Contents -- Preface -- H. Tanaka: An Appreciation -- From Local Times to Random Environments... -- Contributions and Influences of Professor Tanaka in Stochastic Analysis -- Some Comments on My Mathematical Works in Retrospect -- Additive Functionals of the Brownian Path -- 1. INTRODUCTION. -- 2. BROWNIAN MOTION. -- 3. THE ASSOCIATED MEASURE OF AN ADDITIVE FUNCTIONAL. -- 4. UNIQUENESS. -- 5. CONSTRUCTION OF AN ADDITIVE FUNCTIONAL FROM ITS ASSOCIATED MEASURE. -- 6. DIFFUSIONS WITH BROWNIAN HITTING PROBABILITIES. -- 7. SPEED MEASURES -- 8 TWO DIFFUSIONS WITH BROWNIAN HITS AND THE SAME SPEED MEASURE ARE THE SAME. -- 9. SPEED MEASURES ARE POSITIVE ON FINE NEIGHBORHOODS. -- 10. SPEED MEASURES GIVE RISE TO INCREASING ADDITIVE FUNCTIONALS. -- 11. WHEN IS f(+oo)= +oo? -- 12. PERFORMING THE TIME SUBSTITUTION. -- 13. GENERATORS. -- 14. DISCONTINUOUS ADDITIVE FUNCTIONALS. -- REFERENCES -- Note on Continuous Additive Functionals of the 1-Dimensional Brownian Path -- 1. Introduction -- 2. Existence of moments -- 3. Representation -- References -- Existence of Diffusions with Continuous Coefficients -- Introduction. -- S1. Notations and preliminaries. -- S2. Extension of A. -- S3. The equation {/\ - A)u=f and the semigroup on Co(U). -- S4. {Ttn D} and the corresponding: diffusion. -- S5. Construction of the diffusion with generator A. -- References. -- Propagation of Chaos for Certain Purely Discontinuous Markov Processes with Interactions -- S1. Introduction. -- S2. Preliminaries. -- S3. A formula concerning the minimal solution of (1.3). -- S4. The motion of n particles. -- S5. Propagation of chaos. -- References.

An Inequality for a Functional of Probability Distributions and Its Application to Kac's One-Dimensional Model of a Maxwellian Gas -- 1. Introduction -- 2. Basic Properties of e and a Proof of the Central Limit Theorem -- 3. e Decreases along Solutions of Boltzmann's Problem for Kac's Model of a Maxwellian Gas -- References -- On Markov Process Corresponding to Boltzmann's Equation of Maxwellian Gas -- Sl. Introduction. -- S2. Markov processes and stochastic differential equation -- S3. Solving the stochastic differential equation. Main results -- References -- On the Uniqueness of Markov Process Associated with the Boltzmann Equation of Maxwellian Molecules -- S1 . Introduction -- S2. Markov process associated with (1.2) -- S3. Point process {Zt} -- S4. Derivation of stochastic differential equation -- S5. Concluding remark -- References -- Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules -- Introduction -- Chapter I. Associated Markov Process -- S1. Definition of Markov Process Associated with (0.3) -- S2. Preliminaries from Poisson Point Process -- S3. Two Lemmas -- S4. Stochastic Differential Equation -- S5. The Transition Function and the Markov Process Associated with (0.3) -- Chapter II. Trend to Equilibrium -- S6. Some Lemmas Concerning p-metric on P2 -- S7. Non-expansive Property of the Associated Nonlinear Semigroup with Respect to the Metric p -- S8. Theorem of Ikenberry and Truesdell on Time Evolution of Moments -- S9. Proof of the Trend to Equilibrium -- Appendix -- References -- Stochastic Differential Equations with Reflecting Boundary Condition in Convex Regions -- S1. Introduction -- S2. A deterministic problem -- S3. A stochastic version of (2.5) -- S4. Stochastic differential equation with reflection -- References.

Some Probabilistic Problems in the Spatially Homogeneous Boltzmann Equation -- 1. Introduction -- 2. Propagation of chaos -- 3. Fluctuation -- References -- Limit Theorems for Certain Diffusion Processes with Interaction -- Introduction -- S1. A method of Braun and Hepp -- S2. Central limit theorem -- S3. Another expression of /\ -- S4. An infinite dimensional SDE -- S5. Large deviation -- S6. The case of Boltzmann's equation -- References -- Central Limit Theorem for a System of Markovian Particles with Mean Field Interactions -- S0. Introduction -- S1. Symmetric Statistics and Multiple Wiener Integrals -- S2. The Case of Pure Jump Type Markov Processes -- S3. The Case of McKean's Model of Boltzmann's Equation -- S4. The Case of Diffusion Processes -- References -- Propagation of Chaos for Diffusing Particles of Two Types with Singular Mean Field Interaction -- Introduction -- 1. Existence and Uniqueness of Solutions -- 2. A Limit Theorem -- 3. Propagation of Chaos -- References -- Stochastic Differential Equations for Mutually Reflecting Brownian Balls -- Introduction -- 1. Some known results on Skorohod's equation for an N-dimensional domain with reflecting boundary -- 2. D satisfies Condition (B') -- 3. D satisfies Condition (A) -- 4. Mutually reflecting Brownian balls -- 5. Skorohod's SDE for mutually reflecting diffusion balls -- References -- Limit Distribution for 1-Dimensional Diffusion in a Reflected Brownian Medium -- Introduction -- S1. Preliminaries and exit times from valleys -- S2. The limit distribution of X(e/\r /\w) -- S3. Proof of Theorem 1 -- S4. Proof of Theorem 2 -- REFERENCES -- Limit Distributions for One-Dimensional Diffusion Processes in Self-Similar Random Environments -- Introduction -- 1. Preliminaries and the Result of Brox.

2. Nonpositive Reflected Brownian Environment -- 3. Nonnegative Reflected Brownian Environment -- 4. Symmetric Stable Environment -- References -- Stochastic Differential Equation Corresponding to the Spatially Homogeneous Boltzmann Equation of Maxwellian and Non-Cutoff Type -- Introduction -- S1. L1-Lipschitz continuity of b(x x1 0 o) -- S2. Stochastic differential equation-I -- 2.1. Existence theorem -- 2.2. Uniqueness in the law sense -- S3. Stochastic differential equation-II -- References -- Limit Theorem for One-Dimensional Diffusion Process in Brownian Environment -- INTRODUCTION -- S1. OUTLINE OF BROX'S METHOD -- S2. THE LAW OF THE STANDARD VALLEY. -- S3. PROOF OF THE THEOREM -- REFERENCES -- On the Maximum of a Diffusion Process in a Drifted Brownian Environment -- 1. Introduction -- 2. Proof of the theorem -- References -- 86. Recurrence of a Diffusion Process in a Multidimensional Brownian Environment -- Introduction -- S1. Brownian motion with a d-dimensional time. -- S2. Recurrence of Xw. -- References -- Localization of a Diffusion Process in a One-Dimensional Brownian Environment -- 1. Introduction -- 2. Convergence Theorem for Diffusion Processes in One-Dimension -- 3. Valleys -- 4. Proof of Theorem 1.1 -- Bibliography -- Diffusion Processes in Random Environments -- 1 Introduction -- 2 A diffusion in a one-dimensional Brownian environment (with drift) -- 3 Localization by random centering in the case K = 0 -- 4 Limit theorems in the case k # 0 -- 5 A diffusion in a multidimensional Brownian environment -- References -- Environment-Wise Central Limit Theorem for a Diffusion in a Brownian Environment with Large Drift -- Introduction -- S1. Some preliminaries -- S2. Environment-wise central limit theorem -- References.

A Diffusion Process in a Brownian Environment with Drift -- Introduction. -- S1. Kotani's formula. -- S2. Kasahara's continuity theorem for Krein's correspondence. -- S3. Proof of Theorem 1 in the case 01. -- S7. Remark to the case x=0. -- References -- Limit Theorems for a Brownian Motion with Drift in a White Noise Environment -- 1. INTRODUCTION -- 2. MAIN RESULTS -- 3. KEY METHODS -- 4. OUTLINE OF PROOF -- 5. REMARKS -- REFERENCES -- Invariance Principle for a Brownian Motion with Large Drift in a White Noise Environment -- Introduction -- 1. Proof of Theorem 1 -- 2. The proof of Theorem 2 -- 3. Supplement to the proof of (i) of Theorem A -- References -- Some Theorems Concerning Extrema of Brownian Motion with d-Dimensional Time -- Introduction -- 1. A lemma -- 2. Proof of Theorem 1 -- 3. Proof of Theorem 2 -- 4. Proof of Theorem 3 and Theorem 4 -- 5. Remarks on a diffusion process in a rf-dimensional Brownian environment -- References -- Bibliography of Hiroshi Tanaka -- Permissions.