## Abstract

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
- star-product
- probabilistic solutionAMS classification: Primary 45G10; Secondary 60 J80
- 60 J85
- 60 J57

## 1. Introduction

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 question solves the original integral equations (see also [2, 3, 4]).

More precisely, in this chapter we consider the deterministic nonlinear integral equation of the type:

One of the reasons why we are interested in this kind of integral equations consists in its importance in applicatory fields, especially in mathematical physics. For instance, in quantum physics or applied mathematics, a variety of differential equations have been dealt with by many researchers (e.g., [5, 6]), and in most cases, their integral forms have been discussed more than their differential forms on a practical basis. There can be found plenty of integral equations similar to Eq. (1) appearing in mathematical physics.

The purpose of this article is to provide with a quite general method of giving a probabilistic interpretation to deterministic equations. Any deterministic representation of the solutions to Eq. (1) has not been known yet in analysis. The main contents of the study consist in derivation of the probabilistic representation of the solutions to Eq. (1). Our mathematical model is a kind of generalization of the integral equations that were treated in [7], and our kernel appearing in Eq. (1) is given in a more abstract setting. We are aiming at establishment of new probabilistic representations of the solutions.

This paper is organized as follows: In Section 2 we introduce notations which are used in what follows. In Section 3 principal results are stated, where we refer the probabilistic representation of the solutions to a class of deterministic nonlinear integral equations in question. Section 4 deals with branching model and its treelike structure. Section 5 treats construction of star-product functional based upon those tree structures of branching model described in the previous section. The proof of the main theorem will be stated in Sections 6 and 7. Section 6 provides with the proof of existence of the probabilistic solutions to the integral equations. We also consider

We think that it would not be enough to derive simply explicit representations of probabilistic solutions to the equations, but it is extremely important to make use of the formulae practically in the problem of computations. We hope that our result shall be a trigger to further development on the study in this direction.

## 2. Notations

Let

Here,

Suppose that the integral kernel

Moreover, we assume that for every measurable functions

holds, where the measure

The equality (Eq. (4)) is not only a simple integral transform formula. In fact, in the analytical point of view, it merely says that the double integral with respect to

## 3. Main results

In this section we shall introduce our main results, which assert the existence and uniqueness of solutions to the nonlinear integral equation. That is to say, we derive a probabilistic representation of the solutions to Eq. (2) by employing the star-product functional. As a matter of fact, the solution

be a probabilistic representation in terms of tree-based star-product functional with weight

Theorem 1. * sufficiently large*)

* holds. Then*,

there exists a

*★-*weighted tree-based star

product functional

indexed by a set of node labels accordingly to the tree structure which a binary critical branching process

determines. Furthermore, the function

* gives a unique solution to the integral equation* (Eq. (2)).

*,*Here

denotes the expectation with respect to a probability measure

as the time-reversed law of

## 4. Branching model and its associated treelike structure

In this section we consider a continuous time binary critical branching process

by

is a set of all labels, namely, finite sequences of symbols with length

## 5. Star-product functional

This section treats a tree-based star-product functional. First of all, we denote by the symbol

Notice that this product ★ is noncommutative. This property will be the key point in defining the star-product functional below, especially as far as the uniqueness of functional is concerned. We shall define

whereas for the product order in the star-product ★, when we write

especially when

Under these circumstances, we consider a random quantity which is obtained by executing the star-product ★ inductively at each node in

where
* unique* explicit representation of the corresponding star-product functional

Example 2. Let us consider a typical realization

Similarly, for the pair of particles

For the pair of particles

Next, when we take a look at the groups of particles with nodes of the level

Therefore, it follows by a similar argument that the explicit representation of star-product functional for

## 6. The
∗
-product functional and existence

In this section we first begin with constructing a

In what follows we shall give an outline of the existence in Theorem 1. We need the following lemma, which is essentially important for the proof.

Lemma 3.

* Proof of Lemma* 3. By making use of the conditional expectation, we may decompose

We are next going to take into consideration an equivalence between the events

the tree-based

As to the third term, we need to note the following matters. A particle generates two offsprings or descendants

Note that as for the second term, it goes almost similarly as the computation of the above-mentioned third one. Finally, summing up we obtain

On this account, if we multiply both sides of Eq. (18) by

By a glance at the expression Eq. (15) obtained in Lemma 3, it is quite obvious that, for each

holds for

Lemma 4. (

This inequality enables us to govern the behavior of the star-product functional with a very complicated structure by that of the

Next, we are going to derive the space of solutions to Eq. (2). If we define

On this account, from Eq. (15) in Lemma 3, by finiteness of the expectation of tree-based

Hence, taking Eq. (22) into consideration, we define the space

By employing the Markov property [13] with respect to time

Furthermore, we may apply the integral equality Eq. (4) in the assumption on the Markov kernel for Eq. (24) to obtain

because in the above last equality we need to rewrite its double integral relative to the space parameters into a single integral. Finally, we attain that

## 7. Uniqueness

First of all, note that we can choose a proper measurable subset

and

is convergent for a.e.-

satisfying

We are now in a position to introduce a new class

Lemma 5. (

because of the domination property:

Let us now introduce a filtration

for each

Lemma 6.

* Proof.* By its construction, we can conclude the equality of Eq. (30) from the strong Markov property [13] applied at times

Moreover, an application of Lemma 6 with the

Lemma 7. The

* Proof.* When we set

by virtue of the inclusion property of the

holds a.s. By employing the representation formula (Eq. (8)), an conditioning argument leads to Eq. (31), because the establishment is verified by the Markov property applied at

Finally, the uniqueness yields from the following assertion.

Proposition 8. * is a solution to the nonlinear integral equation* (Eq. (2)),

then we have

* and for* a.e.

* Proof.* Our proof is technically due to a martingale method. We need the following lemma.

Lemma 9. * section of* ★-

product functional, and let

*(Eq. (2)).*be a solution of the nonlinear integral equation

Then, we have the following identity: for each

Proof of Lemma 9. Recall that

Next, for the case

We resort to the mathematical induction with respect to

where we made use of the martingale property in the first equality and employed the hypothesis of induction in the last identity. This concludes the assertion.

To go back to the proof of Proposition 8. We define an

Hence, for every

where the symbol

Furthermore, we continue computing

Since

it follows by the bounded convergence theorem of Lebesgue that

Consequently, from Eq. (39) and Eq. (41), we readily obtain

holds for every

Concurrently, this completes the proof of the uniqueness.

## Acknowledgments

This work is supported in part by the Japan MEXT Grant-in-Aids SR(C) 17 K05358 and also by ISM Coop. Res. Program: 2011-CRP-5010.

## References

- 1.
Dôku I. Star-product functional and unbiased estimator of solutions to nonlinear integral equations. Far East Journal of Mathematical Sciences. 2014; 89 :69-128 - 2.
Dôku I. On a limit theorem for environment-dependent models. Institute of Statistical Mathematics Research Reports. 2016; 352 :103-111 - 3.
Dôku I. A recursive inequality of empirical measures associated with EDM. Journal of Saitama University. Faculty of Education (Mathematics for Natural Science). 2016; 65 (2):253-259 - 4.
Dôku I. A support problem for superprocesses in terms of random measure. RIMS Kôkyûroku (Kyoto University). 2017; 2030 :108-115 - 5.
Dôku I. Exponential moments of solutions for nonlinear equations with catalytic noise and large deviation. Acta Applicandae Mathematicae. 2000; 63 :101-117 - 6.
Dôku I. Removability of exceptional sets on the boundary for solutions to some nonlinear equations. Scientiae Mathematicae Japonicae. 2001; 54 :161-169 - 7.
Le Jan Y, Sznitman AS. Stochastic cascades and 3-dimensional Navier-Stokes equations. Probability Theory and Related Fields. 1997; 109 :343-366 - 8.
Kallenberg O. Foundations of Modern Probability. 2nd ed. New York: Springer; 2002. 638 p - 9.
Harris TE. The Theory of Branching Processes. Berlin: Springer-Verlag; 1963. 248 p - 10.
Aldous D. Tree-based models for random distribution of mass. Journal of Statistical Physics. 1993; 73 :625-641 - 11.
Le Gall J-F. Random trees and applications. Probability Surveys. 2005; 2 :245-311 - 12.
Drmota M. Random Trees. Wien: Springer-Verlag; 2009. 458 p - 13.
Dynkin EB. Markov Processes. Vol. 1. Berlin: Springer-Verlag; 1965. 380 p