Stochastic Processes

The following sections are included:

• Preface

• Contents

It is a pleasant thing for me to write this little note to accompany the selected papers of H. Tanaka collected in this volume.

When I was young and fresh out of school, I visited Kyôto for a year (1957/58) at the invitation of K. Itô. My great-grandfather (whom I never knew) was an amateur of Japanese art, generally, and of Okyô in particular. Part of his collection came down to my family, so even when I was quite little, I was fascinated by Japan and things Japanese. The double chance to see Japan and to work with Itô was an irresistible piece of luck and I was pretty well prepared for that…

It is both a pleasure and an honor to have been asked to write a few words about the works of Professor H. Tanaka.

At least two aspects of Professor Tanaka's works have been of much interest to me and, more generally to many probabilists in France, for quite some time…

The aim of this note is to express my hearty admiration and respect to Professor Hiroshi Tanaka for his works in probability theory which have influenced much and are playing important roles in its modern developments. As we can see by his great contributions collected in this volume, his works have been widely ranged in the field of stochastic analysis with two high peaks around main themes; stochastic models associated with nonlinear equations, particularly Boltzmann equations, and diffusion models in random environment. For me it is difficult and also, I think I am not well qualified, to make comments on his works in these main themes. I would rather give some historical account, combined with my own experience, of his contributions and influences in the area of stochastic analysis and the theory of stochastic differential equations (SDE's) in general. I should say, however, that motivations of Professor Tanaka must have been always to challenge new and unsolved problems; developing or organizing general theories may not have been his major concern…

In this short note I would like to give some comments on my works on probability theory, first by mentioning briefly the initial stage of my mathematical development and then by concentrating on those works which are related mainly to Boltzmann's equation and diffusion processes in random environments…

The following sections are included:

• INTRODUCTION

• BROWNIAN MOTION

• THE ASSOCIATED MEASURE OF AN ADDITIVE FUNCTIONAL

• UNIQUENESS

• CONSTRUCTION OF AN ADDITIVE FUNCTIONAL FROM ITS ASSOCIATED MEASURE

• DIFFUSIONS WITH BROWNIAN HITTING PROBABILITIES

• SPEED MEASURES

• TWO DIFFUSIONS WITH BROWNIAN HITS AND THE SAME SPEED MEASURE ARE THE SAME

• SPEED MEASURES ARE POSITIVE ON FINE NEIGHBORHOODS

• SPEED MEASURES GIVE RISE TO INCREASING ADDITIVE FUNCTIONALS

• WHEN IS f(+∞) = +∞?

• PERFORMING THE TIME SUBSTITUTION

• GENERATORS

• DISCONTINUOUS ADDITIVE FUNCTIONALS

• REFERENCES

The following sections are included:

• Introduction

• Existence of moments

• Representation

• References

The following sections are included:

• Introduction

• Notations and preliminaries

• Extension of A

• The equation and the semigroup on

• and the corresponding diffusion

• Construction of the diffusion with generator

• References

The following sections are included:

• Introduction

• Preliminaries

• A formula concerning the minimal solution of (1.3)

• The motion of n particles

• Propagation of chaos

• References

The following sections are included:

• Introduction

• Basic Properties of e and a Proof of the Central Limit Theorem

• e Decreases along Solutions of Boltzmann's Problem for Kac's Model of a Maxwellian Gas

• References

The following sections are included:

• Introduction

• Markov processes and stochastic differential equation

• Solving the stochastic differential equation. Main results

• References

The following sections are included:

• Introduction

• Markov process associated with (1.2)

• Point process {Z_t}

• Derivation of stochastic differential equation

• Concluding remark

• References

The following sections are included:

• Introduction

• Associated Markov Process
  • Definition of Markov Process Associated with (0.3)
  • Preliminaries from Poisson Point Process
  • Two Lemmas
  • Stochastic Differential Equation
  • The Transition Function and the Markov Process Associated with (0.3)

• Trend to Equilibrium
  • Some Lemmas Concerning ρ-metric on 𝒫₂
  • Non-expansive Property of the Associated Nonlinear Semigroup with Respect to the Metric ρ
  • Theorem of Ikenberry and Truesdell on Time Evolution of Moments
  • Proof of the Trend to Equilibrium

• Appendix

• References

The following sections are included:

• Introduction

• A deterministic problem

• A stochastic version of (2.5)

• Stochastic differential equation with reflection

• References

The following sections are included:

• Introduction

• Propagation of chaos

• Fluctuation

• References

The following sections are included:

• Introduction

• A method of Braun and Hepp

• Central limit theorem

• Another expression of Λ

• An infinite dimensional SDE

• Large deviation

• The case of Boltzmann's equation

• References

The following sections are included:

• Introduction

• Symmetric Statistics and Multiple Wiener Integrals

• The Case of Pure Jump Type Markov Processes

• The Case of McKean's Model of Boltzmann's Equation

• The Case of Diffusion Processes

• References

The following sections are included:

• Introduction

• Existence and Uniqueness of Solutions

• A Limit Theorem

• Propagation of Chaos

• References

The following sections are included:

• Introduction

• Some known results on Skorohod's equation for an N-dimensional domain with reflecting boundary

• D satisfies Condition (B')

• D satisfies Condition (A)

• Mutually reflecting Brownian balls

• Skorohod's SDE for mutually reflecting diffusion balls

• References

The following sections are included:

• Introduction

• Preliminaries and exit times from valleys

• The limit distribution of X(e^λr, λW)

• Proof of Theorem 1

• Proof of Theorem 2

• REFERENCES

The following sections are included:

• Introduction

• Preliminaries and the Result of Brox

• Nonpositive Reflected Brownian Environment

• Nonnegative Reflected Brownian Environment

• Symmetric Stable Environment

• References

The following sections are included:

• Introduction

• L¹-Lipschitz continuity of b(x, x₁, θ, σ)

• Stochastic differential equation—I
  • Existence theorem
  • Uniqueness in the law sense

• Stochastic differential equation—II

• References

The following sections are included:

• INTRODUCTION

• OUTLINE OF BROX'S METHOD

• THE LAW OF THE STANDARD VALLEY

• PROOF OF THE THEOREM

• REFERENCES

The following sections are included:

• Introduction

• Proof of the theorem
  • Proof of (i)
  • Proof of (ii)
  • Proof of (iii)

• References

The following sections are included:

• Introduction

• Brownian motion with a d-dimensional time

• Recurrence of X_w

• References

The following sections are included:

• Introduction

• Convergence Theorem for Diffusion Processes in One-Dimension

• Valleys

• Proof of Theorem 1.1

• Bibliography

The following sections are included:

• Introduction

• A diffusion in a one-dimensional Brownian environment (with drift)

• Localization by random centering in the case κ = 0

• Limit theorems in the case κ ≠ 0

• A diffusion in a multidimensional Brownian environment

• References

The following sections are included:

• Introduction

• Some preliminaries

• Environment-wise central limit theorem

• References

The following sections are included:

• Introduction

• Kotani's formula

• Kasahara's continuity theorem for Krein's correspondence

• Proof of Theorem 1 in the case 0 < κ < 1

• Proof of Theorem 2

• Proof of Theorem 1 in the case κ = 1

• Proof of Theorem 1 in the case κ > 1

• Remark to the case κ = 0

• References

This paper discusses limit theorems for a diffusion analogue of Kesten–Kozlov–Spitzer's random walk in a random environment. The results obtained are similar to theirs but can be presented in a more explicit form by the use of Krein's spectral theory for one-dimensional generalized second-order differential operators of the form d/dMd/dx.

This paper discusses an invariance principle for a Brownian motion with drift coefficient κ/4 in a white noise environment under the assumption that κ is large. Our method clarifies the relation between the environment-wise invariance principle discussed in [7] and the present result (the invariance principle in random environment).

The following sections are included:

• Introduction

• A lemma

• Proof of Theorem 1

• Proof of Theorem 2

• Proof of Theorem 3 and Theorem 4

• Remarks on a diffusion process in a d-dimensional Brownian environment

• References

The following sections are included:

• Bibliography of Hiroshi Tanaka

• Permissions