5. Star-product functional

This section treats a tree-based star-product functional. First of all, we denote by the symbol Proj<sup>z</sup> ð Þ� a projection of the objective element onto its orthogonal part of the z component in C<sup>3</sup> , and we define a ★-product of <sup>β</sup>, <sup>γ</sup> for <sup>z</sup><sup>∈</sup> <sup>D</sup><sup>0</sup> as

$$
\beta \star\_{[\!\!\!\!z]} \chi = -i(\beta \cdot e\_{\!\!\!z}) \text{Proj}^x(\chi). \tag{11}
$$

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 <sup>Θ</sup><sup>m</sup>ð Þ <sup>ω</sup> for each <sup>ω</sup><sup>∈</sup> <sup>Ω</sup> realized as follows. When <sup>m</sup> <sup>∈</sup> <sup>N</sup>þð Þ <sup>ω</sup> , then <sup>Θ</sup><sup>m</sup>ð Þ¼ <sup>ω</sup> <sup>~</sup>f tð Þ <sup>m</sup>ð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> , while <sup>Θ</sup><sup>m</sup>ð Þ¼ <sup>ω</sup> <sup>u</sup>0ð Þ xmð Þ <sup>ω</sup> if <sup>m</sup> <sup>∈</sup> <sup>N</sup>�ð Þ <sup>ω</sup> . Then, we define

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

holds. Then, there exists a uð Þ <sup>0</sup>; <sup>f</sup> -weighted tree-based star ★-product functional

★ 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 tð Þ¼ ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

4. Branching model and its associated treelike structure

<sup>Z</sup>Kx ð Þ<sup>t</sup> on <sup>D</sup><sup>0</sup> [9], whose branching rate is given by a parameter <sup>λ</sup>j j <sup>x</sup>

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 <sup>Z</sup>Kx ð Þ<sup>t</sup> determines, we denote the

by <sup>Ω</sup> (see [11]). We also consider the time-reversed law of <sup>Z</sup>Kx ð Þ<sup>t</sup> being a probability measure on Ω as Pt,x ∈Pð Þ Ω . Here, t denotes the birth time of common ancestor, and the particle xm dies when η<sup>m</sup> ¼ 0, while it generates two descendants

> V ¼ ⋃ ℓ≥0

f g <sup>1</sup>; <sup>2</sup> <sup>ℓ</sup>

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 ð Þ ω the totality of nodes being the branching points of tree; let Nþð Þ ω be the set of all nodes m being a member of V \ N ð Þ ω , whose direct predecessor lies in N ð Þ ω and which satisfies the condition tmð Þ ω . 0, and let N�ð Þ ω be the same set as described above

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

<sup>β</sup>★½ � <sup>z</sup> <sup>γ</sup> ¼ �ið Þ <sup>β</sup> � ez Proj<sup>z</sup>

follows. When <sup>m</sup> <sup>∈</sup> <sup>N</sup>þð Þ <sup>ω</sup> , then <sup>Θ</sup><sup>m</sup>ð Þ¼ <sup>ω</sup> <sup>~</sup>f tð Þ <sup>m</sup>ð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> , while

<sup>Θ</sup><sup>m</sup>ð Þ¼ <sup>ω</sup> <sup>u</sup>0ð Þ xmð Þ <sup>ω</sup> if <sup>m</sup> <sup>∈</sup> <sup>N</sup>�ð Þ <sup>ω</sup> . Then, we define

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 <sup>Θ</sup><sup>m</sup>ð Þ <sup>ω</sup> for each <sup>ω</sup><sup>∈</sup> <sup>Ω</sup> realized as

ð Þ� a projection of the objective element onto its orthogonal part of

, and we define a ★-product of <sup>β</sup>, <sup>γ</sup> for <sup>z</sup><sup>∈</sup> <sup>D</sup><sup>0</sup> as

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> .

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

★ ð Þ <sup>ω</sup> h i

ω ¼ ð Þ t;ð Þ tm ;ð Þ xm ;ð Þ η<sup>m</sup> ; m ∈V (9)

Nð Þ¼ ω Nþð Þ ω ∪ N�ð Þ ω : (10)

ð Þγ : (11)

(8)

2 , whose

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

Recent Advances in Integral Equations

space of marked trees

xm1, xm<sup>2</sup> when η<sup>m</sup> ¼ 1. On the other hand,

but satisfying tmð Þ ω ⩽ 0. Finally, we put

5. Star-product functional

the symbol Proj<sup>z</sup>

32

the z component in C<sup>3</sup>

$$\Xi\_{m\_{2},m\_{3}}^{m\_{1}}(\boldsymbol{\alpha}) \equiv \Xi\_{m\_{2},m\_{3}}^{m\_{1}}[\boldsymbol{u}\_{0},\boldsymbol{f}](\boldsymbol{\alpha}) \coloneqq \Theta^{m\_{2}}(\boldsymbol{\alpha}) \star\_{\left[\boldsymbol{x}\_{m\_{1}}\right]} \Theta^{m\_{3}}(\boldsymbol{\alpha}),\tag{12}$$

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 necessarily occupies the left-hand side, and the other Θm<sup>0</sup> labeled by m<sup>0</sup> occupies the right-hand side by all means. And besides, as abuse of notation, we write

$$\Xi\_{m,\mathcal{Q}}^{\mathcal{Q}}(\boldsymbol{\alpha}) \equiv \Xi\_{m,\mathcal{Q}}^{\mathcal{Q}}[\boldsymbol{\mu}\_0, \boldsymbol{f}](\boldsymbol{\alpha}) \coloneqq \Theta^m(\boldsymbol{\alpha}),\tag{13}$$

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 <sup>N</sup> ð Þ <sup>ω</sup> , we call it a treebased ★-product functional, and we express it symbolically as

$$M\_{\star}^{\langle u\_0 f \rangle}(o) = \Pi \star\_{[\times\_{\bar{m}}]} \Xi\_{m\_2 \cdot m\_3}^{m\_1}[u\_0, f](o), \tag{14}$$

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 corresponding star-product functional Mh i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> .

Example 2. Let us consider a typical realization ω∈ Ω. Suppose that we have 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 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 the level ∣m∣ ¼ ℓ ¼ 3) with its pivoting node x11, we have

$$\begin{aligned} \Xi^{11}\_{111,112}(o\_2) &= \Theta^{111}(o\_2) \star\_{[\mathbf{x}\_{11}]} \Theta^{112}(o\_2) \\ &= u\_0(\mathfrak{x}\_{111}(o\_2)) \star\_{[\mathbf{x}\_{11}]} \tilde{f}\left(t\_{112}(o\_2), \mathfrak{x}\_{112}(o\_2)\right). \end{aligned}$$

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

$$\begin{aligned} \Xi^{12}\_{121,122}(a\_2) &= \Theta^{121}(a\_2) \star\_{[\mathfrak{x}\_{12}]} \Theta^{122}(a\_2) \\ &= \mathfrak{u}\_0(\mathfrak{x}\_{121}(a\_2)) \star\_{[\mathfrak{x}\_{12}]} \mathfrak{u}\_0(\mathfrak{x}\_{122}(a\_2)) .\end{aligned}$$

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

$$\begin{aligned} \Xi^{22}\_{221,222}(a\_2) &= \Theta^{221}(a\_2) \star\_{[x\_{22}]} \Theta^{222}(a\_2) \\ &= \tilde{f}\left(t\_{221}(a\_2), \mathfrak{x}\_{221}(a\_2)\right) \star\_{[x\_{22}]} \mu\_0(\mathfrak{x}\_{222}(a\_2)) .\end{aligned}$$

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 pivoting node x1, we get an expression

$$\begin{split} \Xi^{1}\_{11,12}(\boldsymbol{\mu}\_{2}) &= \Theta^{11}(\boldsymbol{\mu}\_{2}) \star\_{[\boldsymbol{\mathbf{x}}\_{1}]} \Theta^{12}(\boldsymbol{\mu}\_{2}) \\ &= \Xi^{11}\_{11,112}(\boldsymbol{\mu}\_{2}) \star\_{[\boldsymbol{\mathbf{x}}\_{1}]} \Xi^{12}\_{121,122}(\boldsymbol{\mu}\_{2}) \\ &= \left(\boldsymbol{\mu}\_{0}(\boldsymbol{\mu}\_{111}) \star\_{[\boldsymbol{\mathbf{x}}\_{11}]} \tilde{f}\left(t\_{112}, \boldsymbol{\mu}\_{112}\right)\right) \star\_{[\boldsymbol{\mathbf{x}}\_{1}]} \left(\boldsymbol{\mu}\_{0}(\boldsymbol{\mu}\_{121}) \star\_{[\boldsymbol{\mathbf{x}}\_{12}]} \boldsymbol{\mu}\_{0}(\boldsymbol{\mu}\_{122})\right). \end{split}$$

Therefore, it follows by a similar argument that the explicit representation of star-product functional for ω<sup>2</sup> is given by

operation at tϕ, the Markov property [13] guarantees that the lower tree structure below the first-generation branching node point x<sup>1</sup> is independent to that below the location x<sup>2</sup> with realized ω∈ Ω; hence, a tree-based <sup>∗</sup>-product functional branched after time s is also probabilistically independent of the other tree-based <sup>∗</sup>-product functional branched after time s, and besides the distributions of x<sup>1</sup> and x<sup>2</sup> are totally controlled by the Markov kernel Kx. Therefore, an easy computation pro-

> 2 ðt 0 dsλj j x 2 e �λj j <sup>x</sup> <sup>2</sup> ð Þ <sup>t</sup>�<sup>s</sup> �

Note that as for the second term, it goes almost similarly as the computation of

On this account, if we multiply both sides of Eq. (18) by exp <sup>λ</sup>t xj j<sup>2</sup> n o, 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

function. Taking the above fact into consideration, we can deduce with ease that

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

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

∣Mh i <sup>u</sup>0;<sup>f</sup>

u tð Þ ; <sup>x</sup> <sup>≔</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

(

holds for ∀t∈½ � 0; T and x∈Ec, where the measurable set Ec denotes the totality

inequality, which is a basic property of the star-product ★. That is to say, we have.

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

<sup>∣</sup>w★½ � <sup>x</sup> <sup>v</sup>∣<sup>⩽</sup> <sup>∣</sup>w<sup>∣</sup> � <sup>∣</sup>v<sup>∣</sup> for every w, v<sup>∈</sup> <sup>C</sup><sup>3</sup> and every <sup>x</sup><sup>∈</sup> <sup>D</sup>0: Next, we are going to derive the space of solutions to Eq. (2). If we define

> <sup>⋆</sup> ð Þ ω h i, on Ec,

0, otherwise,

� ÐÐ Es,x<sup>1</sup> ½ �� <sup>M</sup><sup>∗</sup> Es,x<sup>2</sup> ½ � <sup>M</sup><sup>∗</sup> Kxð Þ <sup>d</sup>x1; <sup>d</sup>x<sup>2</sup> :

ð Þ <sup>t</sup>�<sup>s</sup> F sð Þ ; <sup>x</sup> <sup>d</sup><sup>s</sup>

(18)

ðð V sð Þ ; <sup>y</sup> V sð Þ ; <sup>z</sup> Kxð Þ dy; dz d s:

t

V tð Þ ; x ∈ R<sup>þ</sup> is a nondecreasing

� � <sup>&</sup>lt; <sup>∞</sup> holds for a.e.‐x, namely, it is

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> (19)

★ ð Þ <sup>ω</sup> <sup>∣</sup><sup>⩽</sup> <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> : (20)

vides with an impressive expression:

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

Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ; <sup>t</sup><sup>ϕ</sup> . <sup>0</sup>; ηϕ <sup>¼</sup> <sup>1</sup> � � <sup>¼</sup> <sup>1</sup>

A Probabilistic Interpretation of Nonlinear Integral Equations

V tð Þ¼ ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> � �

<sup>¼</sup> U xð Þr�λt xj j<sup>2</sup>

that, for each <sup>x</sup><sup>∈</sup> <sup>D</sup>0, the mapping 0½ � ; <sup>T</sup> <sup>∍</sup>t↦eλj j <sup>x</sup> <sup>2</sup>

of all the elements x in D<sup>0</sup> such that ET,x Mh i <sup>U</sup>;<sup>F</sup> <sup>∗</sup>

holds Pt,x-a.s.

simple fact:

35

<sup>þ</sup> <sup>Ð</sup><sup>t</sup> 0 λj j x 2 2 e �λj j <sup>x</sup> <sup>2</sup> ð Þ t�s

the above-mentioned third one. Finally, summing up we obtain

<sup>þ</sup> <sup>Ð</sup><sup>t</sup> 0 λj j x 2 <sup>2</sup> <sup>e</sup> �λj j <sup>x</sup> <sup>2</sup>

$$M^{(u\_0 f)}\_{\star}(\boldsymbol{\alpha}\_2) = \left\{ \left( \boldsymbol{u}\_0(\boldsymbol{x}\_{111}) \boldsymbol{\star}\_{[\boldsymbol{x}\_{11}]} \boldsymbol{\tilde{f}}(t\_{112}, \boldsymbol{x}\_{112}) \right) \boldsymbol{\star}\_{[\boldsymbol{x}\_1]}(\boldsymbol{u}\_0(\boldsymbol{x}\_{121}) \boldsymbol{\star}\_{[\boldsymbol{x}\_{12}]} \boldsymbol{u}\_0(\boldsymbol{x}\_{122})) \right\}$$

$$\boldsymbol{\star}\_{[\boldsymbol{x}\_\theta]} \left\{ \boldsymbol{\tilde{f}}(t\_{21}, \boldsymbol{x}\_{21}) \boldsymbol{\star}\_{[\boldsymbol{x}\_\parallel]}(\boldsymbol{u}\_0(\boldsymbol{x}\_{221}) \boldsymbol{\star}\_{[\boldsymbol{x}\_{22}]} \boldsymbol{u}\_0(\boldsymbol{x}\_{222})) \right\}$$
