Silverstein [27] in Very recently in joint work with Z.-Q. Chen [F48] one point extension theory was used to give a direct construction of Walsh s Brownian motion. This construction uses Dirichlet form methods. The Dirichlet form characterization of this process together with the results in [4] and [6] gives the analytic characterization of harmonicity and subharmonicity for Walsh s Brownian motion which recovers the main result of the recent paper of P. Fitzsimmons and K. E. Kutler [14]. Using Dirichlet forms, in [F42]and [F45] Professor Fukushima has revisited Feller s boundary theory for one-dimensional diffusions providing us with an elegant characterization. In the joint monograph [F44] with Z.-Q. Chen many of these and more results are presented. In particular this monograph covers the potential theory for regular symmetric Dirichlet forms as well as quasi-regular symmetric Dirichlet forms initiated by S. Albeverio, Z.-M. Ma and M. Röckner, see [1] and [25], in a unified and systematic way. A further new area of Professor Fukushima s latest research is the study of the interaction between probability theory and complex analysis, more precisely Komatu - Loewner evolution in multiply connected planar domains. Here we refer to the recent papers [F46] with Z.-Q. Chen and S. Rohde, as well as [F50], and [5]. A special case of the many-point extension of absorbing Brownian motion, called Brownian motion with darning, played an important role in such study. References to publications of M. Fukushima [F1] A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4 (1967), [F2] On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities. J. Math. Soc. Japan 21 (1969), [F3] Regular representations of Dirichlet spaces. Trans. Amer. Math. Soc. 155 (1971), [F4] Dirichlet spaces and strong Markov processes. Trans. Amer. Math. Soc. 162 (1971), [F5] Almost polar sets and an ergodic theorem. J. Math. Soc. Japan 26 (1974), [F6] On an asymptotic property of spectra of a random difference operator (with H. Nagai and S. Nakao). Proc. Japan Acad. 51 (1975),

6 8 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura [F7] On the spectra of the Schrödinger operator with a white Gaussian noise potential (with S. Nakao). Z. Wahrscheinlichkeitstheorie und verw. Gebiete 37 (1976/77), [F8] A decomposition of additive functionals of finite energy. Nagoya Math. J. 74 (1979), [F9] (with K. Ishii) Natural Phenomena and Stochastic Processes (Japanese). Nippon-Hyoron-Sha Co., Ltd., Tokyo 1980, (enlarged ed ) [F10] Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, and Kodansha, Tokyo, [F11] Capacity and quantum mechanical tunnelling (with S. Albeverio, W. Karwowski, and L. Streit). Commun. Math. Phys. 81 (1981), [F12] A note on irreducibility and ergodicity of symmetric Markow processes. In : Stochastic Processes and Quantum Theory and Statistical Physics, Marseille Lecture Notes in Physics 173, Springer Verlag, Berlin 1982, [F13] Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Japan 36 (1984), [F14] On conformal martingale diffusions and pluripolar sets (with M. Okada). J. Funct. Anal. 55 (1984), [F15] A transformation of a symmetric Markov process and the Donsker-Varadhan theory (with M. Takeda). Osaka J. Math. 21 (1984), [F16] (r, p)-capacities for general Markovian semigroups (with H. Kaneko). In : Infinite Dimensional Analysis and Stochastic Processes, Bielefeld Research Notes in Mathematics 124, Pitman, Boston MA 1985, [F17] On the continuity of plurisubharmonic functions along conformal diffusions. Osaka J. Math. 23 (1986), [F18] On Dirichlet forms with random data - recurrence and homogenization (with S. Nakao and M. Takeda). In : Stochastic Processes - Mathematics and Physics II, Bielefeld Lecture Notes in Math. 1250, Springer Verlag, Berlin 1987, [F19] On Dirichlet forms for plurisubharmonic functions (with M. Okada). Acta Math 159 (1987), [F20] On a spectral analysis for the Sierpinski gasket (with T. Shima). Potential Analysis 1 (1992), [F21] Two topics related to Dirichlet forms : Quasi-everywhere convergence and additive functionals In : Dirichlet Forms, Varenna Lecture Notes in Math. 1563, Springer Verlag, 1993, [F22] On discontinuity and tail behaviours of the integrated density of states for nested pre-fractals (with T. Shima). Commun. Math. Phys. 163 (1994), [F23] (with Y. Oshima and M. Takeda) Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter Verlag, Berlin [F24] Reflecting diffusions on Lipschitz domains with cusps - analytic construction and Skorohod representation (with M. Tomisaki). Potential Analysis 4 (1995),

7 The mathematical work of Masatoshi Fukushima - An Essay [F25] Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps (with M. Tomisaki). Probab. Theory Related Fields 106 (1996), [F26] On decomposition of additive functionals of reflecting Brownian motions (with M. Tomisaki). In : Ito s Stochastic Calculus and Probability Theory. Springer Verlag, Tokyo 1996, [F27] Large deviations and related LIL s for Brownian motion on nested fractals (with T. Shima and M. Takeda). Osaka J. Math. 36 (1999), [F28] BV functions and distorted Ornstein - Uhlenbeck processes over abstract Wiener space. J. Funct. Anal. 174 (2000), [F29] On the space of BV functions and a related stochastic calculus in infinite dimensions (with M. Hino). J. Funct. Anal. 183 (2001), [F30] On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets (with T. Uemura). Potential Analysis 18 (2003), [F31] Capacitary bounds of measures and ultracontactivity of time changed processes (with T. Uemura). J. Math. Pures Appl. 82 (2003), [F32] On spectral synthesis for contractive p-norms and Besov spaces (with T. Uemura). Potential Analysis 20 (2004), [F33] Time changes of symmetric diffusions and Feller measures (with P. He and J. Ying). Ann. Probab. 32 (2004), [F34] Poisson point processes attached to symmetric diffusions (with H. Tanaka). Ann. Inst. Henri Poincaré Probab. Stat. 41 (2005), [F35] Traces of symmetric Markov processes and their characterizations (with Z.-Q. Chen and J. Ying). Ann. Probab. 34 (2006), [F36] Entrance law, exit system and Lévy system of time changed processes (with Z.-Q. Chen and J. Ying). Illinois J. Math. 50 (2006), [F37] Extending Markov processes in weak duality by Poisson point processes of excursions (with Z.-Q. Chen and J. Ying). The Abel Symposium Springer Verlag, Berlin 2007, [F38] On Feller s boundary problem for Markov processes in weak duality (with Z.-Q. Chen) J. Funct. Anal. 252 (2007), [F39] On the works of Kiyoshi Itô and stochastic analysis. Japanese J. Math. 2 (2007), [F40] One-point extensions of Markov processes by darning (with Z.-Q. Chen). Probab. Theory Related Fields 141 (2008), [F41] Flux and lateral conditions for symmetric Markov processes (with Z.-Q. Chen). Potential Analysis 29 (2008), [F42] From one dimensional diffusions to symmetric Markov processes Stochastic Processes Appl. 120 (2010), [F43] (with Y. Oshima and M. Takeda) Dirichlet Forms and Symmetric Markov Processes. Second revised and extended editions. Walter de Gruyter Verlag,

8 10 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura Berlin [F44] (with Z.-Q. Chen) Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton NJ, [F45] On general boundary conditions for one-dimensional diffusions with symmetry. J. Math Soc. Japan 66 (2014), [F46] Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains (with Z.-Q. Chen and S. Rohde). Trans. Amer. Math. Soc. (to appear). [F47] On Villat s kernels and BMD Schwarz kernels in Komatu - Loewner equations (with H. Kaneko). Stochastic Analysis and Applications 2014, Springer Proceedings in Mathematics. (to appear) [F48] One - point reflection (with Z.-Q. Chen). Stochastic Processes and their Applications. (to appear) [F49] Feller s contributions to the one - dimensional diffusion theory and beyond. In: Selected Works of William Feller, 2 vols. Eds. R. Schilling, Z. Vondracek, W. Woyczynski. Springer Verlag. (to appear) [F50] Stochastic Komatu - Loewner evolution and BMD domain constant (with Z.- Q. Chen), arxiv: [math PR] (preprint) Further References [1] Albeverio, S., and Ma, Z.-M., Necessary and sufficient conditions for the existence of m-perfect processes associated with Dirichlet forms. In:Séminaire de Probabilités. XXV. Lecture Notes in Mathematics, Vol.1485, Springer Verlag, Berlin 1991, [2] Beurling, A., and Deny, J., Espace de Dirichlet I. Le cas élémentaire. Acta Math. 99 (1958), [3] Beurling, A., and Deny, J., Dirichlet spaces. Proc. Nat. Acad. Sci. U.S.A. 45 (1959), [4] Chen, Z.-Q., On notions of harmonicity. Proc. Amer. Math. Soc. 137 (2009), [5] Chen, Z.-Q., Brownian Motion with Darning. Lecture notes for talks given at RIMS, Kyoto University, [6] Chen, Z.-Q., and Kuwae, K., On subharmonicity for symmetric Markov processes. J. Math. Soc. Japan 64 (2012), [7] Deny, J., and Lions, J.-L., Les espaces du type de Beppo Levi. Ann. Inst. Fourier (Grenoble) 5 (1953/54), [8] Doob, J.L., Stochastic processes depending on a continuous parameter. Trans. Amer. Math. Soc. 42 (1937), [9] Doob, J.L., Boundary properties of functions with finite Dirichlet integrals. Ann. Inst. Fourier (Grenoble) 12 (1962), [10] Dynkin, E.B., Markov Processes. Vol. I. Springer Verlag, Berlin 1965.

9 The mathematical work of Masatoshi Fukushima - An Essay 11 [11] Dynkin, E.B., Markov Processes. Vol. II. Springer Verlag, Berlin [12] Feller, W., Zur Theorie der stochastischen Prozesse (Existenz-und Eindeutigkeitss atze). Math. Ann. 113 (1936), [13] Feller, W., The parabolic differential equations and the associated semi-groups of transformations. Ann.ofMath.(2)55 (1952), [14] Fitzsimmons, P. J., and Kuter, K.E., Harmonic functions on Walsh s Brownian motion. Stochastic Processes Appl. 124 (2014), [15] Föllmer, H., On Kiyosi Itô s work and its impact. In : International Congress of Mathematicians, Vol. I. Madrid European Mathematical Society, Zürich 2007, [16] Hunt, G.A., Markov processes and potentials I. Illinois J. Math. 1 (1957), [17] Hunt, G.A.,Markov processes and potentials II. Illinois J. Math. 1 (1957), [18] Hunt, G.A., Markov processes and potentials III. Illinois J. Math. 2 (1958), [19] Itô, K., On stochastic processes. I. (Infinitely divisible laws of probability), (doctoral thesis), Jap. J. Math. 18 (1942), [20] Itô, K., Differential equations determining a Markoff process. (In Japanese). J. Pan-Japan Math. Coll (1942), English translation in: Kiyosi Itô. Selected Papers (Eds. D.W. Stroock and S.R.S. Varadhan.) Springer Verlag, Berlin 1986, [21] Itô, K., and McKean, H.P., Diffusion Processes and Their Sample Paths. Springer Verlag, Berlin [22] Kakutani, S., Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20 (1944), [23] Kaneko, H., On (r, p) - capacities for Markov processes. Osaka J. Math. 23 (1986), [24] Lévy, P., Théorie de l Addition des Variables Aléatoires. Gauthier - Villars, Paris [25] Ma, Z.-M., and Röckner, M., Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext. Springer Verlag, Berlin [26] Malliavin, P., Implicit functions in finite co-rank on the Wiener space. In: Stochastic Analysis, Katata / Kyoto North - Holland Publisher, Amsterdam 1984, [27] Silverstein, M.L., Symmetric Markov Processes. Lecture Notes in Mathematics, Vol Springer Verlag, Berlin [28] Silverstein, M.L., Boundary Theory for Symmetric Markov Processes. Lecture Notes in Mathematics, Vol. 516, Springer Verlag, [29] Stroock, D.W., and Varadhan, S.R.S., Introduction. In: Kiyosi Itô. Selected Papers (Eds. D.W. Stroock and S.R.S. Varadhan.) Springer Verlag, Berlin 1986, vii - xii.

10 12 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura [30] Watanabe, S., Markov Processes (Comments), and Potential Theory (Comments). In: Kôsaku Yosida. Collected Papers. Ed. K. Itô. Springer Verlag, Berlin 1992, and

11 The mathematical work of Masatoshi Fukushima - An Essay 13 M. Fukushima with S. Watanabe and H. Kunita in London, 1980.

12 14 Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda and Toshihiro Uemura M. Fukushima at study, 2011.

13 The mathematical work of Masatoshi Fukushima - An Essay 15 M. Fukushima with N. Jacob and Z.-Q. Chen at SAA-Osaka, 2010.