Nonlinear stochastic operator equations.

*(English)*Zbl 0609.60072
Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers). XV, 287 p. $ 64.50; £54.00 (1986).

The author expresses his basic principle as follows: ”It is not always wise to follow overzealously the footsteps of the masters”.

The book contains a really new and original method for solving nonlinear (and linear as a particular case) stochastic (and deterministic as a particular case) equations of general type (differential, algebraic, etc.). This is the so called decomposition method involving the \(A_ n\)- polynomials-technique which in brief could be explained as follows: Assume a function f is to be found which obeys the equation \(Lf=Nf\), where L is an invertible linear operator and N is a nonlinear one. We decompose f as \(f=f_ 0+f_ 1+..\). and write Nf as \(Nf=A_ 0(f_ 0)+A_ 1(f_ 0,f_ 1)+...+A_ n(f_ 0,f_ 1,...,f_ n)+...\), where \(A_ n\) for \(n\geq 0\) is a polynomial of degree n. Then \(f_ n\) is obtained according to the following recursive scheme: \(Lf_ 0=0\) and \(f_ n=L^{-1}A_{n-1}(f_ 0,...,f_{n-1})\) for \(n\geq 1\). The calculation of the \(A_ n\)-polynomials is made explicit in the most important cases of nonlinear operators N. When one deals with a differential equation L is a differential operator and its inversion means to calculate the corresponding Green’s function. This problem is broadly discussed.

The author proposes a principally new treatment of deterministic and stochastic problems in which both cases are considered simultaneously. It should be pointed out that such an approach contradicts to the tradition. However it might be that in this joint treatment ”stochastic differentiation” is loosing its typical nature. The author illustrates his new theory with many examples and comparisons with traditional methods. Some general and very interesting remarks concerned with the exploration of the present computer generation are given. The convergence question is also considered and answered at least for the most important cases of appearing nonlinearities.

The book is easy to read meaning that it is within the grasp of any graduate student in mathematics. However, because of its principally new approach, it addresses researchers mainly. It is apparent that the author has used successfully his method for solving important practical problems. The book really deserves to be noticed by anyone whose main topics are equations (stochastic or deterministic) from both - practical and purely theoretic point of view. Finally the reader can hardly avoid the impression that the author is a master too.

The book contains a really new and original method for solving nonlinear (and linear as a particular case) stochastic (and deterministic as a particular case) equations of general type (differential, algebraic, etc.). This is the so called decomposition method involving the \(A_ n\)- polynomials-technique which in brief could be explained as follows: Assume a function f is to be found which obeys the equation \(Lf=Nf\), where L is an invertible linear operator and N is a nonlinear one. We decompose f as \(f=f_ 0+f_ 1+..\). and write Nf as \(Nf=A_ 0(f_ 0)+A_ 1(f_ 0,f_ 1)+...+A_ n(f_ 0,f_ 1,...,f_ n)+...\), where \(A_ n\) for \(n\geq 0\) is a polynomial of degree n. Then \(f_ n\) is obtained according to the following recursive scheme: \(Lf_ 0=0\) and \(f_ n=L^{-1}A_{n-1}(f_ 0,...,f_{n-1})\) for \(n\geq 1\). The calculation of the \(A_ n\)-polynomials is made explicit in the most important cases of nonlinear operators N. When one deals with a differential equation L is a differential operator and its inversion means to calculate the corresponding Green’s function. This problem is broadly discussed.

The author proposes a principally new treatment of deterministic and stochastic problems in which both cases are considered simultaneously. It should be pointed out that such an approach contradicts to the tradition. However it might be that in this joint treatment ”stochastic differentiation” is loosing its typical nature. The author illustrates his new theory with many examples and comparisons with traditional methods. Some general and very interesting remarks concerned with the exploration of the present computer generation are given. The convergence question is also considered and answered at least for the most important cases of appearing nonlinearities.

The book is easy to read meaning that it is within the grasp of any graduate student in mathematics. However, because of its principally new approach, it addresses researchers mainly. It is apparent that the author has used successfully his method for solving important practical problems. The book really deserves to be noticed by anyone whose main topics are equations (stochastic or deterministic) from both - practical and purely theoretic point of view. Finally the reader can hardly avoid the impression that the author is a master too.

Reviewer: O.Enchev

##### MSC:

60H25 | Random operators and equations (aspects of stochastic analysis) |

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

34A34 | Nonlinear ordinary differential equations and systems |

34F05 | Ordinary differential equations and systems with randomness |

35R60 | PDEs with randomness, stochastic partial differential equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |