A Probabilistic Interpretation of Nonlinear Integral Equations

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.


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 starproduct 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.

Notations
For every α, β ∈ C 3 , the symbol α Á β means the inner product, and we define e x ≔ x=|x| for every x ∈ D 0 . We consider the following deterministic nonlinear integral equation: Here, u u t; x ð Þ is an unknown function: R þ Â D 0 ! C 3 , λ . 0, and u 0 : D 0 ! C 3 are the initial data such that u t; x (2) is actually given by p t; x; y; u ð Þ¼u t; y ð ÞÁ e x u t; x À y ð ÞÀe x u t; x À y ð ÞÁe x ð Þ f g : Suppose that the integral kernel n x; y ð Þ is bounded and measurable with respect to x Â y. On the other hand, we consider a Markov kernel K: D 0 ! D 0 Â D 0 . Namely, for every z ∈ D 0 , K z x; y ð Þ lies in the space P D 0 Â D 0 ð Þof all probability measures on a product space D 0 Â D 0 . When the kernel k is given by k x; y ð Þ ¼ i x j j À2 n x; y ð Þ, then we define K z as a Markov kernel satisfying that for any positive measurable function Moreover, we assume that for every measurable functions f , g . 0 on R þ , holds, where the measure ν is given by ν dz ð Þ ¼ z j j À3 dz. 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 K z is changed into a single integral with respect to x just after the execution of iterative integration of h x; y ð Þwith 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.

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 u t; x ð Þ can 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 u 0 ; f ð Þ(see Section 5 for the details of the definition). On the other hand, M U;F h i * ω ð Þ denotes the associated * -product functional with weight U; F ð Þ, which is indexed by the nodes x m ð Þ of a binary tree. Here, we suppose that is a nonnegative measurable function on D 0 (resp. R þ Â D 0 ), respectively, and also that F Á; x ð Þ∈ L 1 R þ ð Þ 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 |u 0 x ð Þ| ⩽ U x ð Þ for every x and | f t; x ð Þ| ⩽ F t; x ð Þ for every t, x and also that for some T . 0 (T . . 1, sufficiently large) holds. Then, there exists a u 0 ; f ð Þ-weighted tree-based star ★-product functional indexed by a set of node labels accordingly to the tree structure which a binary critical branching process Z K x t ð Þ determines. Furthermore, the function gives a unique solution to the integral equation (Eq. (2)). Here, E t, x denotes the expectation with respect to a probability measure P t, x as the time-reversed law of Z K x t ð Þ.

Branching model and its associated treelike structure
In this section we consider a continuous time binary critical branching process Z K x t ð Þ on D 0 [9], whose branching rate is given by a parameter λ x j j 2 , whose branching mechanism is binary with equiprobability, and whose descendant branching particle behavior is determined by the kernel K x (cf. [10]). Next, taking notice of the tree structure which the process Z K x t ð Þ determines, we denote the space of marked trees by Ω (see [11]). We also consider the time-reversed law of Z K x t ð Þ being a probability measure on Ω as P t, x ∈ P Ω ð Þ. Here, t denotes the birth time of common ancestor, and the particle x m dies when η m ¼ 0, while it generates two descendants x m1 , x m2 when η m ¼ 1. On the other hand, is a set of all labels, namely, finite sequences of symbols with length ℓ, which describe the whole tree structure given [12]. For ω ∈ Ω we denote by N ω ð Þ the totality of nodes being the branching points of tree; let N þ ω ð Þ be the set of all nodes m being a member of V \ N ω ð Þ, whose direct predecessor lies in N ω ð Þ and which satisfies the condition t m ω ð Þ. 0, and let N À ω ð Þ be the same set as described above but satisfying t m ω ð Þ⩽ 0. Finally, we put

Star-product functional
This section treats a tree-based star-product functional. First of all, we denote by the symbol Proj z Á ð Þ a projection of the objective element onto its orthogonal part of the z component in C 3 , and we define a ★-product of β, γ for z ∈ D 0 as 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 Θ m ω ð Þ for each ω ∈ Ω realized as follows.
whereas for the product order in the star-product ★, when we write m≺m 0 lexicographically with respect to the natural order ≺, the term Θ m labeled by m necessarily occupies the left-hand side, and the other Θ m 0 labeled by m 0 occupies the right-hand side by all means. And besides, as abuse of notation, we write especially when m ∈ V is 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 N ω ð Þ, we call it a treebased ★-product functional, and we express it symbolically as where m 1 ∈ N ω ð Þ and m 2 , m 3 ∈ N ω ð Þ, and by the symbol Q ★ (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 ∈ N ω ð Þ∩ jmj ¼ ℓ À 1g f when |m 1 | ¼ ℓ. At any rate it is of the extreme importance that once a branching pattern ω ∈ Ω ð Þis realized, its tree structure is uniquely determined, and there can be found the unique explicit representation of the corresponding star-product functional M Example 2. Let us consider a typical realization ω ∈ Ω. Suppose that we have N ω 2 ð Þ ¼ ϕ; 1; 2; 11; 12; 22 f g , N þ ω 2 ð Þ ¼ 21; 112; 221 f g , and N À ω 2 ð Þ ¼ 111; 121; 122; 222 f g . This case is nothing but an all-the-members participating type of game. For the case of particle located at x 111 and x 112 (with nodes of the level |m| ¼ ℓ ¼ 3) with its pivoting node x 11 , we have Similarly, for the pair of particles x 121 and x 122 , we have For the pair of particles x 221 and x 222 , we also have Next, when we take a look at the groups of particles with nodes of the level |m| ¼ ℓ ¼ 2. For instance, as to a pair of particles located at x 11 and x 12 with its pivoting node x 1 , we get an expression Therefore, it follows by a similar argument that the explicit representation of star-product functional for ω 2 is given by 6. The * -product functional and existence In this section we first begin with constructing a U; F ð Þ-weighted tree-based * -product functional M U;F h i * ω ð Þ, which is indexed by the nodes x m ð Þ of a binary tree.
is a nonnegative measurable function on D 0 (resp. R þ Â D 0 ), respectively, and also that F Á; x ð Þ∈ L 1 R þ ð Þfor 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 0 ⩽ t ⩽ T and x ∈ D 0 , the function V t; Proof of Lemma 3. By making use of the conditional expectation, we may decompose V t; x ð Þ as follows: We are next going to take into consideration an equivalence between the events t ϕ ⩽ 0 and T∉ 0; t ½ . Indeed, as to the first term in the third line of Eq. (16), since the condition t ϕ ⩽ 0 implies that T never lies in an interval 0; t ½ , and since m ¼ ϕ ∈ N À ω ð Þ leads 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 x 1 , x 2 with probability 1 2 under the condition η ϕ ¼ 1; since t ϕ . 0, when the branching occurs at t ϕ ¼ s, then, under the conditioning operation at t ϕ , the Markov property [13] guarantees that the lower tree structure below the first-generation branching node point x 1 is independent to that below the location x 2 with realized ω ∈ Ω; hence, a tree-based * -product functional branched after time s is also probabilistically independent of the other tree-based * -product functional branched after time s, and besides the distributions of x 1 and x 2 are totally controlled by the Markov kernel K x . 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 □ By a glance at the expression Eq. (15) obtained in Lemma 3, it is quite obvious that, for each x ∈ D 0 , the mapping 0; T ½ ∍t↦e λ x j j 2 t V t; x ð Þ∈ R þ is a nondecreasing function. Taking the above fact into consideration, we can deduce with ease that holds for ∀t ∈ 0; T ½ and x ∈ E c , where the measurable set E c denotes the totality of all the elements x in D 0 such that E T, x M U;F h i * Â Ã < ∞ 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 P t, 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: |w★ x ½ v| ⩽ |w| Á |v| for every w, v ∈ C 3 and every x ∈ D 0 : Next, we are going to derive the space of solutions to Eq. (2). If we define then u t; x ð Þ is well defined on the whole space D 0 under 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 M U;F h i * ω ð Þ, 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 D of solutions to Eq. (2) as follows: holds a:e: À x É By employing the Markov property [13] with respect to time t ϕ and by a similar technique as in the proof of Lemma 3, we may proceed in rewriting and calculating the expectation for ∀t . 0 and x ∈ E c : 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 i satisfies the integral equation Eq. (2), and this u t; x ð Þ is a solution lying in the space D. This completes the proof of the existence.

Uniqueness
First of all, note that we can choose a proper measurable subset and is convergent for a.e.-x ∈ F 0 ð Þ, and u t; x ð Þ satisfies the nonlinear integral equation (Eq. (2)) for a.e.-x ∈ F 0 . 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 u in 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 þ be the set of all the elements ω's corresponding to time t m ω ð Þ. 0 for the label m. Next, we consider a kind of the notion like n-section of the set of labels for n ∈ N 0 ≔ N ∪ 0 f g. We define several families of Ω in what follows, in order to facilitate the extraction of its martingale part from our star-product functional M u 0 ;f h i ★ ω ð Þ. For each realized tree ω,Ñ n ω ð Þ is the totality of the labels m ∈ ⋃ 0 ⩽ ℓ ⩽ n 1; 2 f g ℓ satisfying t m ω ð Þ. 0 and η m ω ð Þ ¼ 1. Namely, this familyÑ n ω ð Þ is a subset of labels restricted up to the nth generation and limited to the nodes related to branching at positive time. Moreover, letÑ n ω ð Þ be the set of labels lying in N \Ñ n ω ð Þ whose direct predecessor belongs toÑ n ω ð Þ. By convention, we definẽ We shall introduce a new familyÑ cut n ω ð Þ of cutoff labels, which is determined by the set of labels m ∈ V whose direct predecessor belongs toÑ n ω ð Þ and has length |m| ¼ n, and we call this familyÑ cut n ω ð Þ the cutoff part ofÑ n ω ð Þ, while N n nct ω ð Þ is the non-cutoff part ofÑ n ω ð Þ, which is defined by N n nct ω ð Þ≔Ñ n ω ð Þ \ Ñ cut n ω ð Þ: We are now in a position to introduce a new class M n, u 0 ;f ;u h i ★ ω ð Þ of ★-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 In such a way, we can construct a new ★-product functional M n, u 0 ;f ;u h i ★ ω ð Þ by the almost sure procedure, and we call it the n-section ★-product functional. Similarly, we can also define the corresponding n-section ★-product functional M n, U;F;V Simply enough, to get the * -product counterpart, we have only to replace those functions u 0 ,f and u by U, F and V in the definition of ★-product functional. As easily imagined, we can also derive an n-section version of M n * -control inequality: Lemma 5. (M n * -control inequality) The following inequality holds P t, x -a.s. because of the domination property: |u t; x ð Þ| ⩽ V t; x ð Þ for 0; T ½ ÂD 0 , |u 0 x ð Þ| ⩽ U x ð Þ for ∀x, |f t; x ð Þ| ⩽ F t; x ð Þ for ∀t, x, and a simple inequality |w★ x ½ v| ⩽ |w| Á |v| for ∀w, v ∈ C 3 and ∀x ∈ D 0 . Let us now introduce a filtration F n f g for n ∈ N 0 on Ω þ , according to the discussion in Example 2. As a matter of fact, we define holds P t, x -a.s. for every t ∈ 0; T ½ and every x ∈ F 0 . Proof. By its construction, we can conclude the equality of Eq. (30) from the strong Markov property [13] applied at times t m ð Þs for m ∈ V of length n on the set m ∈Ñ n ω ð Þ n o ∈ F n . □ Moreover, an application of Lemma 6 with the n-section M n * -control inequality (Eq. (28)) shows the P t, x -integrability of M n, u 0 ;f ;u h i ★ ω ð Þ for every t ∈ 0; T ½ and every x ∈ F 0 . 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- Lemma 7. The n-section M n, u 0 ;f ;u h i ★ ω ð Þ of ★-product functional with weight functions u 0 and f is an F n f g-martingale [8]. Proof. When we set ¼ E t, x M n ★ ω ð ÞjF n Â Ã , then ξ n turns out to be a F n f gmartingale, since E t, x ξ n jF nÀ1 ½ ¼E t, x E t, x M n ★ jF n