Functional Integration and Partial Differential Equations. (AM-109), Volume 109

✍ Scribed by Mark Iosifovich Freidlin

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 556
• Series: Annals of Mathematics Studies; 109
• Category: Library

✦ Synopsis

This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

✦ Table of Contents

CONTENTS
PREFACE
INTRODUCTION
I. STOCHASTIC DIFFERENTIAL EQUATIONS AND RELATED TOPICS
§1.1 Preliminaries
§1.2 The Wiener measure
§1.3 Stochastic differential equations
§1.4 Markov processes and semi-groups of operators
§1.5 Measures in the space of continuous functions corresponding to diffusion processes
§1.6 Diffusion processes with reflection
§1.7 Limit theorems. Action functional
II. REPRESENTATION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS AS FUNCTIONAL INTEGRALS AND THE STATEMENT OF BOUNDARY VALUE PROBLEMS
§2.1 The Feynman-Kac formula for the solution of Cauchy’s problem
§2.2 Probabilistic representation of the solution of Dirichlet’s problem
§2.3 On the correct statement of Dirichlet’s problem
§2.4 Dirichlet’s problem in unbounded domain
§2.5 Probabilistic representation of solutions of boundary problems with reflection conditions
III.BOUNDARY VALUE PROBLEMS FOR EQUATIONS WITH NON-NEGATIVE CHARACTERISTIC FORM
§3.1 On peculiarities in the statement of boundary value problems for degenerate equations
§3.2 On factorization of non-negative definite matrices
§3.3 The exit of process from domain
§3.4 Classification of boundary points
§3.5 First boundary value problem. Existence and uniqueness theorems for generalized solutions
§3.6 The Hölder continuity of generalized solutions. Existence conditions for derivatives
§3.7 Second boundary value problem
IV. SMALL PARAMETER IN SECOND-ORDER EL L IPTIC DIFFERENTIAL EQUATIONS
§4.1 Classical case. Problem statement
§4.2 The generalized Levinson conditions
§4.3 Averaging principle
§4.4 Leaving a domain at the expense of large deviations
§4.5 Large deviations. Continuation
§4.6 Small parameter in problems with mixed boundary conditions
V. QUASI-LINEAR PARABOLIC EQUATIONS WITH NONNEGATIVE CHARACTERISTIC FORM
§5.1 Generalized solution of Cauchy’s problem. Local solvability
§5.2 Solvability in the large at the expense of absorption. The existence conditions for derivatives
§5.3 On equations with subordinate non-linear terms
§5.4 On a class of systems of differential equations
§5.5 Parabolic equations and branching diffusion processes
VI. QUASI-LINEAR PARABOLIC EQUATIONS WITH SMALL PARAMETER. WAVE FRONT PROPAGATION
§6.1 Statement of problem
§6.2 Generalized KPP equation
§6.3 Some remarks and refinements
§6.4 Other forms of non-linear terms
§6.5 Other kinds of random movements
§6.6 Wave front propagation due to non-linear boundary effects
§6.7 On wave front propagation in a diffusion-reaction system
VII. WAVE FRONT PROPAGATION IN PERIODIC AND RANDOM MEDIA
§7.1 Introduction
§7.2 Calculation of the action functional
§7.3 Asymptotic velocity of wave front propagation in periodic medium
§7.4 Kolmogorov-Petrovskii-Piskunov equation with random multiplication coefficient
§7.5 The definition and basic properties of the function μ(z)
§7.6 Asymptotic wave front propagation velocity in random media
§7.7 The function μ(z) and the one-dimensional Schrödinger equation with random potential
LIST OF NOTATIONS
REFERENCES