A σ-additive mapping $\mu :\mathcal{B}\to {\mathsf{L}_{0}}$ is called stochastic measure (SM).

Let ${\phi _{l}}:\hspace{2.5pt}\mathbb{R}\to \mathbb{R}$, $l\ge 1$, be measurable functions such that $\tilde{\phi }(x)={\textstyle\sum _{l=1}^{\infty }}|{\phi _{l}}(x)|$ is integrable w.r.t. μ on $\mathbb{R}$. Then

Let Z be an arbitrary set, and the function $q(z,s):Z\times [j,j+1]\to \mathbb{R}$ be such that all paths $q(z,\cdot )$ are continuous on $[j,j+1]$. Denote Then the random function has a version such that for all $\beta >0$, $\omega \in \Omega $, $z\in Z$

Functions $a(t)$, $b(t)$, $c(t)$ are continuous and bounded on $[0,+\infty )$, and where $t,{t_{1}},{t_{2}}\in [0,+\infty )$, L, λ, δ are positive constants.

There exists a constant ${c_{0}}>0$ such that $c(t)\le -{c_{0}}$ $\forall t\ge 0$.

${u_{0}}:\mathbb{R}\times \Omega \to \mathbb{R}$ is measurable and for all $y,{y_{1}},{y_{2}}\in \mathbb{R}$ where $C(\omega )$, ${L_{{u_{0}}}}(\omega )$ are random constants, $\beta ({u_{0}})\ge 1/2$.

$f:{\mathbb{R}_{+}}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous, bounded, and for some constant ${L_{f}}$ and all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}},{v_{1}},{v_{2}}\in \mathbb{R}$.

$\sigma :{\mathbb{R}_{+}}\times \mathbb{R}\to \mathbb{R}$ is measurable and for some constant $1/2<\beta (\sigma )<1$, all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}}\in \mathbb{R}$ and bounded functions ${C_{\sigma }},{L_{\sigma }}:{\mathbb{R}_{+}}\to \mathbb{R}$ such that

Assume that the function $v(t,x):[0,T]\times \mathbb{R}\to \mathbb{R}$ is bounded, Assumptions 1, 2 hold, $|v(0,x)|\le {M_{1}}$, $|\mathcal{L}v(t,x)|\le {M_{2}}$. Then

There exist positive constants ν, η, C such that for each $x,y\in \mathbb{R}$, $t>s>0$ the following estimates hold: where ${E_{\alpha ,\beta }}(z)={\textstyle\sum _{k=0}^{\infty }}\frac{{z^{k}}}{\Gamma (\alpha k+\beta )}$ is the Mittag-Leffler function, $\phi <1$.

We represent p as (see, for example, [6, (4.18)]). Here the function $\Phi (t,x;s,y)$ is a solution of the integral equation It is easy to calculate that Using Assumption 1 and boundedness of the function ${x^{\alpha }}{e^{-x}}$ on $[0,+\infty )$ for arbitrary $\alpha >0$, we easily obtain that where $0<\nu <{(4{\sup _{t\in \mathbb{R}}}a(t))^{-1}}$. The solution of (13) can be rewritten as where Using (14)–(17), we obtain that Analogously to [6, (4.58)] we show that where constants C, $\tilde{C}$ depend on λ. Taking the sum of (18) for each $k\ge 1$, we get the inequality The following inequality plays an important role in further estimates Now we use (19) and (20) to obtain (10). where the last equality is a consequence of [8, chapter III, (1.15)]. From (15), (21) and the inequalities $t\ge s$ ${e^{\eta (t-s)}}\ge 1$, ${E_{\lambda ,\lambda +1}}(\tilde{C}{(t-s)^{\lambda }})\ge \frac{1}{\Gamma (\lambda +1)}$ we obtain (10). We prove (11) analogously, using [8, chapter III, (1.16) with $\alpha =1/2$] instead of (22).

In the proof of (12) we use where $A>0$. Firstly assume that ${({x^{\prime }}-x)^{2}}<A(t-s)$; we prove that for such t, x, s Notice that where $0\le l<1/2$, ${x^{\ast }}=\epsilon x+(1-\epsilon ){x^{\prime }}$, $0\le \epsilon \le 1$, $0<{\nu _{1}}<\nu $; here we used the fact that the function ${x^{\alpha }}{e^{-x}}$ is bounded on $[0,+\infty )$ for arbitrary $\alpha >0$. Now we show that ${({x^{\prime }}-x)^{2}}<A(t-s)$ implies where ${\nu _{1}}>{\nu _{2}}>0$, C, ${\nu _{2}}$ do not depend on y and ${x^{\ast }}$, but C depends on A. We consider two cases. Now we set $C=\max \{{C_{1}},{C_{2}}\}$, ${\nu _{2}}=\min \{{\nu _{21}},{\nu _{22}}\}$ and obtain (25). Therefore, the following estimate holds: Consider the expression We estimate ${J_{1}}$ in the following way: Here we used the inequality $t-s-\frac{|{x^{\prime }}-x{|^{2}}}{2A}>\frac{t-s}{2}$ and (20). We estimate ${J_{2}}$ in analogous way using (25): We apply (20) and (26) with $\tilde{A}=2A$ to prove the estimation for ${J_{3}}$: for arbitrary $l\in (0,1/2)$. Now (24) follows from (26), (27), (28) and (29).

Let ${({x^{\prime }}-x)^{2}}\ge A(t-s)$. This implies where $l\in (0,1)$. On the other hand, and the same estimates hold for ${\textstyle\int _{s}^{t}}d\theta {\textstyle\int _{\mathbb{R}}}\frac{\partial W(t,{x^{\prime }};\theta ,\zeta )}{\partial {x^{\prime }}}\Phi (\theta ,\zeta ;s,y)\hspace{0.1667em}d\zeta $. Using (30), (24) and (23), we obtain (12); in (23) we can set, for example, $A=1$.  □

Assume Assumptions 1, 2, 5 hold. Then there exists a version ${\nu _{1}}(t,x)$ of integral (32) such that for each $\omega \in \Omega $

It follows from Lemma 2 and (6) that a version ${\nu _{1}}(t,x)$ of integral (32) exists such that for each $x\in \mathbb{R}$, $t\ge 0$, $\omega \in \Omega $ where Next we prove that ${\nu _{1}}(t,x)$ satisfies (33). In order to estimate the Besov norm on $[j,j+1]$, we consider where $y,\hspace{2.5pt}y+h\in [j,j+1]$. Denote where M, N, $\gamma >0$. To estimate ${I_{2}}$ we introduce the function where It is obvious that, for each $0<\gamma <\min \{M/2,N/2\}$ and arbitrary fixed pair $(s,y)$, $\tilde{p}$ belongs to a class ${C^{1,2}}([0,+\infty )\times \mathbb{R})$ as a function of $(t,x)$. It is easy to obtain that $\tilde{p}(t,x;s,y)=p(t,x;s,y)$ if $(t,x)\in {\Omega _{MN}}$ and $\tilde{p}(t,x;s,y)=0$ if $(t,x)\in \big([0,+\infty )\times \mathbb{R}\big)\setminus {\Omega _{MN}^{2\gamma }}\cup \{s\ge t\}$. Moreover, boundedness of $p(t,x;s,y)$ on ${\Omega _{MN}^{2\gamma }}\cap \{t\le T\}$ for each $T>0$ and the fact that $|\Psi |\le 1$ imply boundedness of $\tilde{p}(t,x;s,y)$ on $([0,T]\times \mathbb{R})$. Now we estimate $\mathcal{L}\tilde{p}$. Taking into consideration properties of $\tilde{p}$, it is easy to see that $\mathcal{L}\tilde{p}=0$ outside the set ${\Theta _{MN}^{\gamma }}$. And inside ${\Theta _{MN}^{\gamma }}$, For the derivatives of ${\eta _{\varepsilon }}(v)$ we have the following well-known inequalities: where the constant C does not depend on ε. Let us prove, for example, the first inequality in (36): where we used that ${\eta _{1}}\in {C^{\infty }}({\mathbb{R}^{2}})$. From (36) and the fact that ${\eta _{\varepsilon }}=0$ outside the ball with radius ε it follows that Using estimates (35), (10) and (11) we obtain the inequality for each $(t,x)\in {\Theta _{MN}^{\gamma }}$. Let $\gamma \in (0,1/3)$ be fixed; consider $t>3\gamma $. Thus, ${I_{2}}$ can be rewritten in following way: We estimate the first summand using the function $\tilde{p}(t,x;s,y)$ for $M=N=3\gamma $. Note that $t-s<3\gamma $ on the set ${\Theta _{MN}^{\gamma }}$; moreover, $t-s>\gamma $ or $|x-y|>\gamma $. In the first case we have the following consequence of (37): In the second case, Anyway, Using Lemma 3 for arbitrary $T>t$, we obtain where the constant C does not depend on T. On the other hand, taking into account that $\tilde{p}(t,x;s,y)=p(t,x;s,y)$ if $t-s>3\gamma $ we obtain Now we estimate ${I_{22}}$. We get For further estimates we use the function Let $M>2\gamma $, $N>1+2\gamma $. Then function $\hat{p}$ has properties, which are analogouos to the properties of $\tilde{p}$. For example, $\hat{p}(t,x;s,y,h)=p(t,x;s,y+h)-p(t,x;s,y)$ when $(t,x)\in {\Omega _{MN}}$, $\hat{p}$ is bounded on $([0,T]\times \mathbb{R})$, where $T>0$, $\hat{p}=0$ when $(t,x)\in \big([0,+\infty )\times \mathbb{R}\big)\setminus {\Omega _{MN}^{2\gamma }}\cup \{s\ge t\}$. Now we estimate $\mathcal{L}\hat{p}$. Notice that $\mathcal{L}\tilde{p}=0$ outside the set ${\Theta _{MN}^{\gamma }}$, and for each $(t,x)\in {\Theta _{MN}^{\gamma }}$ the following estimates hold: where for convenience we denote ${p_{y}}=p(t,x;s,y)$. (11) and (12) imply that Therefore, for each $(t,x)\in {\Theta _{MN}^{\gamma }}$, We rewrite ${I_{1}}$ as Consider the function $\hat{p}(t,x;s,y,h)$ for $M=3\gamma $, $N=3\gamma +1$. We estimate ${I_{11}}$ analogouosly to ${I_{21}}$. It follows from (40) and Lemma 3 that On the other hand, the equality $\hat{p}(t,x;s,y,h)=p(t,x;s,y+h)-p(t,x;s,y)$ for $t-s>3\gamma $ implies that ${I_{12}}$ is estimated in the following way: where $l\in (0,1/2)$. We can choose l, ϕ such that $1-2l=\beta (\sigma )$, $\phi =\beta (\sigma )$.

Now assume that for each $y\in [j,j+1]$ and some $m\in \mathbb{N}$ the following inequality holds: Then we consider the functions $\tilde{p}(t,x;s,y)$ and $\hat{p}(t,x;s,y,h)$ with $M=3\gamma $ and $N=3\gamma +m$. For such M and N, provided that (43) holds, $(t,x)\in {\Omega _{MN}}$. Moreover, using (37), (40), the fact that (43) implies and Lemma 3, we obtain: Now it is easy to estimate ${I_{1}}$, ${I_{2}}$: Note that we estimate $|q(t,x,y)|$ analogously to ${I_{2}}$. From (38), (39), (41), (42), (44), (45) it follows that there exists ${G_{\gamma }}(t):[0,+\infty )\to [0,+\infty )$ such that ${G_{\gamma }}(t)\to 0$, $t\to \infty $, and From this it follows that for each $\alpha \in (1/2,\beta (\sigma ))$ the following inequalities hold: These estimates imply for each $x\in \mathbb{R}$. On the other hand, Therefore, for each version that satisfies (34) we have Taking a supremum on x and sending t to infinity, we obtain the statement of the lemma.  □

We use the iteration process (9). For each $n\in \mathbb{N}$ we consider the function From [6, Theorem 2 §4] it follows that the function ${\nu _{2}^{(n)}}$ is a solution, bounded on $[0,T]\times \mathbb{R}$, of the Cauchy problem for each $\omega \in \Omega $, $T>0$. Using Lemma 3, we obtain Now from (8) and (46) it follows that Taking a supremum on x, sending t to infinity and using Lemma 5, we obtain the statement of the theorem.  □