6. The <sup>∗</sup>-product functional and existence

In this section we first begin with constructing a ð Þ U; F -weighted tree-based <sup>∗</sup>-product functional <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> , which is indexed by the nodes ð Þ xm of a binary tree. Recall 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> <sup>L</sup><sup>1</sup> ð Þ R<sup>þ</sup> for each x. Moreover, in construction of the functional, the product is taken as ordinary multiplication <sup>∗</sup> 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 <sup>0</sup>⩽t⩽T and x<sup>∈</sup> <sup>D</sup>0, the function 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> � � satisfies

$$e^{it|\mathbf{x}|^{2}}V(t,\mathbf{x}) = U(\mathbf{x}) + \int\_{0}^{t} \mathbf{d}\mathbf{s} \frac{|\mathbf{x}|^{2}}{2} e^{k|\mathbf{x}|^{2}} \left\{ F(s,\mathbf{x}) + \int V(s,\mathbf{y}) V(s,\mathbf{z}) \mathbf{K}\_{\mathbf{x}}(\mathbf{d}\mathbf{y},\mathbf{z}) \right\}. \tag{15}$$

Proof of Lemma 3. By making use of the conditional expectation, we may decompose V tð Þ ; x as follows:

$$\begin{split} V(t, \boldsymbol{x}) &= \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{U, F}(\boldsymbol{w}) \right] \\ &= \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{(U, F)}(\boldsymbol{w}), \ t\_{\phi} \leqslant \mathbf{0} \right] + \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{(U, F)}(\boldsymbol{w}), t\_{\phi} > \mathbf{0} \right] \\ &= \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{U, F}(\boldsymbol{w}), \ t\_{\phi} \leqslant \mathbf{0} \right] + \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{(U, F)}(\boldsymbol{w}), t\_{\phi} > \mathbf{0}, \eta\_{\phi} = \mathbf{0} \right] \\ &\quad + \mathbb{E}\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\ast}^{(U, F)}(\boldsymbol{w}), \ t\_{\phi} > \mathbf{0}, \eta\_{\phi} = \mathbf{1} \right]. \end{split} \tag{16}$$

We are next going to take into consideration an equivalence between the events t<sup>ϕ</sup> ⩽0 and T∉½ � 0; t . Indeed, as to the first term in the third line of Eq. (16), since the condition t<sup>ϕ</sup> ⩽0 implies that T never lies in an interval 0½ � ; t , and since m ¼ ϕ∈ N�ð Þ ω leads to a nonrandom expression

$$M\_\* = \Theta^\phi = U(\mathfrak{x}),$$

the tree-based <sup>∗</sup>-product functional is allowed to possess a simple representation:

$$\begin{split} E\_{t,\mathbf{x}}\left[M\_{\ast}^{(U,F)},t\_{\boldsymbol{\theta}}\leqslant\mathbf{0}\right] &= E\_{t,\mathbf{x}}\left[M\_{\ast}^{(U,F)}\cdot\mathbf{1}\_{\left\{t\_{\boldsymbol{\theta}}\leqslant\leqslant 0\right\}}\right] = U(\mathbf{x})\cdot P\_{t,\mathbf{x}}\left(t\_{\boldsymbol{\theta}}\leqslant\mathbf{0}\right) \\ &= U(\mathbf{x})\cdot P(T\notin[\mathbf{0},t]) = U(\mathbf{x})\cdot P(T\in(t,\infty)) \\ &= U(\mathbf{x})\int\_{t}^{\infty}f\_{T}(\mathbf{s})\mathbf{d}\mathbf{s} = U(\mathbf{x})\int\_{t}^{\infty}\lambda|\mathbf{x}|^{2}e^{-\lambda t|\mathbf{x}|^{2}}\mathbf{d}\mathbf{s} \\ &= U(\mathbf{x})\cdot\exp\left\{-\lambda t|\mathbf{x}|^{2}\right\}. \end{split} \tag{17}$$

As to the third term, we need to note the following matters. A particle generates two offsprings or descendants x1, x<sup>2</sup> with probability <sup>1</sup> <sup>2</sup> under the condition ηϕ ¼ 1; since t<sup>ϕ</sup> . 0, when the branching occurs at t<sup>ϕ</sup> ¼ s, then, under the conditioning

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

Therefore, it follows by a similar argument that the explicit representation of

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

In this section we first begin with constructing a ð Þ U; F -weighted tree-based <sup>∗</sup>-product functional <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> , which is indexed by the nodes ð Þ xm of a binary tree. Recall that U ¼ U xð Þ (resp. F ¼ F tð Þ ; x ) is a nonnegative measurable function on D<sup>0</sup>

construction of the functional, the product is taken as ordinary multiplication <sup>∗</sup>

�

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

<sup>¼</sup> Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> <sup>⩽</sup><sup>0</sup> � � <sup>þ</sup> Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> . <sup>0</sup> � �

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

m ¼ ϕ∈ N�ð Þ ω leads to a nonrandom expression

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

two offsprings or descendants x1, x<sup>2</sup> with probability <sup>1</sup>

¼ U xð Þ

condition t<sup>ϕ</sup> ⩽0 implies that T never lies in an interval 0½ � ; t , and since

ð<sup>∞</sup> t

<sup>¼</sup> U xð Þ� exp �λt xj j<sup>2</sup> n o

since t<sup>ϕ</sup> . 0, when the branching occurs at t<sup>ϕ</sup> ¼ s, then, under the conditioning

<sup>¼</sup> Et,x MU,F <sup>∗</sup> ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> <sup>⩽</sup><sup>0</sup> � � <sup>þ</sup> Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> . <sup>0</sup>; ηϕ <sup>¼</sup> <sup>0</sup> � �

We are next going to take into consideration an equivalence between the events t<sup>ϕ</sup> ⩽0 and T∉½ � 0; t . Indeed, as to the first term in the third line of Eq. (16), since the

<sup>M</sup><sup>∗</sup> <sup>¼</sup> <sup>Θ</sup><sup>ϕ</sup> <sup>¼</sup> U xð Þ,

h i

the tree-based <sup>∗</sup>-product functional is allowed to possess a simple representation:

¼ U xð Þ� P Tð Þ¼ ∉½ � 0; t U xð Þ� P Tð Þ ∈ ð Þ t; ∞

ð<sup>∞</sup> t λj j x 2 e �λs xj j<sup>2</sup> ds

:

f <sup>T</sup>ð Þs ds ¼ U xð Þ

As to the third term, we need to note the following matters. A particle generates

In what follows we shall give an outline of the existence in Theorem 1. We need

Lemma 3. For <sup>0</sup>⩽t⩽T and x<sup>∈</sup> <sup>D</sup>0, the function 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> � � satisfies

F sð Þþ ; x

ð

<sup>~</sup>f tð Þ <sup>21</sup>; <sup>x</sup><sup>21</sup> ★½ � <sup>x</sup><sup>2</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>221</sup> ★½ � <sup>x</sup><sup>22</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>222</sup> n o � �

n o � �

★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>

V sð Þ ; y Vðs; zÞKxðdy; zÞ

<sup>¼</sup> U xð Þ� Pt,x <sup>t</sup><sup>ϕ</sup> <sup>⩽</sup> <sup>0</sup> � �

<sup>2</sup> under the condition ηϕ ¼ 1;

ð Þ R<sup>þ</sup> for each x. Moreover, in

� :

(15)

(16)

(17)

� �

star-product functional for ω<sup>2</sup> is given by

Recent Advances in Integral Equations

★ ð Þ¼ <sup>ω</sup><sup>2</sup> <sup>u</sup>0ð Þ <sup>x</sup><sup>111</sup> ★½ � <sup>x</sup><sup>11</sup>

★½ � <sup>x</sup><sup>ϕ</sup>

6. The <sup>∗</sup>-product functional and existence

(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> <sup>L</sup><sup>1</sup>

<sup>0</sup> ds j j x 2 2 e <sup>λ</sup>s xj j<sup>2</sup>

the following lemma, which is essentially important for the proof.

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

instead of the star-product ★.

V tð Þ¼ ; <sup>x</sup> U xð Þþ <sup>Ð</sup><sup>t</sup>

V tð Þ¼ ; <sup>x</sup> Et,x MU,F <sup>∗</sup> ð Þ <sup>ω</sup> � �

decompose V tð Þ ; x as follows:

eλt xj j<sup>2</sup>

34

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 provides with an impressive expression:

$$\begin{split} E\_{t,\mathbf{x}}\left[\mathcal{M}^{\langle U,F\rangle}\_{\*},t\_{\boldsymbol{\phi}} > \mathbf{0}, \eta\_{\boldsymbol{\phi}} = \mathbf{1}\right] &= \frac{1}{2} \int\_{0}^{t} \mathbf{ds} \,\lambda \left|\mathbf{x}\right|^{2} e^{-\lambda |\mathbf{x}|^{2}(t-s)}. \\ &\quad \times \iint E\_{s,\mathbf{x}\_{1}}[\mathcal{M}\_{\*}] \cdot E\_{s,\mathbf{x}\_{2}}[\mathcal{M}\_{\*}] K\_{\mathbf{x}}(\mathbf{dx}\_{1}, \mathbf{dx}\_{2}). \end{split}$$

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

$$\begin{split} V(t, \mathbf{x}) &= E\_{t, \mathbf{x}} \left[ M\_{\ast}^{(U, F)}(\boldsymbol{w}) \right] \\ &= U(\mathbf{x}) r^{-\dot{\boldsymbol{\mu}} \left| \mathbf{x} \right|^{2}} + \int\_{0}^{t} \frac{\lambda \left| \mathbf{x} \right|^{2}}{2} e^{-\lambda \left| \mathbf{x} \right|^{2} (t - s)} F(s, \mathbf{x}) \, \mathrm{d}s \\ &\quad + \int\_{0}^{t} \frac{\lambda \left| \mathbf{x} \right|^{2}}{2} e^{-\lambda \left| \mathbf{x} \right|^{2} (t - s)} \iint V(s, \mathbf{y}) V(s, \mathbf{z}) K\_{\mathbf{x}}(d\mathbf{y}, dz) \, \mathrm{d}s. \end{split} \tag{18}$$

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 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> t V tð Þ ; x ∈ R<sup>þ</sup> is a nondecreasing function. Taking the above fact into consideration, we can deduce with ease that

$$E\_{t, \mathbf{x}} \left[ \mathbf{M}\_{\ast}^{\langle U, F \rangle} (a) \right] < \infty \tag{19}$$

holds for ∀t∈½ � 0; T and x∈Ec, where the measurable set Ec denotes the totality of all the elements x in D<sup>0</sup> such that ET,x Mh i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> � � <sup>&</sup>lt; <sup>∞</sup> 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

$$|\mathcal{M}\_{\star}^{\langle u\_{0}f\rangle}(\boldsymbol{\alpha})| \leqslant \mathcal{M}\_{\ast}^{\langle U,\boldsymbol{F}\rangle}(\boldsymbol{\alpha}).\tag{20}$$

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 <sup>∗</sup>-product functional with a rather simplified structure. In fact, the M∗-control inequality yields immediately from a simple fact:

$$|w \star\_{[x]} v| \leqslant |w| \cdot |v| \quad \text{for every} \quad w, v \in \mathbb{C}^3 \quad \text{and every} \quad x \in D\_0.$$

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

$$u(t,x) := \begin{cases} E\_{t,x} \left[ M^{\langle u\_0 f \rangle}\_{\star}(o) \right], & \text{on} \quad E\_{co}, \\ 0, & \text{otherwise}, \end{cases}$$

then u tð Þ ; x is well defined on the whole space D<sup>0</sup> under the assumptions of the main theorem (Theorem 1). Moreover, it follows from the M∗-control inequality (Eq. (20)) that

$$|u(t, \mathbf{x})| \leqslant V(t, \mathbf{x}) \quad \text{on} \quad [\mathbf{0}, T] \times D\_0. \tag{21}$$

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

First of all, note that we can choose a proper measurable subset F<sup>0</sup> ⊂ D<sup>0</sup> with

solution lying in the space D. This completes the proof of the existence.

� � <sup>¼</sup> m Dð Þ¼ <sup>0</sup> \ <sup>F</sup><sup>0</sup> 0 (meaning that its complement <sup>F</sup><sup>c</sup>

martingale part from our star-product functional Mh i <sup>u</sup>0;<sup>f</sup>

m ∈ ⋃ 0⩽ℓ⩽n

labels restricted up to the nth generation and limited to the nodes related to branching at positive time. Moreover, let <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> be the set of labels lying in <sup>N</sup> \ <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> whose direct predecessor belongs to <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> . By convention, we define

labels, which is determined by the set of labels m ∈ V whose direct predecessor

part of <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> , while <sup>N</sup>� nnctð Þ <sup>ω</sup> is the non-cutoff part of <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> , which is defined by

<sup>N</sup>� nnctð Þ <sup>ω</sup> <sup>≔</sup> <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> \ <sup>N</sup><sup>~</sup> cut

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

<sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup> f g <sup>∅</sup> if <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup> <sup>∅</sup>. We shall introduce a new family <sup>N</sup><sup>~</sup> cut

We are now in a position to introduce a new class Mn,h i <sup>u</sup>0;f;<sup>u</sup>

belongs to <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> and has length <sup>∣</sup>m<sup>∣</sup> <sup>¼</sup> <sup>n</sup>, and we call this family <sup>N</sup><sup>~</sup> cut

satisfying tmð Þ <sup>ω</sup> . 0 and <sup>η</sup>mð Þ¼ <sup>ω</sup> 1. Namely, this family <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> is a subset of

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

<sup>ω</sup>, <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> is the totality of the labels

h i satisfies the integral equation Eq. (2), and this u tð Þ ; <sup>x</sup> is a

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> on <sup>F</sup><sup>0</sup> (26)

∣u sð Þ ; y ∣ � ∣u sð Þ ; z ∣Kxð Þ dy; dz , for ∀T . 0,

is convergent for a.e.-x ð Þ ∈ F<sup>0</sup> , and u tð Þ ; x satisfies the nonlinear integral equation (Eq. (2)) for a.e.‐x<sup>∈</sup> <sup>F</sup>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<sup>þ</sup> be the set of all the elements ω's corresponding to time tmð Þ ω . 0 for the label m. Next, we consider a kind of the notion like n-section of the set of labels for n∈ N<sup>0</sup> ≔ N ∪ f g0 . We define several families of Ω in what follows, in order to facilitate the extraction of its

<sup>0</sup> is a null set with respect

★ ð Þ <sup>ω</sup> . For each realized tree

<sup>n</sup> ð Þ ω of cutoff

<sup>n</sup> ð Þ ω : (27)

★ ð Þ <sup>ω</sup> of ★-product

<sup>n</sup> ð Þ ω the cutoff

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

7. Uniqueness

m F<sup>c</sup> 0

and

37

★ ð Þ <sup>ω</sup>

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

A Probabilistic Interpretation of Nonlinear Integral Equations

to Lebesgue measure m xð Þ), such that

ðT 0 ds ðð e <sup>λ</sup>j j <sup>x</sup> <sup>2</sup> s

On this account, from Eq. (15) in Lemma 3, by finiteness of the expectation of tree-based <sup>∗</sup>-product functional <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> , by the <sup>M</sup>∗-control inequality, and from Eq. (21), it is easy to see that

$$\int\_{0}^{T} \mathrm{d}s \int |u(s,y)| \cdot |u(s,z)| K\_{x}(\mathrm{d}y, \mathrm{d}z) < \infty \qquad \text{for} \quad x \in E\_{c}. \tag{22}$$

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

$$\mathcal{D} \coloneqq \left\{ \boldsymbol{\varrho} : \mathbb{R}\_{+} \times D\_{0} \to \mathbb{C}^{3} ; \; \boldsymbol{\varrho} \quad \text{is continuous in} \quad t$$

$$\text{and measurable such that} \begin{aligned} &\int\_{0}^{\infty} \text{d}s \int e^{i|\mathbf{x}|^{2}s} |\boldsymbol{\varrho}(s, \boldsymbol{\jmath})| \cdot |(\boldsymbol{\varsigma}, \boldsymbol{z})| K\_{\boldsymbol{x}}(\mathbf{d} \boldsymbol{\jmath}, \mathbf{d} \boldsymbol{z}) < \infty \\ &\qquad \qquad \qquad \text{holds a.e.} - \boldsymbol{\varkappa} \right\} \end{aligned} \tag{23}$$

By employing the Markov property [13] with respect to time t<sup>ϕ</sup> 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∈ Ec:

$$\begin{split} u(t,x) &= E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a) \right] \\ &= E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a), t\_{\phi} \lessgtr 0 \right] + E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a) t\_{\phi} > 0 \right] \\ &= E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a), t\_{\phi} \lessgtr 0 \right] + E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a) t\_{\phi} > 0, \eta\_{\phi} = 0 \right] \\ &+ E\_{t,x} \left[ M^{(u\_0 f)}\_{\star \star}(a) t\_{\phi} > 0 \right. \left. \left. \right. \left. \right. \left. \right. \right] \\ &= e^{-t|x|^{2}} u\_0(x) + \int\_{0}^{t} s \left| x \right|^{2} e^{-(t-t)|x|^{2}} \\ &\times \frac{1}{2} \left[ \tilde{f}(s,x) + \iint E\_{t,x\_1} [M \star] \star \_{[x]} E\_{s,x\_2} [M \star] K\_{\star}(\text{d}x\_1, \text{d}x\_2) \right]. \end{split} \tag{24}$$

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

Et,x Mh i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> h i <sup>¼</sup> <sup>e</sup> �λt xj j<sup>2</sup> u0ð Þþ x ðt 0 <sup>d</sup>s<sup>λ</sup> j j <sup>x</sup> <sup>2</sup> e �λð Þ <sup>t</sup>�<sup>s</sup> j j <sup>x</sup> <sup>2</sup> � 1 <sup>2</sup> <sup>~</sup>f sð Þþ ; <sup>x</sup> ðð Es,x<sup>1</sup> ½ � <sup>M</sup>★ ★½ � <sup>x</sup> Es,x<sup>2</sup> ½ � <sup>M</sup>★ Kxðdx1; <sup>d</sup>x2<sup>Þ</sup> � � ¼ e �λt xj j<sup>2</sup> u0ð Þþ x ðt 0 <sup>d</sup><sup>s</sup> <sup>λ</sup>j j <sup>x</sup> <sup>2</sup> e �λð Þ <sup>t</sup>�<sup>s</sup> j j <sup>x</sup> <sup>2</sup> � 1 <sup>2</sup> <sup>~</sup>f sð Þþ ; <sup>x</sup> ðð <sup>u</sup>ðs; <sup>y</sup>Þ★½ � <sup>x</sup> <sup>u</sup>ðs; <sup>z</sup>ÞKxðdy; <sup>d</sup>z<sup>Þ</sup> � � ¼ e �λt xj j<sup>2</sup> <sup>u</sup>0ð Þþ <sup>x</sup> <sup>λ</sup> 2 ðt 0 e <sup>λ</sup>s xj j<sup>2</sup> fðs; xÞds þ λ 2 ðt 0 ds ð e <sup>λ</sup>s xj j<sup>2</sup> p sð Þ ; x; y; u nðx; yÞdy � , � (25)

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

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 u tð Þ¼ ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> h i satisfies the integral equation Eq. (2), and this u tð Þ ; <sup>x</sup> is a solution lying in the space D. This completes the proof of the existence.

## 7. Uniqueness

then u tð Þ ; x is well defined on the whole space D<sup>0</sup> under the assumptions of the main theorem (Theorem 1). Moreover, it follows from the M∗-control inequality

On this account, from Eq. (15) in Lemma 3, by finiteness of the expectation of tree-based <sup>∗</sup>-product functional <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þ <sup>ω</sup> , by the <sup>M</sup>∗-control inequality, and from

Hence, taking Eq. (22) into consideration, we define the space D of solutions to

∣φð Þ s; y ∣ � ∣ð Þ s; z ∣Kxð Þ dy; dz < ∞ holds a:e: � x

By employing the Markov property [13] with respect to time t<sup>ϕ</sup> and by a similar technique as in the proof of Lemma 3, we may proceed in rewriting and calculating

<sup>þ</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

<sup>þ</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

� �

Furthermore, we may apply the integral equality Eq. (4) in the assumption on

� �

<sup>u</sup>ðs; <sup>y</sup>Þ★½ � <sup>x</sup> <sup>u</sup>ðs; <sup>z</sup>ÞKxðdy; <sup>d</sup>z<sup>Þ</sup>

fðs; xÞds þ

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

Es,x<sup>1</sup> ½ � <sup>M</sup>★ ★½ � <sup>x</sup> Es,x<sup>2</sup> ½ � <sup>M</sup>★ Kxðdx1; <sup>d</sup>x2<sup>Þ</sup>

Es,x<sup>1</sup> ½ � <sup>M</sup>★ ★½ � <sup>x</sup> Es,x<sup>2</sup> ½ � <sup>M</sup>★ Kxðdx1; <sup>d</sup>x2<sup>Þ</sup>

<sup>φ</sup> : <sup>R</sup><sup>þ</sup> � <sup>D</sup><sup>0</sup> ! <sup>C</sup><sup>3</sup>

and measurable such that

★ ð Þ <sup>ω</sup> h i

<sup>u</sup>0ð Þþ <sup>x</sup> <sup>Ð</sup><sup>t</sup>

u0ð Þþ x

u0ð Þþ x

<sup>2</sup> <sup>~</sup>f sð Þþ ; <sup>x</sup>

<sup>2</sup> <sup>~</sup>f sð Þþ ; <sup>x</sup>

�

<sup>2</sup> <sup>~</sup>f sð Þþ ; <sup>x</sup>

★ ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> <sup>⩽</sup><sup>0</sup> h i

★ ð Þ <sup>ω</sup> ; <sup>t</sup><sup>ϕ</sup> <sup>⩽</sup><sup>0</sup> h i

★ ð Þ <sup>ω</sup> <sup>t</sup><sup>ϕ</sup> . <sup>0</sup> ηϕ <sup>¼</sup> <sup>1</sup> h i

e�ð Þ <sup>t</sup>�<sup>s</sup> j j <sup>x</sup> <sup>2</sup>

<sup>0</sup> s xj j<sup>2</sup>

ðð

ðt 0

ðð

ðt 0

ðð

<sup>u</sup>0ð Þþ <sup>x</sup> <sup>λ</sup> 2 ðt 0 e <sup>λ</sup>s xj j<sup>2</sup>

<sup>d</sup>s<sup>λ</sup> j j <sup>x</sup> <sup>2</sup> e �λð Þ <sup>t</sup>�<sup>s</sup> j j <sup>x</sup> <sup>2</sup>

<sup>d</sup><sup>s</sup> <sup>λ</sup>j j <sup>x</sup> <sup>2</sup> e �λð Þ <sup>t</sup>�<sup>s</sup> j j <sup>x</sup> <sup>2</sup>

� �

∣u tð Þ ; x ∣⩽V tð Þ ; x on 0½ �� ; T D0: (21)

∣u sð Þ ; y ∣ � ∣u sð Þ ; z ∣Kxð Þ dy; dz < ∞ for x∈Ec: (22)

; φ is continuous in t

★ ð Þ <sup>ω</sup> <sup>t</sup><sup>ϕ</sup> . <sup>0</sup> h i

★ ð Þ <sup>ω</sup> <sup>t</sup><sup>ϕ</sup> . <sup>0</sup>; ηϕ <sup>¼</sup> <sup>0</sup> h i

:

p sð Þ ; x; y; u nðx; yÞdy

� ,

(25)

(23)

(24)

�

(Eq. (20)) that

Eq. (2) as follows:

Eq. (21), it is easy to see that

ðT 0 ds ð

Recent Advances in Integral Equations

D ≔ �

Ð <sup>∞</sup> <sup>0</sup> ds Ð eλj j <sup>x</sup> <sup>2</sup> s

the expectation for ∀t . 0 and x∈ Ec:

<sup>¼</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

<sup>¼</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

the Markov kernel for Eq. (24) to obtain

<sup>þ</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

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

<sup>¼</sup> <sup>e</sup>�t xj j<sup>2</sup>

� 1

¼ e �λt xj j<sup>2</sup>

¼ e �λt xj j<sup>2</sup>

¼ e �λt xj j<sup>2</sup>

� 1

� 1

Et,x Mh i <sup>u</sup>0;<sup>f</sup>

36

★ ð Þ <sup>ω</sup> h i

First of all, note that we can choose a proper measurable subset F<sup>0</sup> ⊂ D<sup>0</sup> with m F<sup>c</sup> 0 � � <sup>¼</sup> m Dð Þ¼ <sup>0</sup> \ <sup>F</sup><sup>0</sup> 0 (meaning that its complement <sup>F</sup><sup>c</sup> <sup>0</sup> is a null set with respect to Lebesgue measure m xð Þ), such that

$$E\_{t,x}\left[\mathcal{M}\_\*^{(U,F)}(a)\right] < \infty \quad \text{on} \quad F\_0 \tag{26}$$

and

$$\int\_0^T \mathrm{d}s \int \left[ e^{i|\mathbf{x}|^2 s} |u(s, \mathbf{y})| \cdot |u(s, \mathbf{z})| \mathcal{K}\_\mathbf{x}(\mathbf{d}\mathbf{y}, \mathbf{d}\mathbf{z}), \quad \text{for} \quad \forall T > \mathbf{0}, \mathbf{z}$$

is convergent for a.e.-x ð Þ ∈ F<sup>0</sup> , and u tð Þ ; x satisfies the nonlinear integral equation (Eq. (2)) for a.e.‐x<sup>∈</sup> <sup>F</sup>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<sup>þ</sup> be the set of all the elements ω's corresponding to time tmð Þ ω . 0 for the label m. Next, we consider a kind of the notion like n-section of the set of labels for n∈ N<sup>0</sup> ≔ N ∪ f g0 . We define several families of Ω in what follows, in order to facilitate the extraction of its martingale part from our star-product functional Mh i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> . For each realized tree <sup>ω</sup>, <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> is the totality of the labels

$$m \in \bigcup\_{0 \leqslant \ell \leqslant n} \{1, 2\}^{\ell}$$

satisfying tmð Þ <sup>ω</sup> . 0 and <sup>η</sup>mð Þ¼ <sup>ω</sup> 1. Namely, this family <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> is a subset of labels restricted up to the nth generation and limited to the nodes related to branching at positive time. Moreover, let <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> be the set of labels lying in <sup>N</sup> \ <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> whose direct predecessor belongs to <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> . By convention, we define <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup> f g <sup>∅</sup> if <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ¼ <sup>ω</sup> <sup>∅</sup>. We shall introduce a new family <sup>N</sup><sup>~</sup> cut <sup>n</sup> ð Þ ω of cutoff labels, which is determined by the set of labels m ∈ V whose direct predecessor belongs to <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> and has length <sup>∣</sup>m<sup>∣</sup> <sup>¼</sup> <sup>n</sup>, and we call this family <sup>N</sup><sup>~</sup> cut <sup>n</sup> ð Þ ω the cutoff part of <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> , while <sup>N</sup>� nnctð Þ <sup>ω</sup> is the non-cutoff part of <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> , which is defined by

$$
\check{N}\_n nct(o) \coloneqq \check{N}\_n(o) \backslash \check{N}\_n^{cut}(o). \tag{27}
$$

We are now in a position to introduce a new class Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> 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

n-section as follows. In fact, if the label m is a member of the cutoff family N~ cut <sup>n</sup> ð Þ ω , the input data of the functional attached to <sup>m</sup> is given by u tp mð Þð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> � � instead of the usual initial data <sup>u</sup>0ð Þ xmð Þ <sup>ω</sup> or <sup>~</sup>f tð Þ <sup>m</sup>ð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> , where p mð Þ indicates the direct ancestor m<sup>0</sup> of m having length n. On the other hand, if m lies in the non-cutoff family <sup>N</sup>� nnctð Þ <sup>ω</sup> , then the input data of the functional attached to <sup>m</sup> is completely the same as before with no change, that is, we use <sup>u</sup>0ð Þ xm if tm <sup>⩽</sup>0 and use <sup>~</sup>f tð Þ <sup>m</sup>; xm if tm . 0. In such a way, we can construct a new ★-product functional Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> by the almost sure procedure, and we call it the <sup>n</sup>-section ★-product functional. Similarly, we can also define the corresponding <sup>n</sup>-section ★-product functional <sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ <sup>ω</sup> . Simply enough, to get the <sup>∗</sup>-product counterpart, we have only to replace those functions <sup>u</sup>0, <sup>~</sup><sup>f</sup> and <sup>u</sup> by U, F and <sup>V</sup> in the definition of ★-product functional. As easily imagined, we can also derive an n-section version of M<sup>n</sup> <sup>∗</sup>-control inequality:

Lemma 5. (M<sup>n</sup> <sup>∗</sup>-control inequality) The following inequality

$$|M^{n,\langle u\_0 f, u\rangle}\_{\star}(o)| \lessapprox M^{n,\langle U, F, V\rangle}\_{\ast}(o) \tag{28}$$

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

h i <sup>¼</sup> <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

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

Proposition 8. When u tð Þ ; x is a solution to the nonlinear integral equation (Eq. (2)),

Proof. Our proof is technically due to a martingale method. We need the follow-

be a solution of the nonlinear integral equation (Eq. (2)). Then, we have the following

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

n ¼ 0, it follows from the identity (Eq. (31)) and by the martingale property that

<sup>¼</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

<sup>¼</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

<sup>¼</sup> Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

We resort to the mathematical induction with respect to n ∈ N0. If we assume

h i <sup>¼</sup> Et,x Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

h i <sup>¼</sup> Et,x Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

the identity (Eq. (33)) for the case of n, then the case of n þ 1 reads at once

h i <sup>¼</sup> Et,x Et,x <sup>M</sup><sup>n</sup>þ1,h i <sup>u</sup>0;f;<sup>u</sup>

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. □

★ ð Þ <sup>ω</sup>

★ ð Þ <sup>ω</sup> be the n-section of ★-product functional, and let u tð Þ ; <sup>x</sup>

★ ð Þ <sup>ω</sup>

★ ð Þ <sup>ω</sup> h i <sup>¼</sup> u tð Þ ; <sup>x</sup> :

★ ð Þ <sup>ω</sup> h i <sup>¼</sup> u tð Þ ; <sup>x</sup> :

★ ð Þ <sup>ω</sup> h i <sup>¼</sup> u tð Þ ; <sup>x</sup> ,

n

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

★ ð Þ <sup>ω</sup> (31)

being <sup>F</sup>n-measurable. □

h i (32)

h i (33)

★ ð Þ <sup>ω</sup> is a martingale relative to f g <sup>F</sup><sup>n</sup> . For

(34)

(35)

(36)

★ ð Þj <sup>ω</sup> <sup>F</sup><sup>0</sup> h i h i

★ ð Þj <sup>ω</sup> <sup>F</sup><sup>1</sup> h i h i

★ ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup> h i h i

Et,x Mh i <sup>u</sup>0;<sup>f</sup>

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

Finally, the uniqueness yields from the following assertion.

property applied at tm and on the event <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> <sup>n</sup><sup>g</sup>

holds for every t<sup>∈</sup> ½ � <sup>0</sup>; <sup>T</sup> and for a.e.‐x.

holds for every t ð Þ 0⩽ t⩽T and every x∈F0. Proof of Lemma 9. Recall that Mn,h i <sup>u</sup>0;f;<sup>u</sup>

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

Et,x <sup>M</sup><sup>1</sup>,h i <sup>u</sup>0;f;<sup>u</sup>

Et,x <sup>M</sup><sup>n</sup>þ1,h i <sup>u</sup>0;f;<sup>u</sup>

★ ð Þ <sup>ω</sup>

★ ð Þ <sup>ω</sup>

Next, for the case n ¼ 1, by the same reason, we can get

★ ð Þ <sup>ω</sup>

Lemma 9. Let Mn,h i <sup>u</sup>0;f;<sup>u</sup>

identity: for each n∈ N<sup>0</sup>

★ ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup>

show that

then we have

ing lemma.

39

holds Pt,x-a.s.

because of the domination property: ∣u tð Þ ; x ∣⩽V tð Þ ; x for 0½ �� ; T D0, ∣u0ð Þ x ∣⩽ U xð Þ for ∀x, ∣ <sup>~</sup>f tð Þ ; <sup>x</sup> <sup>∣</sup> <sup>⩽</sup>F tð Þ ; <sup>x</sup> for <sup>∀</sup>t, x, and a simple inequality <sup>∣</sup>w★½ � <sup>x</sup> <sup>v</sup>∣⩽∣w<sup>∣</sup> � <sup>∣</sup>v<sup>∣</sup> for <sup>∀</sup>w, v<sup>∈</sup> <sup>C</sup><sup>3</sup> and <sup>∀</sup>x<sup>∈</sup> <sup>D</sup>0.

Let us now introduce a filtration f g F<sup>n</sup> for n ∈ N<sup>0</sup> on Ωþ, according to the discussion in Example 2. As a matter of fact, we define

$$\mathcal{F}\_n \coloneqq \sigma \Big( \check{\mathcal{N}}\_n(\boldsymbol{\alpha}); (t\_m, \boldsymbol{\chi}\_m), \boldsymbol{m} \in \check{\mathcal{N}}\_n(\boldsymbol{\alpha}) \cup \check{\mathcal{N}}\_n^{\mathrm{act}}(\boldsymbol{\alpha}); (\boldsymbol{\eta}\_m), \boldsymbol{m} \in \check{\mathcal{N}}\_n^{\mathrm{cut}}(\boldsymbol{\alpha}) \Big) \tag{29}$$

for each <sup>n</sup><sup>∈</sup> <sup>N</sup>0. Notice that <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> itself determines the other two families N~ cut <sup>n</sup> ð Þ <sup>ω</sup> and <sup>N</sup>� nnctð Þ <sup>ω</sup> . Then, it is readily observed that both functionals Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> and <sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ <sup>ω</sup> are <sup>F</sup>n-adapted.

Lemma 6. For each n∈ N0, the equality

$$M\_\*^{n,\langle U,F,V\rangle}(o) = E\_{t,x} \left[ M\_\*^{\langle U,F\rangle}(o) | \mathcal{F}\_n \right] \tag{30}$$

holds Pt,x-a.s. for every t∈ ½ � 0; T and every x ∈F0.

Proof. By its construction, we can conclude the equality of Eq. (30) from the strong Markov property [13] applied at times ð Þ tm s for m ∈V of length n on the set <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> n o <sup>∈</sup> <sup>F</sup>n. □

Moreover, an application of Lemma 6 with the n-section M<sup>n</sup> <sup>∗</sup>-control inequality (Eq. (28)) shows the Pt,x-integrability of <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> for every <sup>t</sup> <sup>∈</sup>½ � <sup>0</sup>; <sup>T</sup> and every x∈F0. 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 nsection ★-product functional Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> .

Lemma 7. The n-section Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> of ★-product functional with weight functions u<sup>0</sup> and f is an f g F<sup>n</sup> -martingale [8].

Proof. When we set <sup>¼</sup> Et,x <sup>M</sup><sup>n</sup> ★ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup> � �, then <sup>ξ</sup><sup>n</sup> turns out to be a f g <sup>F</sup><sup>n</sup> martingale, since

$$E\_{t, \mathbf{x}}[\xi\_n | \mathcal{F}\_{n-1}] = E\_{t, \mathbf{x}}\left[E\_{t, \mathbf{x}}\left[M^n\_{\mathbf{x}} | \mathcal{F}\_n\right] | \mathcal{F}\_{n-1}\right] = E\_{t, \mathbf{x}}\left[M^n\_{\mathbf{x}}\big|\_{n-1}\right] = \xi\_{n-1}$$

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

n-section as follows. In fact, if the label m is a member of the cutoff family N~ cut

In such a way, we can construct a new ★-product functional Mn,h i <sup>u</sup>0;f;<sup>u</sup>

easily imagined, we can also derive an n-section version of M<sup>n</sup>

∣Mn,h i <sup>u</sup>0;f;<sup>u</sup>

Lemma 5. (M<sup>n</sup>

Recent Advances in Integral Equations

holds Pt,x-a.s.

N~ cut

Mn,h i <sup>u</sup>0;f;<sup>u</sup>

<sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> n o

martingale, since

38

∣u0ð Þ x ∣⩽ U xð Þ for ∀x, ∣

<sup>∣</sup>w★½ � <sup>x</sup> <sup>v</sup>∣⩽∣w<sup>∣</sup> � <sup>∣</sup>v<sup>∣</sup> for <sup>∀</sup>w, v<sup>∈</sup> <sup>C</sup><sup>3</sup> and <sup>∀</sup>x<sup>∈</sup> <sup>D</sup>0.

discussion in Example 2. As a matter of fact, we define

<sup>F</sup><sup>n</sup> <sup>≔</sup> <sup>σ</sup> <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> ;ð Þ tm; xm ; <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> <sup>∪</sup> <sup>N</sup>� nct

★ ð Þ <sup>ω</sup> and <sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ <sup>ω</sup> are <sup>F</sup>n-adapted. Lemma 6. For each n∈ N0, the equality

holds Pt,x-a.s. for every t∈ ½ � 0; T and every x ∈F0.

(Eq. (28)) shows the Pt,x-integrability of <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

section ★-product functional Mn,h i <sup>u</sup>0;f;<sup>u</sup>

Proof. When we set <sup>¼</sup> Et,x <sup>M</sup><sup>n</sup>

Lemma 7. The n-section Mn,h i <sup>u</sup>0;f;<sup>u</sup>

functions u<sup>0</sup> and f is an f g F<sup>n</sup> -martingale [8].

Et,x <sup>ξ</sup><sup>n</sup> ½ �¼ <sup>j</sup>F<sup>n</sup>�<sup>1</sup> Et,x Et,x <sup>M</sup><sup>n</sup>

Moreover, an application of Lemma 6 with the n-section M<sup>n</sup>

the input data of the functional attached to <sup>m</sup> is given by u tp mð Þð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> � � instead of the usual initial data <sup>u</sup>0ð Þ xmð Þ <sup>ω</sup> or <sup>~</sup>f tð Þ <sup>m</sup>ð Þ <sup>ω</sup> ; xmð Þ <sup>ω</sup> , where p mð Þ indicates the direct ancestor m<sup>0</sup> of m having length n. On the other hand, if m lies in the non-cutoff family <sup>N</sup>� nnctð Þ <sup>ω</sup> , then the input data of the functional attached to <sup>m</sup> is completely the same as before with no change, that is, we use <sup>u</sup>0ð Þ xm if tm <sup>⩽</sup>0 and use <sup>~</sup>f tð Þ <sup>m</sup>; xm if tm . 0.

almost sure procedure, and we call it the <sup>n</sup>-section ★-product functional. Similarly, we can also define the corresponding <sup>n</sup>-section ★-product functional <sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ <sup>ω</sup> . Simply enough, to get the <sup>∗</sup>-product counterpart, we have only to replace those functions <sup>u</sup>0, <sup>~</sup><sup>f</sup> and <sup>u</sup> by U, F and <sup>V</sup> in the definition of ★-product functional. As

<sup>∗</sup>-control inequality) The following inequality

because of the domination property: ∣u tð Þ ; x ∣⩽V tð Þ ; x for 0½ �� ; T D0,

Let us now introduce a filtration f g F<sup>n</sup> for n ∈ N<sup>0</sup> on Ωþ, according to the

� �

for each <sup>n</sup><sup>∈</sup> <sup>N</sup>0. Notice that <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> itself determines the other two families

<sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ¼ <sup>ω</sup> Et,x <sup>M</sup>h i <sup>U</sup>;<sup>F</sup> <sup>∗</sup> ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup>

Proof. By its construction, we can conclude the equality of Eq. (30) from the strong Markov property [13] applied at times ð Þ tm s for m ∈V of length n on the set

x∈F0. 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-

★ ð Þ <sup>ω</sup> .

★ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup>

★jF<sup>n</sup>

<sup>∈</sup> <sup>F</sup>n. □

★ ð Þ <sup>ω</sup> of ★-product functional with weight

� �, then <sup>ξ</sup><sup>n</sup> turns out to be a f g <sup>F</sup><sup>n</sup> -

� �jF<sup>n</sup>�<sup>1</sup>� ¼ Et,x <sup>M</sup><sup>n</sup>

h i

<sup>n</sup> ð Þ <sup>ω</sup> and <sup>N</sup>� nnctð Þ <sup>ω</sup> . Then, it is readily observed that both functionals

<sup>~</sup>f tð Þ ; <sup>x</sup> <sup>∣</sup> <sup>⩽</sup>F tð Þ ; <sup>x</sup> for <sup>∀</sup>t, x, and a simple inequality

<sup>n</sup> ð Þ ω ,

★ ð Þ <sup>ω</sup> by the

<sup>∗</sup>-control inequality:

<sup>n</sup> ð Þ ω

� � (30)

★ ð Þ <sup>ω</sup> for every <sup>t</sup> <sup>∈</sup>½ � <sup>0</sup>; <sup>T</sup> and every

★ � � � n�1

<sup>∗</sup>-control inequality

¼ ξ<sup>n</sup>�<sup>1</sup>

(29)

★ ð Þ <sup>ω</sup> <sup>∣</sup><sup>⩽</sup> <sup>M</sup>n,h i <sup>U</sup>;F;<sup>V</sup> <sup>∗</sup> ð Þ <sup>ω</sup> (28)

<sup>n</sup> ð Þ <sup>ω</sup> ;ð Þ <sup>η</sup><sup>m</sup> ; <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> cut

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

$$E\_{\mathfrak{t},\mathfrak{x}}\left[\mathcal{M}^{\langle u\_0 f \rangle}\_{\star}(o)|\mathcal{F}\_{\mathfrak{n}}\right] = \mathcal{M}^{\mathfrak{t},\langle u\_0 f,\mathfrak{u} \rangle}\_{\star}(o) \tag{31}$$

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 tm and on the event <sup>m</sup> <sup>∈</sup> <sup>N</sup><sup>~</sup> <sup>n</sup><sup>g</sup> n being <sup>F</sup>n-measurable. □

Finally, the uniqueness yields from the following assertion.

Proposition 8. When u tð Þ ; x is a solution to the nonlinear integral equation (Eq. (2)), then we have

$$u(t, \boldsymbol{x}) = E\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}\_{\star}^{\langle u\_0 f \rangle}(o) \right] \tag{32}$$

holds for every t<sup>∈</sup> ½ � <sup>0</sup>; <sup>T</sup> and for a.e.‐x.

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

Lemma 9. Let Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> be the n-section of ★-product functional, and let u tð Þ ; <sup>x</sup> be a solution of the nonlinear integral equation (Eq. (2)). Then, we have the following identity: for each n∈ N<sup>0</sup>

$$u(t, \boldsymbol{x}) = E\_{t, \boldsymbol{x}} \left[ \boldsymbol{M}^{u, \langle u\_0 f, \boldsymbol{u} \rangle}\_{\star}(o) \right] \tag{33}$$

holds for every t ð Þ 0⩽ t⩽T and every x∈F0.

Proof of Lemma 9. Recall that Mn,h i <sup>u</sup>0;f;<sup>u</sup> ★ ð Þ <sup>ω</sup> is a martingale relative to f g <sup>F</sup><sup>n</sup> . For n ¼ 0, it follows from the identity (Eq. (31)) and by the martingale property that

$$\begin{split} E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{0,\langle u\_{0}f,\boldsymbol{u}\rangle}\_{\star}(\boldsymbol{\alpha})\right] &= E\_{t,\boldsymbol{x}}\left[E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{\langle u\_{0}f\rangle}\_{\star}(\boldsymbol{\alpha})|\mathcal{F}\_{0}\right]\right] \\ &= E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{\langle u\_{0}f\rangle}\_{\star}(\boldsymbol{\alpha})\right] = \boldsymbol{u}(t,\boldsymbol{x}). \end{split} \tag{34}$$

Next, for the case n ¼ 1, by the same reason, we can get

$$\begin{split} E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{1,(u\_0f,\boldsymbol{u})}\_{\star}(\boldsymbol{\alpha})\right] &= E\_{t,\boldsymbol{x}}\left[E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{(u\_0f)}\_{\star}(\boldsymbol{\alpha})|\mathcal{F}\_1\right]\right] \\ &= E\_{t,\boldsymbol{x}}\left[\boldsymbol{M}^{(u\_0f)}\_{\star}(\boldsymbol{\alpha})\right] = \boldsymbol{u}(t,\boldsymbol{x}). \end{split} \tag{35}$$

We resort to the mathematical induction with respect to n ∈ N0. If we assume the identity (Eq. (33)) for the case of n, then the case of n þ 1 reads at once

$$\begin{split} E\_{t,\mathbf{x}}\left[\mathcal{M}^{n+1,\langle\boldsymbol{u}\_{0}f,\boldsymbol{u}\rangle}\_{\star}(\boldsymbol{\alpha})\right] &= E\_{t,\mathbf{x}}\left[E\_{t,\mathbf{x}}\left[\mathcal{M}^{n+1,\langle\boldsymbol{u}\_{0}f,\boldsymbol{u}\rangle}\_{\star}(\boldsymbol{\alpha})|\mathcal{F}\_{\boldsymbol{u}}\right]\right] \\ &= E\_{t,\mathbf{x}}\left[\mathcal{M}^{n,\langle\boldsymbol{u}\_{0}f,\boldsymbol{u}\rangle}\_{\star}(\boldsymbol{\alpha})\right] = \boldsymbol{u}(t,\boldsymbol{x}), \end{split} \tag{36}$$

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 An as the set of <sup>ω</sup> <sup>∈</sup> <sup>Ω</sup><sup>þ</sup> such that <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> contains some label <sup>m</sup> of length <sup>n</sup>. From the definition, it holds immediately that

$$\mathcal{M}^{\langle u\_0 f \rangle}\_{\star}(o) = \mathcal{M}^{n, \langle u\_0 f, u \rangle}\_{\star}(o) \quad \text{on} \quad \mathfrak{Q}\_+ \ A\_{\mathfrak{n}}.\tag{37}$$

Acknowledgements

Author details

Isamu Dôku

Japan

41

This work is supported in part by the Japan MEXT Grant-in-Aids SR(C)

Department of Mathematics, Faculty of Education, Saitama University, Saitama,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: idoku@mail.saitama-u.ac.jp

provided the original work is properly cited.

17 K05358 and also by ISM Coop. Res. Program: 2011-CRP-5010.

A Probabilistic Interpretation of Nonlinear Integral Equations

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

Hence, for every x∈ F<sup>0</sup> and 0 ⩽t⩽T and ∀n∈ N0, we may apply Lemma 9 for the expression below with the identity (Eq. (31)) to obtain

$$\begin{split} & \left| u(t,x) - E\_{t,x} \left[ M^{\left[u\_{0}f\right]}\_{\star}(o) \right] \right| \\ &= \left| E\_{t,x} \left[ M^{u\_{\star}\left(u\_{0}f,u\right)}\_{\star}(o) \right] - E\_{t,x} \left[ M^{\left[u\_{0}f\right]}\_{\star}(o) \right] \right| \\ & \quad \leqslant \left| E\_{t,x} \left[ M^{u\_{\star}\left(u\_{0}f,u\right)}\_{\star}(o) - M^{\left[u\_{0}f\right]}\_{\star}(o); A\_{n} \right] \right| \\ & \quad + \left| E\_{t,x} \left[ M^{u\_{\star}\left(u\_{0}f,u\right)}\_{\star}(o) - M^{\left[u\_{0}f\right]}\_{\star}(o); A\_{n}^{c} \right] \right| \\ & = \left| E\_{t,x} \left[ \left( M^{u\_{\star}\left(u\_{0}f,u\right)}\_{\star}(o) - M^{\left[u\_{0}f\right]}\_{\star}(o) \right) \cdot \mathbf{1}\_{A\_{n}} \right] \right| \end{split} \tag{38}$$

where the symbol Et,x½ � Xð Þ ω ; A denotes the integral of Xð Þ ω over a measurable event A with respect to the probability measure Pt,xð Þ dω , namely,

$$E\_{t, \mathbf{x}}[X(\boldsymbol{\omega}); \mathbf{A}] = E\_{t, \mathbf{x}}[X(\boldsymbol{\omega}) \cdot \mathbf{1}\_A] = \int\_A X(\boldsymbol{\omega}) P\_{t, \mathbf{x}}(d\boldsymbol{\omega}).$$

Furthermore, we continue computing

$$\begin{split} \mathbb{P}(\mathfrak{B}) &\leqslant \left| E\_{t,\boldsymbol{x}} \left[ M^{u\_{t}\left(u\_{0}f,\boldsymbol{w}\right)}\_{\star}(\boldsymbol{w}) \mathbbm{1}\_{A\_{\boldsymbol{u}}} \right] \right| + \left| E\_{t,\boldsymbol{x}} \left[ M^{u\_{0}f}\_{\star}(\boldsymbol{w}) \mathbbm{1}\_{A\_{\boldsymbol{u}}} \right] \right| \\ &= \left| E\_{t,\boldsymbol{x}} \left[ E\_{t,\boldsymbol{x}} \left[ M^{u\_{0}f}\_{\star}(\boldsymbol{w}) | \mathcal{F}\_{\boldsymbol{n}} \right] \mathbbm{1}\_{A\_{\boldsymbol{u}}} \right] \right| + \left| E\_{t,\boldsymbol{x}} \left[ M^{\langle u\_{0}f \rangle}\_{\star}(\boldsymbol{w}) \mathbbm{1}\_{A\_{\boldsymbol{u}}} \right] \right| \\ &= 2 \left| E\_{t,\boldsymbol{x}} \left[ M^{\langle u\_{0}f \rangle}\_{\star}(\boldsymbol{w}) \mathbbm{1}\_{A\_{\boldsymbol{u}}} \right] \right|. \end{split} \tag{39}$$

Since ∩nAn ¼ ∅ by the binary critical tree structure [12], and since we have an natural estimate

$$\begin{aligned} &|\mathcal{M}\_{\star}^{\langle u\_{0}f\rangle}(o)\mathbf{1}\_{A\_{\pi}}(o)| \lhd \mathcal{M}\_{\star}^{\langle U,F\rangle}(o), \quad \text{a.s.}\\ &\text{and} \quad \lim\_{n\to\infty} \mathcal{M}\_{\star}^{\langle u\_{0}f\rangle}(o)\mathbf{1}\_{A\_{\pi}}(o) = \mathbf{0}, \quad \text{a.s.}\end{aligned} \tag{40}$$

it follows by the bounded convergence theorem of Lebesgue that

$$\lim\_{n \to \infty} |E\_{t,x} \left[ \mathcal{M}\_{\star}^{(u\_0 f)}(o) \mathbf{1}\_{A\_u} \right]| = \mathbf{0}.\tag{41}$$

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

$$|u(t, \mathfrak{x}) - E\_{t, \mathfrak{x}} \left[ M^{(u\_0 f)}\_{\star}(\
o) \right]| \to \mathbf{0} \quad (\text{as} \quad n \to \infty) \tag{42}$$

holds for every ð Þ <sup>t</sup>; <sup>x</sup> <sup>∈</sup> ½ �� <sup>0</sup>; <sup>T</sup> <sup>F</sup>0. Thus, we attain that u tð Þ¼ ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup> ★ ð Þ <sup>ω</sup> h i, a.e.‐x∈F0. This finishes the proof of Proposition 8. □

Concurrently, this completes the proof of the uniqueness.
