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Chapter 3

Isamu Dôku

star-product, probabilistic solution

1. Introduction

equation of the type:

29

e <sup>λ</sup>t xj j<sup>2</sup>

Abstract

A Probabilistic Interpretation of

We study a probabilistic interpretation of solutions to a class of nonlinear integral equations. By considering a branching model and defining a star-product, we construct a tree-based star-product functional as a probabilistic solution of the integral equation. Although the original integral equation has nothing to do with a stochastic world, some probabilistic technique enables us not only to relate the deterministic world with the stochastic one but also to interpret the equation as a random quantity. By studying mathematical structure of the constructed functional, we prove that the function given by expectation of the functional with respect to the law of a branching process satisfies the original integral equation.

Keywords: nonlinear integral equation, branching model, tree structure,

AMS classification: Primary 45G10; Secondary 60 J80, 60 J85, 60 J57

question solves the original integral equations (see also [2–4]).

λ 2 ðt 0

u tð Þ¼ ; x u0ð Þþ x

þ λ 2 ðt 0 e <sup>λ</sup>s xj j<sup>2</sup>

This chapter treats a topic on probabilistic representations of solutions to a certain class of deterministic nonlinear integral equations. Indeed, this is a short review article to introduce the star-product functional and a probabilistic construction of solutions to nonlinear integral equations treated in [1]. The principal parts for the existence and uniqueness of solutions are taken from [1] with slight modification. Since the nonlinear integral equations which we handle are deterministic, they have nothing to do with random world. Hence, we assume that an integral formula may hold, which plays an essential role in connecting a deterministic world with a random one. Once this relationship has been established, we begin with constructing a branching model and we are able to construct a star-product functional based upon the model. At the end we prove that the function provided by the expectation of the functional with respect to the law of a branching process in

More precisely, in this chapter we consider the deterministic nonlinear integral

ds e<sup>λ</sup>s xj j<sup>2</sup>

f sð Þ ; x d s:

ð

p sð Þ ; x; y; u n xð Þ ; y dy

(1)

Nonlinear Integral Equations
