Open access peer-reviewed chapter

A Probabilistic Interpretation of Nonlinear Integral Equations

By Isamu Dôku

Submitted: July 1st 2018Reviewed: September 13th 2018Published: November 19th 2018

DOI: 10.5772/intechopen.81501

Downloaded: 160


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.


  • 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 -product functional, which is a sister functional of the star-product functional. This newly presented functionals play an essential role in governing the behaviors of star-product functionals via control inequality. Section 7 deals with the proof of uniqueness for the constructed solutions, in terms of the martingale theory [8].

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 D0R3\0and R+0. For every α,βC3, the symbol αβmeans the inner product, and we define exx/xfor every xD0. We consider the following deterministic nonlinear integral equation:


Here, uutxis an unknown function: R+×D0C3, λ>0, and u0:D0C3are the initial data such that utxt=0=u0x. Moreover, ftx: R+×D0C3is a given function satisfying ftx/x2=:f˜L1R+for each xD0. The integrand pin Eq. (2) is actually given by


Suppose that the integral kernel nxyis bounded and measurable with respect to x×y. On the other hand, we consider a Markov kernel K: D0D0×D0. Namely, for every zD0, Kzxylies in the space PD0×D0of all probability measures on a product space D0×D0. When the kernel kis given by kxy=ix2nxy, then we define Kzas a Markov kernel satisfying that for any positive measurable function h=hxyon D0×D0,


Moreover, we assume that for every measurable functions f,g>0on R+,


holds, where the measure νis given by νdz=z3dz.

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 Kzis changed into a single integral with respect to xjust after the execution of iterative integration of hxywith respect to the second parameter y. However, our point here consists in establishing a great bridge between a deterministic world and a stochastic world. The validity of the assumed equality (Eq. (5)) means that a sort of symmetry in a wide sense may be posed on our kernel K.

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 utxcan be expressed as the expectation of a star-product functional, which is nothing but a probabilistic solution constructed by making use of the below-mentioned branching particle systems and branching models. Let


be a probabilistic representation in terms of tree-based star-product functional with weight u0f(see Section 5 for the details of the definition). On the other hand, MUFωdenotes the associated -product functional with weight UF, which is indexed by the nodes xmof a binary tree. Here, we suppose that U=Ux(resp. F=Ftx) is a nonnegative measurable function on D0(resp. R+×D0), respectively, and also that FxL1R+for each x. Indeed, in construction of the -product functional, the product in question is taken as ordinary multiplication instead of the star-product ★ in the definition of star-product functional.

Theorem 1. Suppose that u0xUxfor every xand  f˜txFtxfor every t,xand also that for some T>0(T>>1, sufficiently large)


holds. Then, there exists a u0f-weighted tree-based star ★-product functional Mu0fω, indexed by a set of node labels accordingly to the tree structure which a binary critical branching process ZKxtdetermines. Furthermore, the function


gives a unique solution to the integral equation (Eq. (2)). Here, Et,xdenotes the expectation with respect to a probability measure Pt,xas the time-reversed law of ZKxt.

4. Branching model and its associated treelike structure

In this section we consider a continuous time binary critical branching process ZKxton D0[9], whose branching rate is given by a parameter λx2, whose branching mechanism is binary with equiprobability, and whose descendant branching particle behavior is determined by the kernel Kx(cf. [10]). Next, taking notice of the tree structure which the process ZKxtdetermines, we denote the space of marked trees


by Ω(see [11]). We also consider the time-reversed law of ZKxtbeing a probability measure on Ωas Pt,xPΩ. Here, tdenotes the birth time of common ancestor, and the particle xmdies when ηm=0, while it generates two descendants xm1,xm2when ηm=1. On the other hand,


is a set of all labels, namely, finite sequences of symbols with length , which describe the whole tree structure given [12]. For ωΩwe denote by Nωthe totality of nodes being the branching points of tree; let N+ωbe the set of all nodes mbeing a member of V \ Nω, whose direct predecessor lies in Nωand which satisfies the condition tmω>0, and let Nωbe the same set as described above but satisfying tmω0. Finally, we put


5. Star-product functional

This section treats a tree-based star-product functional. First of all, we denote by the symbol Projza projection of the objective element onto its orthogonal part of the zcomponent in C3, and we define a ★-product of β,γfor zD0as


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 Θmωfor each ωΩrealized as follows. When mN+ω, then Θmω=f˜tmωxmω, while Θmω=u0xmωif mNω. Then, we define


whereas for the product order in the star-product ★, when we write mmlexicographically with respect to the natural order , the term Θmlabeled by mnecessarily occupies the left-hand side, and the other Θmlabeled by moccupies the right-hand side by all means. And besides, as abuse of notation, we write


especially when mVis a label of single terminal point in the restricted tree structure in question.

Under these circumstances, we consider a random quantity which is obtained by executing the star-product ★ inductively at each node in Nω, we call it a tree-based ★-product functional, and we express it symbolically as


where m1Nωand m2,m3Nω, and by the symbol (as a product relative to the star-product), we mean that the star-products ★’s should be succeedingly executed in a lexicographical manner with respect to xm˜such that m˜Nωm˜=1when m1=. At any rate it is of the extreme importance that once a branching pattern ωΩis realized, its tree structure is uniquely determined, and there can be found the unique explicit representation of the corresponding star-product functional Mu0fω.

Example 2. Let us consider a typical realization ωΩ. Suppose that we have Nω2=ϕ,1,2,11,12,22, N+ω2=21,112,221, and Nω2=111,121,122,222. This case is nothing but an all-the-members participating type of game. For the case of particle located at x111and x112(with nodes of the level m==3) with its pivoting node x11, we have


Similarly, for the pair of particles x121and x122, we have


For the pair of particles x221and x222, we also have


Next, when we take a look at the groups of particles with nodes of the level m==2. For instance, as to a pair of particles located at x11and x12with its pivoting node x1, we get an expression


Therefore, it follows by a similar argument that the explicit representation of star-product functional for ω2is given by


6. The -product functional and existence

In this section we first begin with constructing a UF-weighted tree-based -product functional MUFω, which is indexed by the nodes xmof a binary tree. Recall that U=Ux(resp. F=Ftx) is a nonnegative measurable function on D0(resp. R+×D0), respectively, and also that FxL1R+for each x. Moreover, in construction of the functional, the product is taken as ordinary multiplication instead of the star-product ★.

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. For 0tTand xD0, the function Vtx=Et,xMUFωsatisfies


Proof of Lemma 3. By making use of the conditional expectation, we may decompose Vtxas follows:


We are next going to take into consideration an equivalence between the events tϕ0and T0t. Indeed, as to the first term in the third line of Eq. (16), since the condition tϕ0implies that Tnever lies in an interval 0t, and since m=ϕNωleads to a nonrandom expression


the tree-based -product functional is allowed to possess a simple representation:


As to the third term, we need to note the following matters. A particle generates two offsprings or descendants x1,x2with probability 12under the condition ηϕ=1; since tϕ>0, when the branching occurs at tϕ=s, then, under the conditioning operation at tϕ, the Markov property [13] guarantees that the lower tree structure below the first-generation branching node point x1is independent to that below the location x2with realized ωΩ; hence, a tree-based -product functional branched after time sis also probabilistically independent of the other tree-based -product functional branched after time s, and besides the distributions of x1and x2are totally controlled by the Markov kernel Kx. Therefore, an easy computation provides with an impressive expression:


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 expλtx2, then the required expression Eq. (15) in Lemma 3 can be derived, which completes the proof.

By a glance at the expression Eq. (15) obtained in Lemma 3, it is quite obvious that, for each xD0, the mapping 0Tteλx2tVtxR¯+is a nondecreasing function. Taking the above fact into consideration, we can deduce with ease that


holds for t0Tand xEc, where the measurable set Ecdenotes the totality of all the elements xin D0such that ET,xMUF<holds for a.e.x, namely, it is the same condition Eq. (7) appearing in the assertion of Theorem 1. Another important aspect for the proof consists in establishment of the M-control inequality, which is a basic property of the star-product ★. That is to say, we have.

Lemma 4. (M-control inequality) The following inequality


holds Pt,x-a.s.

This inequality enables us to govern the behavior of the star-product functional with a very complicated structure by that of the -product functional with a rather simplified structure. In fact, the M-control inequality yields immediately from a simple fact:

wxvwvfor everyw,vC3and everyxD0.

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

then utxis well defined on the whole space D0under the assumptions of the main theorem (Theorem 1). Moreover, it follows from the M-control inequality (Eq. (20)) that

On this account, from Eq. (15) in Lemma 3, by finiteness of the expectation of tree-based -product functional MUFω, by the M-control inequality, and from Eq. (21), it is easy to see that


Hence, taking Eq. (22) into consideration, we define the space Dof solutions to Eq. (2) as follows:

D{φ:R+×D0C3;φis continuous intand measurable such that0dseλx2sφsyszKxdydz<holdsa.e.x}E23

By employing the Markov property [13] with respect to time tϕand by a similar technique as in the proof of Lemma 3, we may proceed in rewriting and calculating the expectation for t>0and xEc:


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 utx=Et,xMu0fωsatisfies the integral equation Eq. (2), and this utxis a solution lying in the space D. This completes the proof of the existence.

7. Uniqueness

First of all, note that we can choose a proper measurable subset F0D0with mF0c=mD0 \ F0=0(meaning that its complement F0cis a null set with respect to Lebesgue measure mx), such that




is convergent for a.e.-xF0, and utxsatisfies the nonlinear integral equation (Eq. (2)) for a.e.xF0. Suggested by the argument in [7], we adopt here a martingale method in order to prove the uniqueness of the solutions to Eq. (2). The leading philosophy for the proof of uniqueness consists in extraction of the martingale part from the realized tree structure and in representation of the solution uin terms of martingale language. In so doing, we need to construct a martingale term from the given functional and to settle down the required σ-algebra with respect to which its constructed term may become a martingale. Let Ω+be the set of all the elements ω's corresponding to time tmω>0for the label m. Next, we consider a kind of the notion like n-section of the set of labels for nN0N0. We define several families of Ωin what follows, in order to facilitate the extraction of its martingale part from our star-product functional Mu0fω. For each realized tree ω, N˜nωis the totality of the labels


satisfying tmω>0and ηmω=1. Namely, this family N˜nωis a subset of labels restricted up to the nth generation and limited to the nodes related to branching at positive time. Moreover, let N˜nωbe the set of labels lying in N\N˜nωwhose direct predecessor belongs to N˜nω. By convention, we define N˜nω=if N˜nω=. We shall introduce a new family N˜ncutωof cutoff labels, which is determined by the set of labels mVwhose direct predecessor belongs to N˜nωand has length m=n, and we call this family N˜ncutωthe cutoff part of N˜nω, while Nnnctωis the non-cutoff part of N˜nω, which is defined by

NnnctωN˜nω \ N˜ncutω.E27

We are now in a position to introduce a new class Mn,u0fuωof ★-product functional, which should be called the n-section of the star-product functional. In fact, by taking the above argument in Example 2 into account, we can define its n-section as follows. In fact, if the label mis a member of the cutoff family N˜ncutω, the input data of the functional attached to mis given by utpmωxmωinstead of the usual initial data u0xmωor f˜tmωxmω, where pmindicates the direct ancestor mof mhaving length n. On the other hand, if mlies in the non-cutoff family Nnnctω, then the input data of the functional attached to mis completely the same as before with no change, that is, we use u0xmif tm0and use f˜tmxmif tm>0. In such a way, we can construct a new ★-product functional Mn,u0fuωby the almost sure procedure, and we call it the n-section ★-product functional. Similarly, we can also define the corresponding n-section ★-product functional Mn,UFVω. Simply enough, to get the -product counterpart, we have only to replace those functions u0,f˜and uby U,Fand Vin the definition of ★-product functional. As easily imagined, we can also derive an n-section version of Mn-control inequality:

Lemma 5. (Mn-control inequality) The following inequality


holds Pt,x-a.s.

because of the domination property: utxVtxfor 0T×D0, u0xUxfor x, f˜txFtxfor t,x, and a simple inequality wxvwvfor w,vC3and xD0.

Let us now introduce a filtration Fnfor nN0on Ω+, according to the discussion in Example 2. As a matter of fact, we define


for each nN0. Notice that N˜nωitself determines the other two families N˜ncutωand Nnnctω. Then, it is readily observed that both functionals Mn,u0fuωand Mn,UFVωare Fn-adapted.

Lemma 6. For each nN0, the equality


holds Pt,x-a.s. for every t0Tand every xF0.

Proof. By its construction, we can conclude the equality of Eq. (30) from the strong Markov property [13] applied at times tms for mVof length non the set mN˜nωFn.

Moreover, an application of Lemma 6 with the n-section Mn-control inequality (Eq. (28)) shows the Pt,x-integrability of Mn,u0fuωfor every t0Tand every xF0. Actually, it proves to be true that a martingale part, in question, extracted by the star-product functional relative to those n-section families, is given by the n-section ★-product functional Mn,u0fuω.

Lemma 7. The n-section Mn,u0fuωof ★-product functional with weight functions u0and fis an Fn-martingale [8].

Proof. When we set =Et,xMnωFn, then ξnturns out to be a Fn-martingale, since


by virtue of the inclusion property of the σ-algebras. Consequently, it suffices to show that


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 tmand on the event mN˜nbeing Fn-measurable.

Finally, the uniqueness yields from the following assertion.

Proposition 8. When utxis a solution to the nonlinear integral equation (Eq. (2)), then we have


holds for every t0Tand for a.e.x.

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

Lemma 9. Let Mn,u0fuωbe the n-section of ★-product functional, and let utxbe a solution of the nonlinear integral equation (Eq. (2)). Then, we have the following identity: for each nN0


holds for every t0tTand every xF0.

Proof of Lemma 9. Recall that Mn,u0fuωis a martingale relative to Fn. For n=0, it follows from the identity (Eq. (31)) and by the martingale property that


Next, for the case n=1, by the same reason, we can get


We resort to the mathematical induction with respect to nN0. If we assume the identity (Eq. (33)) for the case of n, then the case of n+1reads at once


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 Fn-measurable event Anas the set of ωΩ+such that N˜nωcontains some label mof length n. From the definition, it holds immediately that


Hence, for every xF0and 0tTand nN0, we may apply Lemma 9 for the expression below with the identity (Eq. (31)) to obtain


where the symbol Et,xXωAdenotes the integral of Xωover a measurable event Awith respect to the probability measure Pt,xdω, namely,


Furthermore, we continue computing


Since nAn=by the binary critical tree structure [12], and since we have an natural estimate


it follows by the bounded convergence theorem of Lebesgue that


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


holds for every tx0T×F0. Thus, we attain that utx=Et,xMu0fω, a.e.xF0. This finishes the proof of Proposition 8.

Concurrently, this completes the proof of the uniqueness.


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.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Isamu Dôku (November 19th 2018). A Probabilistic Interpretation of Nonlinear Integral Equations, Recent Advances in Integral Equations, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.81501. Available from:

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