## Stochastic Processes: Selected Papers of Hiroshi TanakaHiroshi Tanaka is noted for his discovery of the ?Tanaka formula?, which is a generalization of the It formula in stochastic analysis. This important book is a selection of his brilliant works on stochastic processes and related topics. It contains Tanaka's papers on (i) Brownian motion and stochastic differential equations (additive functionals of Brownian paths and stochastic differential equations with reflecting boundaries), (ii) the probabilistic treatment of nonlinear equations (Boltzmann equation, propagation of chaos and McKean-Vlasov limit), and (iii) stochastic processes in random environments (especially limit theorems on the stochastic processes in one-dimensional random environments and their refinements). The book also includes essays by Henry McKean, Marc Yor, Shinzo Watanabe and Hiroshi Tanaka on Tanaka's works. |

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### Contents

An Appreciation | 1 |

From Local Times to Random Environments | 2 |

Contributions and Influences of Professor Tanaka in Stochastic Analysis | 5 |

Some Comments on My Mathematical Works in Retrospect | 10 |

Additive Functionals of the Brownian Path | 19 |

Note on Continuous Additive Functionals of the 1Dimensional Brownian Path | 47 |

Existence of Diffusions with Continuous Coefficients | 54 |

Propagation of Chaos for Certain Purely Discontinuous Markov Processes with Interactions ... | 69 |

Stochastic Differential Equations for Mutually Reflecting Brownian Balls | 238 |

Limit Distribution for 1Dimensional Diffusion in a Reflected Brownian Medium | 254 |

Limit Distributions for OneDimensional Diffusion Processes in SelfSimilar Random Environments ... | 270 |

Stochastic Differential Equation Corresponding to the Spatially Homogeneous Boltzmann Equation of Maxwellian and NonCutoff Type ... | 292 |

Limit Theorem for OneDimensional Diffusion Process in Brownian Environment | 311 |

On the Maximum of a Diffusion Process in a Drifted Brownian Environment | 328 |

86 Recurrence of a Diffusion Process in a Multidimensional Brownian Environment | 336 |

Localization of a Diffusion Process in a OneDimensional Brownian Environment | 341 |

An Inequality for a Functional of Probability Distributions and Its Application to Kacs OneDimensional Model of a Maxwellian Gas ... | 83 |

On Markov Process Corresponding to Boltzmanns Equation of Maxwellian Gas | 89 |

On the Uniqueness of Markov Process Associated with the Boltzmann Equation of Maxwellian Molecules ... | 101 |

Probabilistic Treatment of the Boltzmann Equation of Maxwellian Molecules | 118 |

Stochastic Differential Equations with Reflecting Boundary Condition in Convex Regions | 157 |

Some Probabilistic Problems in the Spatially Homogeneous Boltzmann Equation | 172 |

Limit Theorems for Certain Diffusion Processes with Interaction | 182 |

Central Limit Theorem for a System of Markovian Particles with Mean Field Interactions | 202 |

Propagation of Chaos for Diffusing Particles of Two Types with Singular Mean Field Interaction ... | 223 |

Diffusion Processes in Random Environments | 353 |

EnvironmentWise Central Limit Theorem for a Diffusion in a Brownian Environment with Large Drift ... | 361 |

A Diffusion Process in a Brownian Environment with Drift | 373 |

Limit Theorems for a Brownian Motion with Drift in a White Noise Environment | 396 |

Invariance Principle for a Brownian Motion with Large Drift in a White Noise Environment ... | 406 |

Some Theorems Concerning Extrema of Brownian Motion with dDimensional Time | 415 |

Bibliography of Hiroshi Tanaka | 425 |

429 | |

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### Common terms and phrases

1-dimensional additive functional assume Boltzmann equation bounded Brownian environment Brownian motion Brownian path Brox central limit theorem completes the proof condition const constant continuous functions converges in law defined denote diffusion process distribution f Equation of Maxwellian equivalent in law finite fixed formula given H. P. McKean hand side hence Hiroshi TANAKA implies independent inequality initial distribution Japan Kawazu limit distribution Markov process Markov process associated Maxwellian Maxwellian gas Maxwellian Molecules notation obtained particles Poisson point process Poisson random measure probability distribution probability measure probability space problem Proc process X(t proof of Theorem Propagation of chaos prove random environment random variable random walk reflected Brownian resp respect result sample paths satisfying Skorohod ſº solution stochastic differential equation stochastic integral theory uniformly