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 solution

AMS 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 [2–4]).

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

$$\begin{split} e^{\frac{i}{\hbar}|\mathbf{x}|^{2}}u(t,\mathbf{x}) &= u\_{0}(\mathbf{x}) + \frac{\lambda}{2} \int\_{0}^{t} ds \, e^{\mathbf{s}|\mathbf{x}|^{2}} \int p(s,\mathbf{x},\mathbf{y};\mathbf{u}) n(\mathbf{x},\mathbf{y}) \, d\mathbf{y} \\ &+ \frac{\lambda}{2} \int\_{0}^{t} e^{\mathbf{s}|\mathbf{x}|^{2}} f(s,\mathbf{x}) \, ds. \end{split} \tag{1}$$

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.

Suppose that the integral kernel n xð Þ ; y is bounded and measurable with respect

ð

Moreover, we assume that for every measurable functions f,g . 0 on Rþ,

ð

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 Kz is changed into a single integral with respect to x just after the execution of iterative integration of h xð Þ ; y with 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

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 tð Þ ; x 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

m2:m<sup>3</sup>

be a probabilistic representation in terms of tree-based star-product functional with weight ð Þ u0; f (see Section 5 for the details of the definition). On the other hand, <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> denotes the associated <sup>∗</sup>-product functional with weight ð Þ <sup>U</sup>; <sup>F</sup> , which is indexed by the nodes ð Þ xm of a binary tree. Here, we suppose that U ¼ U xð Þ (resp. F ¼ F tð Þ ; x ) is a nonnegative measurable function on D<sup>0</sup> (resp.

struction of the <sup>∗</sup>-product functional, the product in question is taken as ordinary multiplication <sup>∗</sup> instead of the star-product ★ in the definition of star-product

Theorem 1. Suppose that <sup>∣</sup>u0ð Þ <sup>x</sup> <sup>∣</sup><sup>⩽</sup> U xð Þ for every x and <sup>∣</sup> <sup>~</sup>f tð Þ ; <sup>x</sup> <sup>∣</sup>⩽F tð Þ ; <sup>x</sup> for every

ET,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> � � <sup>&</sup>lt; <sup>∞</sup>, <sup>a</sup>:e: � <sup>x</sup> (7)

<sup>⋆</sup> ð Þ¼ <sup>ω</sup> <sup>Y</sup>⋆½ � xm<sup>~</sup> <sup>Ξ</sup><sup>m</sup><sup>1</sup>

n xð Þ ; y , then we define Kz as a Markov kernel satisfying that for any

gð Þ jzj νð Þz

ð

h xð Þ ; z � x k xð Þ ; z dx: (4)

hð Þ jyj Kzð Þ dx; dy (5)

½ � u0; f ð Þ ω , (6)

ð Þ R<sup>þ</sup> for each x. Indeed, in con-

to x � y. On the other hand, we consider a Markov kernel K: D<sup>0</sup> ! D<sup>0</sup> � D0. Namely, for every z∈ D0, Kzð Þ x; y lies in the space Pð Þ D<sup>0</sup> � D<sup>0</sup> of all probability

measures on a product space D<sup>0</sup> � D0. When the kernel k is given by

positive measurable function h ¼ h xð Þ ; y on D<sup>0</sup> � D0,

A Probabilistic Interpretation of Nonlinear Integral Equations

DOI: http://dx.doi.org/10.5772/intechopen.81501

ð

ðð h xð Þ ; <sup>y</sup> Kzð Þ¼ <sup>x</sup>; <sup>y</sup>

gð Þ jxj Kzð Þ¼ x; y

holds, where the measure <sup>ν</sup> is given by <sup>ν</sup>ð Þ¼ dz j j <sup>z</sup> �<sup>3</sup> dz.

of symmetry in a wide sense may be posed on our kernel K.

k xð Þ¼ ; <sup>y</sup> i xj j�<sup>2</sup>

ð

3. Main results

functional.

31

systems and branching models. Let

Mh i <sup>u</sup>0;<sup>f</sup>

<sup>R</sup><sup>þ</sup> � <sup>D</sup>0), respectively, and also that <sup>F</sup>ð Þ �; <sup>x</sup> <sup>∈</sup>L<sup>1</sup>

t, x and also that for some T . 0 (T . . 1, sufficiently large)

hð Þ jzj νð Þz

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 <sup>∗</sup>-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.

## 2. Notations

Let <sup>D</sup><sup>0</sup> <sup>≔</sup> <sup>R</sup><sup>3</sup> f g<sup>0</sup> and <sup>R</sup><sup>þ</sup> <sup>≔</sup> ½ Þ <sup>0</sup>; <sup>∞</sup> . For every <sup>α</sup>, <sup>β</sup> <sup>∈</sup> <sup>C</sup><sup>3</sup> , the symbol α � β means the inner product, and we define ex ≔ x=∣x∣ for every x∈ D0. We consider the following deterministic nonlinear integral equation:

$$\begin{split} \boldsymbol{e}^{\boldsymbol{\beta}t\left|\mathbf{x}\right|^{2}}\boldsymbol{u}(t,\mathbf{x}) &= \boldsymbol{u}\_{0}(\mathbf{x}) + \frac{\lambda}{2} \int\_{0}^{t} \mathrm{d}\mathbf{s} \, e^{\boldsymbol{\beta}t\left|\mathbf{x}\right|^{2}} \int p(\boldsymbol{s},\mathbf{x},\boldsymbol{y};\boldsymbol{u}) n(\boldsymbol{x},\mathbf{y}) d\mathbf{y} \\ &+ \frac{\lambda}{2} \int\_{0}^{t} e^{\boldsymbol{\beta}t\left|\mathbf{x}\right|^{2}} \boldsymbol{f}(\boldsymbol{s},\mathbf{x}) \mathbf{s}, \quad \text{for} \quad \forall (t,\mathbf{x}) \in \mathbb{R}\_{+} \times D\_{0}. \end{split} \tag{2}$$

Here, <sup>u</sup> � u tð Þ ; <sup>x</sup> is an unknown function: <sup>R</sup><sup>þ</sup> � <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup> , λ . 0, and <sup>u</sup><sup>0</sup> : <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup> are the initial data such that u tð Þj ; <sup>x</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>u</sup>0ð Þ <sup>x</sup> . Moreover, f tð Þ ; <sup>x</sup> : <sup>R</sup><sup>þ</sup> � <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup> is a given function satisfying f tð Þ ; <sup>x</sup> <sup>=</sup>j j <sup>x</sup> <sup>2</sup> <sup>¼</sup>: <sup>~</sup><sup>f</sup> <sup>∈</sup>L<sup>1</sup> ð Þ R<sup>þ</sup> for each x∈ D0. The integrand p in Eq. (2) is actually given by

$$p(t, \mathbf{x}, \boldsymbol{y}; \boldsymbol{u}) = \boldsymbol{u}(t, \boldsymbol{y}) \cdot \boldsymbol{e}\_{\mathbf{x}} \{ \boldsymbol{u}(t, \boldsymbol{x} - \boldsymbol{y}) - \boldsymbol{e}\_{\mathbf{x}}(\boldsymbol{u}(t, \boldsymbol{x} - \boldsymbol{y}) \cdot \boldsymbol{e}\_{\mathbf{x}}) \}. \tag{3}$$

A Probabilistic Interpretation of Nonlinear Integral Equations DOI: http://dx.doi.org/10.5772/intechopen.81501

Suppose that the integral kernel n xð Þ ; y is bounded and measurable with respect to x � y. On the other hand, we consider a Markov kernel K: D<sup>0</sup> ! D<sup>0</sup> � D0. Namely, for every z∈ D0, Kzð Þ x; y lies in the space Pð Þ D<sup>0</sup> � D<sup>0</sup> of all probability measures on a product space D<sup>0</sup> � D0. When the kernel k is given by k xð Þ¼ ; <sup>y</sup> i xj j�<sup>2</sup> n xð Þ ; y , then we define Kz as a Markov kernel satisfying that for any positive measurable function h ¼ h xð Þ ; y on D<sup>0</sup> � D0,

$$\iint h(\mathbf{x}, \mathbf{y}) K\_x(\mathbf{x}, \mathbf{y}) = \int h(\mathbf{x}, z - \mathbf{x}) k(\mathbf{x}, z) d\mathbf{x}.\tag{4}$$

Moreover, we assume that for every measurable functions f,g . 0 on Rþ,

$$\int h(|x|)\nu(x)\left\{\mathbf{g}(|x|)K\_x(x,y) = \int \mathbf{g}(|x|)\nu(x)\right\} h(|y|)K\_x(\mathbf{dx}, \mathbf{d}y) \tag{5}$$

holds, where the measure <sup>ν</sup> is given by <sup>ν</sup>ð Þ¼ dz j j <sup>z</sup> �<sup>3</sup> 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 Kz is changed into a single integral with respect to x just after the execution of iterative integration of h xð Þ ; y with 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

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)

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 probabilis-

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 <sup>∗</sup>-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 unique-

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

inner product, and we define ex ≔ x=∣x∣ for every x∈ D0. We consider the following

ds e<sup>λ</sup>s xj j<sup>2</sup>

<sup>u</sup><sup>0</sup> : <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup> are the initial data such that u tð Þj ; <sup>x</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>u</sup>0ð Þ <sup>x</sup> . Moreover, f tð Þ ; <sup>x</sup> :

ð

p tð Þ¼ ; x; y; u u tð Þ� ; y exf g u tð Þ� ; x � y exð Þ u tð Þ� ; x � y ex : (3)

p sð Þ ; x; y; u n xð Þ ; y dy

<sup>2</sup> <sup>¼</sup>: <sup>~</sup><sup>f</sup> <sup>∈</sup>L<sup>1</sup>

f sð Þ ; x s, for ∀ð Þ t; x ∈ R<sup>þ</sup> � D0:

, the symbol α � β means the

, λ . 0, and

ð Þ R<sup>þ</sup> for each

(2)

ness for the constructed solutions, in terms of the martingale theory [8].

shall be a trigger to further development on the study in this direction.

λ 2 ðt 0

Let <sup>D</sup><sup>0</sup> <sup>≔</sup> <sup>R</sup><sup>3</sup> f g<sup>0</sup> and <sup>R</sup><sup>þ</sup> <sup>≔</sup> ½ Þ <sup>0</sup>; <sup>∞</sup> . For every <sup>α</sup>, <sup>β</sup> <sup>∈</sup> <sup>C</sup><sup>3</sup>

u tð Þ¼ ; x u0ð Þþ x

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

<sup>R</sup><sup>þ</sup> � <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup> is a given function satisfying f tð Þ ; <sup>x</sup> <sup>=</sup>j j <sup>x</sup>

x∈ D0. The integrand p in Eq. (2) is actually given by

Here, <sup>u</sup> � u tð Þ ; <sup>x</sup> is an unknown function: <sup>R</sup><sup>þ</sup> � <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup>

deterministic nonlinear integral equation:

e <sup>λ</sup>t xj j<sup>2</sup>

appearing in mathematical physics.

Recent Advances in Integral Equations

tic representations of the solutions.

2. Notations

30

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 tð Þ ; x 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

$$M^{\langle u\_0 f \rangle}\_{\star}(o) = \prod \star\_{[\times \tilde{m}]} \Xi^{m\_1}\_{m\_2, m\_3}[u\_0, f](o), \tag{6}$$

be a probabilistic representation in terms of tree-based star-product functional with weight ð Þ u0; f (see Section 5 for the details of the definition). On the other hand, <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> denotes the associated <sup>∗</sup>-product functional with weight ð Þ <sup>U</sup>; <sup>F</sup> , which is indexed by the nodes ð Þ xm of a binary tree. Here, we suppose that U ¼ U xð Þ (resp. F ¼ F tð Þ ; x ) is a nonnegative measurable function on D<sup>0</sup> (resp. <sup>R</sup><sup>þ</sup> � <sup>D</sup>0), respectively, and also that <sup>F</sup>ð Þ �; <sup>x</sup> <sup>∈</sup>L<sup>1</sup> ð Þ R<sup>þ</sup> for each x. Indeed, in construction of the <sup>∗</sup>-product functional, the product in question is taken as ordinary multiplication <sup>∗</sup> instead of the star-product ★ in the definition of star-product functional.

Theorem 1. Suppose that <sup>∣</sup>u0ð Þ <sup>x</sup> <sup>∣</sup><sup>⩽</sup> U xð Þ for every x and <sup>∣</sup> <sup>~</sup>f tð Þ ; <sup>x</sup> <sup>∣</sup>⩽F tð Þ ; <sup>x</sup> for every t, x and also that for some T . 0 (T . . 1, sufficiently large)

$$E\_{T, \mathbf{x}} \left[ \mathbf{M}\_\*^{\langle U, F \rangle} (\boldsymbol{\alpha}) \right] < \infty, \quad \mathbf{a.e.} - \boldsymbol{\infty} \tag{7}$$

holds. Then, there exists a uð Þ <sup>0</sup>; <sup>f</sup> -weighted tree-based star ★-product functional Mh i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> , indexed by a set of node labels accordingly to the tree structure which a binary critical branching process ZKx ð Þ<sup>t</sup> determines. Furthermore, the function

$$u(t, \mathbf{x}) = E\_{t, \mathbf{x}} \left[ \mathbf{M}\_{\star}^{\langle u\_0 f \rangle}(a) \right] \tag{8}$$

Ξm<sup>1</sup> m2:m<sup>3</sup>

DOI: http://dx.doi.org/10.5772/intechopen.81501

structure in question.

ð Þ� <sup>ω</sup> <sup>Ξ</sup>m<sup>1</sup>

A Probabilistic Interpretation of Nonlinear Integral Equations

necessarily occupies the left-hand side, and the other Θm<sup>0</sup>

Ξ∅

corresponding star-product functional Mh i <sup>u</sup>0;<sup>f</sup>

Ξ<sup>11</sup>

Ξ<sup>22</sup>

pivoting node x1, we get an expression

<sup>¼</sup> <sup>Ξ</sup><sup>11</sup>

<sup>11</sup>, <sup>12</sup>ð Þ¼ <sup>ω</sup><sup>2</sup> <sup>Θ</sup><sup>11</sup>ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>1</sup> <sup>Θ</sup><sup>12</sup>ð Þ <sup>ω</sup><sup>2</sup>

<sup>¼</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>111</sup> ★½ � <sup>x</sup><sup>11</sup>

<sup>111</sup>;112ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>1</sup> <sup>Ξ</sup><sup>12</sup>

� �

Ξ1

33

Ξ<sup>12</sup>

N ð Þ¼ ω<sup>2</sup> f g ϕ; 1; 2; 11; 12; 22 , Nþð Þ¼ ω<sup>2</sup> f g 21; 112; 221 , and

the level ∣m∣ ¼ ℓ ¼ 3) with its pivoting node x11, we have

Similarly, for the pair of particles x<sup>121</sup> and x122, we have

For the pair of particles x<sup>221</sup> and x222, we also have

<sup>111</sup>;112ð Þ¼ <sup>ω</sup><sup>2</sup> <sup>Θ</sup><sup>111</sup>ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>11</sup> <sup>Θ</sup><sup>112</sup>ð Þ <sup>ω</sup><sup>2</sup>

<sup>221</sup>;222ð Þ¼ <sup>ω</sup><sup>2</sup> <sup>Θ</sup><sup>221</sup>ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>22</sup> <sup>Θ</sup><sup>222</sup>ð Þ <sup>ω</sup><sup>2</sup>

<sup>¼</sup> <sup>u</sup>0ð Þ <sup>x</sup>111ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>11</sup>

<sup>121</sup>;122ð Þ¼ <sup>ω</sup><sup>2</sup> <sup>Θ</sup><sup>121</sup>ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>12</sup> <sup>Θ</sup><sup>122</sup>ð Þ <sup>ω</sup><sup>2</sup>

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<sup>11</sup> and x<sup>12</sup> with its

<sup>121</sup>;122ð Þ ω<sup>2</sup>

<sup>~</sup>fðt112; <sup>x</sup>112<sup>Þ</sup>

<sup>¼</sup> <sup>u</sup>0ð Þ <sup>x</sup>121ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>12</sup> <sup>u</sup>0ð Þ <sup>x</sup>122ð Þ <sup>ω</sup><sup>2</sup> :

<sup>¼</sup> <sup>~</sup>f tð Þ <sup>221</sup>ð Þ <sup>ω</sup><sup>2</sup> ; <sup>x</sup>221ð Þ <sup>ω</sup><sup>2</sup> ★½ � <sup>x</sup><sup>22</sup> <sup>u</sup>0ð Þ <sup>x</sup>222ð Þ <sup>ω</sup><sup>2</sup> :

m2,m<sup>3</sup>

m,∅ð Þ� <sup>ω</sup> <sup>Ξ</sup><sup>∅</sup>

based ★-product functional, and we express it symbolically as

<sup>M</sup>★h i <sup>u</sup>0;<sup>f</sup> ð Þ¼ <sup>ω</sup> <sup>Π</sup>★½ � xm<sup>~</sup> <sup>Ξ</sup><sup>m</sup><sup>1</sup>

whereas for the product order in the star-product ★, when we write <sup>m</sup>≺m<sup>0</sup> lexicographically with respect to the natural order ≺, the term Θ<sup>m</sup> labeled by m

especially when m ∈V is a label of single terminal point in the restricted tree

executing the star-product ★ inductively at each node in <sup>N</sup> ð Þ <sup>ω</sup> , we call it a tree-

where <sup>m</sup><sup>1</sup> <sup>∈</sup> <sup>N</sup> ð Þ <sup>ω</sup> and <sup>m</sup>2, m<sup>3</sup> <sup>∈</sup> <sup>N</sup>ð Þ <sup>ω</sup> , and by the symbol <sup>Q</sup> ★ (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 ð Þ ω ∩fjm~ j ¼ ℓ � 1g when ∣m1∣ ¼ ℓ. At any rate it is of the extreme importance that once a branching pattern ωð Þ ∈ Ω is realized, its tree structure is uniquely determined, and there can be found the unique explicit representation of the

N�ð Þ¼ ω<sup>2</sup> f g 111; 121; 122; 222 . This case is nothing but an all-the-members participating type of game. For the case of particle located at x<sup>111</sup> and x<sup>112</sup> (with nodes of

Under these circumstances, we consider a random quantity which is obtained by

m2�m<sup>3</sup>

★ ð Þ <sup>ω</sup> . Example 2. Let us consider a typical realization ω∈ Ω. Suppose that we have

right-hand side by all means. And besides, as abuse of notation, we write

½ � <sup>u</sup>0; <sup>f</sup> ð Þ <sup>ω</sup> <sup>≔</sup> <sup>Θ</sup>m<sup>2</sup> ð Þ <sup>ω</sup> ★ xm<sup>1</sup> ½ �Θm<sup>3</sup> ð Þ <sup>ω</sup> , (12)

m,∅½ � <sup>u</sup>0; <sup>f</sup> ð Þ <sup>ω</sup> <sup>≔</sup> <sup>Θ</sup>mð Þ <sup>ω</sup> , (13)

<sup>~</sup>f tð Þ <sup>112</sup>ð Þ <sup>ω</sup><sup>2</sup> ; <sup>x</sup>112ð Þ <sup>ω</sup><sup>2</sup> :

★½ � <sup>x</sup><sup>1</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>121</sup> ★½ � <sup>x</sup><sup>12</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>122</sup> � �:

labeled by m<sup>0</sup> occupies the

½ � u0; f ð Þ ω , (14)

gives a unique solution to the integral equation (Eq. (2)). Here, Et,x denotes the expectation with respect to a probability measure Pt,x as the time-reversed law of ZKx ð Þ<sup>t</sup> .
