Functional integration and partial differential equations.

*(English)*Zbl 0568.60057
Annals of Mathematics Studies, No. 109. Princeton, New Jersey: Princeton University Press. IX, 545 p. hbk: $ 60.00; pbk: $ 19.95 (1985).

The author considers some problems of the theory of partial differential equations. The probabilistic approach is used. It is based on the representation of the solutions of certain equations by functional integrals. In chapter I, certain statements concerning stochastic differential equations, Markov processes and Laplace type asymptotics of functional integrals are presented. They are necessary, for the analysis of processes connected with differential operators.

In chapter II, the formulas representing solutions of second order parabolic and elliptic equations are studied. They are expressed in form of the mean values of the functionals of corresponding processes. In chapter III, the peculiarities of the statement of boundary value problems for degenerate elliptic and parabolic operators are examined and the existence, uniqueness and smoothness of generalized solutions investigated. Chapter IV deals with elliptic operators depending on a small parameter. Using the statements of chapter III and results of large deviation type and of averaging principle type, the author studies the limit behavior of solutions of the Dirichlet problem.

The last three chapters are devoted to the analysis of quasilinear equations. In chapter V, the existence of a continuous solution for Cauchy’s problem and for some mixed problems is proved. Chapters VI and VII consider some generalizations of the Kolmogorov-Petrovskij-Piskunov equation (wave front propagation). Introducing a small parameter and using asymptotic bounds of the Laplace type for functional integrals, the author examines some new effects such as the appearance of ”new sources” in space, non homogeneous media, and wave front propagation due to nonlinear boundary conditions. In chapter VII, the asymptotic velocity of wave front propagation in periodic and random media is investigated.

This book undoubtedly shows that the probabilistic interpretation can help one to carry out exact proofs and to discover new facts.

In chapter II, the formulas representing solutions of second order parabolic and elliptic equations are studied. They are expressed in form of the mean values of the functionals of corresponding processes. In chapter III, the peculiarities of the statement of boundary value problems for degenerate elliptic and parabolic operators are examined and the existence, uniqueness and smoothness of generalized solutions investigated. Chapter IV deals with elliptic operators depending on a small parameter. Using the statements of chapter III and results of large deviation type and of averaging principle type, the author studies the limit behavior of solutions of the Dirichlet problem.

The last three chapters are devoted to the analysis of quasilinear equations. In chapter V, the existence of a continuous solution for Cauchy’s problem and for some mixed problems is proved. Chapters VI and VII consider some generalizations of the Kolmogorov-Petrovskij-Piskunov equation (wave front propagation). Introducing a small parameter and using asymptotic bounds of the Laplace type for functional integrals, the author examines some new effects such as the appearance of ”new sources” in space, non homogeneous media, and wave front propagation due to nonlinear boundary conditions. In chapter VII, the asymptotic velocity of wave front propagation in periodic and random media is investigated.

This book undoubtedly shows that the probabilistic interpretation can help one to carry out exact proofs and to discover new facts.

Reviewer: R.Mikulevičius

##### MSC:

60Hxx | Stochastic analysis |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

60F10 | Large deviations |

60H05 | Stochastic integrals |