1 Introduction
In this paper, we consider a stochastic heat equation that can formally be written as
where $(t,x)\in [0,T]\times \mathbb{R}$, $a\in \mathbb{R}$, $a\ne 0$, and μ is a stochastic measure (SM) defined on the Borel σ-algebra of $[0,T]$. We consider a solution to the formal equation (1) in the mild sense (see Eq. (5)). We prove the existence and uniqueness of the solution and obtain Hölder regularity of its paths under some general conditions for the stochastic part of equation.
(1)
\[\left\{\begin{array}{l}du(t,x)={a}^{2}\displaystyle \frac{{\partial }^{2}u(t,x)}{\partial {x}^{2}}\hspace{0.1667em}dt+f\big(t,x,u(t,x)\big)\hspace{0.1667em}dt+\sigma (t,x)\hspace{0.1667em}d\mu (t),\\{} u(0,x)=u_{0}(x),\end{array}\right.\]A similar problem for μ dependent on the spatial variable x was considered in [5]. The stochastic heat equation on fractals was studied in [7], and a review of results on equations driven by SMs is given in [6].
For equations driven by white noise, the regularity of paths of solutions was considered in [10, Chapter 3]. Equations driven by fractional noise were studied in [9, Chapter 2]. In many papers, the regularity of solutions was considered in appropriate function spaces; see, for example, [2] and references therein.
2 Preliminaries
Let $\mathsf{L}_{0}=\mathsf{L}_{0}(\varOmega ,\mathcal{F},\mathsf{P})$ be the set of (equivalence classes of) all real-valued random variables defined on a complete probability space $(\varOmega ,\mathcal{F},\mathsf{P})$. The convergence in $\mathsf{L}_{0}$ is understood as the convergence in probability. Let $\mathsf{X}$ be an arbitrary set, and $\mathcal{B}$ be a σ-algebra of subsets of $\mathsf{X}$.
Definition 1.
Any σ-additive mapping $\mu :\hspace{2.5pt}\mathcal{B}\to \mathsf{L}_{0}$ is called a stochastic measure (SM).
In other words, μ is a vector measure with values in $\mathsf{L}_{0}$. In [3], such μ is called a general SM.
Examples of SMs are the following. Let $\mathsf{X}=[0,T]\subset \mathbb{R}_{+}$, $\mathcal{B}$ be the σ-algebra of Borel subsets of $[0,T]$, and $N(t)$ be a square-integrable martingale. Then $\mu (\mathsf{A})={\int _{0}^{T}}\mathbf{1}_{\mathsf{A}}(t)\hspace{0.1667em}dN(t)$ is an SM. If ${W}^{H}(t)$ is a fractional Brownian motion with Hurst index $H>1/2$ and $f:[0,T]\to \mathbb{R}$ is a bounded measurable function, then $\mu (\mathsf{A})={\int _{0}^{T}}f(t)\mathbf{1}_{\mathsf{A}}(t)\hspace{0.1667em}d{W}^{H}(t)$ is also an SM, as follows from [4, Theorem 1.1]. An α-stable random measure defined on a σ-algebra is an SM [8, Chapter 3]. Theorem 8.3.1 of [3] states the conditions under which the increments of a real-valued Lévy process generate an SM.
For a deterministic measurable function $g:\mathsf{X}\to \mathbb{R}$ and SM μ, an integral of the form $\int _{\mathsf{X}}g\hspace{0.1667em}d\mu $ is defined and studied in [3, Chapter 7]; see also [1]. In particular, every bounded measurable g is integrable w.r.t. any μ. An analogue of the Lebesgue dominated convergence theorem holds for this integral [3, Proposition 7.1.1].
We consider the Besov spaces ${B_{22}^{\alpha }}([c,d])$. Recall that the norm in this classical space for $0<\alpha <1$ may be introduced by
where
For all $n\ge 1$, $1\le k\le {2}^{n}$, put ${\varDelta _{kn}^{(t)}}=((k-1){2}^{-n}t,k{2}^{-n}t]$.
The following estimate is a key tool for the proof of Hölder regularity of the stochastic integral. In our estimates, C and $C(\omega )$ will denote a constant and a random constant, respectively, which may be different from formula to formula.
Lemma 1 (Lemma 3.2 [5]).
Let SM μ be defined on the Borel σ-algebra of $[0,t]$, $\mathsf{Z}$ be an arbitrary set, and $q(z,s):\mathsf{Z}\times [0,t]\to \mathbb{R}$ be a function such that for some $1/2<\alpha <1$ and for each $z\in \mathsf{Z}$, $q(z,\cdot )\in {B_{22}^{\alpha }}([0,t])$. Then the random function
has a version $\tilde{\eta }(z)$ such that for some constant C (independent of z, ω) and each $\omega \in \varOmega $,
(3)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big|\tilde{\eta }(z)\big|\le & \displaystyle \big|q(z,0)\mu \big([0,t]\big)\big|\\{} & \displaystyle +C\big\| q(z,\cdot )\big\| _{{B_{22}^{\alpha }}([0,t])}{\bigg\{\sum \limits_{n\ge 1}{2}^{n(1-2\alpha )}\sum \limits_{1\le k\le {2}^{n}}{\big|\mu \big({\varDelta _{kn}^{(t)}}\big)\big|}^{2}\bigg\}}^{1/2}.\end{array}\]From Lemma 3.1 [5] it follows that, for $\varepsilon >0$,
3 The problem
Consider equation (1) in the following mild sense:
(5)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle u(t,x)=& \displaystyle \int _{\mathbb{R}}p(t,x-y)u_{0}(y)\hspace{0.1667em}dy+{\int _{0}^{t}}\hspace{0.1667em}ds\int _{\mathbb{R}}p(t-s,x-y)f\big(s,y,u(s,y)\big)\hspace{0.1667em}dy\\{} & \displaystyle +\int _{(0,t]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}dy.\end{array}\]Here
is the Gaussian heat kernel, $u(t,x)=u(t,x,\omega ):[0,T]\times \mathbb{R}\times \varOmega \to \mathbb{R}$ is an unknown measurable random function, and μ is an SM defined on the Borel σ-algebra of $[0,T]$. The integrals of random functions w.r.t. $dy$ and $ds$ are taken for each fixed $\omega \in \varOmega $.
Throughout this paper, we will use the following assumptions.
-
A1. $u_{0}(y)=u_{0}(y,\omega ):\mathbb{R}\times \varOmega \to \mathbb{R}$ is measurable and ω-wise bounded, $|u_{0}(y,\omega )|\le C(\omega )$.
-
A3. $f(s,y,v):[0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is measurable and bounded: $|f(s,y,v)|\le C$.
-
A5. $\sigma (s,y):[0,T]\times \mathbb{R}\to \mathbb{R}$ is measurable and bounded: $|\sigma (s,y)|\le C$.
Recall that $\int _{\mathbb{R}}p(t,x)\hspace{0.1667em}dx=1$.
4 Hölder continuity in x
Consider the regularity of paths of the stochastic integral from (5).
Lemma 2.
Let Assumptions A5 and A6 hold. Then, for any fixed $t\in [0,T]$ and $\gamma _{1}<\beta (\sigma )-1/2$, the stochastic function
\[\vartheta (x)=\int _{(0,t]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}dy,\hspace{1em}x\in \mathbb{R},\]
has a Hölder continuous version with exponent $\gamma _{1}$.
Proof.
Denote
Using (6) and the change of variables
we get
By a similar way, we can estimate $|D_{2}|$ and obtain
\[q(z,s)=\int _{\mathbb{R}}\big(p(t-s,x_{1}-y)-p(t-s,x_{2}-y)\big)\sigma (s,y)\hspace{0.1667em}dy,\hspace{1em}z=(x_{1},x_{2},t),\]
and apply (3) to $\eta (z)=\vartheta (x_{1})-\vartheta (x_{2})$. We will estimate the Besov space norm in (3). Consider the difference
(7)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle q(z,s+h)-q(z,s)\\{} & \displaystyle \hspace{1em}=\bigg(\int _{\mathbb{R}}p(t-s-h,x_{1}-y)\sigma (s+h,y)\hspace{0.1667em}dy-\int _{\mathbb{R}}p(t-s,x_{1}-y)\sigma (s,y)\hspace{0.1667em}dy\bigg)\\{} & \displaystyle \hspace{2em}-\bigg(\int _{\mathbb{R}}p(t-s-h,x_{2}-y)\sigma (s+h,y)\hspace{0.1667em}dy-\int _{\mathbb{R}}p(t-s,x_{2}-y)\sigma (s,y)\hspace{0.1667em}dy\bigg)\\{} & \displaystyle \hspace{1em}:=D_{1}-D_{2}.\end{array}\](8)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle |D_{1}|& \displaystyle =C\Bigg|\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s+h,x_{1}-2av\sqrt{t-s-h})\hspace{0.1667em}dv\\{} & \displaystyle \hspace{1em}-\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s,x_{1}-2av\sqrt{t-s})\hspace{0.1667em}dv\Bigg|\\{} & \displaystyle \stackrel{\mathrm{A}\mathrm{6}}{\le }C\int _{\mathbb{R}}{e}^{-{v}^{2}}\big(|h{|}^{\beta (\sigma )}+{\big|v(\sqrt{t-s-h}-\sqrt{t-s})\big|}^{\beta (\sigma )}\big)\hspace{0.1667em}dv\\{} & \displaystyle =C\int _{\mathbb{R}}{e}^{-{v}^{2}}\bigg(|h{|}^{\beta (\sigma )}+\frac{|v{|}^{\beta (\sigma )}|h{|}^{\beta (\sigma )}}{|\sqrt{t-s-h}+\sqrt{t-s}{|}^{\beta (\sigma )}}\bigg)\hspace{0.1667em}dv\\{} & \displaystyle \le C|h{|}^{\beta (\sigma )}\int _{\mathbb{R}}{e}^{-{v}^{2}}\bigg(1+\frac{|v{|}^{\beta (\sigma )}}{{\sqrt{t-s}}^{\beta (\sigma )}}\bigg)\hspace{0.1667em}dv=C|h{|}^{\beta (\sigma )}{(t-s)}^{-\beta (\sigma )/2}.\end{array}\]Further, consider
\[\begin{array}{r@{\hskip0pt}l}\displaystyle q(z,s+h)-q(z,s)=& \displaystyle \bigg(\int _{\mathbb{R}}p(t-s-h,x_{1}-y)\sigma (s+h,y)\hspace{0.1667em}dy\\{} & \displaystyle -\int _{\mathbb{R}}p(t-s-h,x_{2}-y)\sigma (s+h,y)\hspace{0.1667em}dy\bigg)\\{} & \displaystyle -\bigg(\int _{\mathbb{R}}p(t-s,x_{1}-y)\sigma (s,y)\hspace{0.1667em}dy\\{} & \displaystyle -\int _{\mathbb{R}}p(t-s,x_{2}-y)\sigma (s,y)\hspace{0.1667em}dy\bigg):=E_{1}-E_{2}.\end{array}\]
Using (6) and the substitutions
we get
\[\begin{array}{r@{\hskip0pt}l}\displaystyle |E_{1}|& \displaystyle =C\Bigg|\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s+h,x_{1}-2av\sqrt{t-s-h})\hspace{0.1667em}dv\\{} & \displaystyle \hspace{1em}-\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s+h,x_{2}-2av\sqrt{t-s-h})\hspace{0.1667em}dv\Bigg|\\{} & \displaystyle \stackrel{\mathrm{A}\mathrm{6}}{\le }C\int _{\mathbb{R}}{e}^{-{v}^{2}}|x_{1}-x_{2}{|}^{\beta (\sigma )}\hspace{0.1667em}dv=C|x_{1}-x_{2}{|}^{\beta (\sigma )}.\end{array}\]
Similarly, we can estimate $|E_{2}|$ (we consider $|E_{1}|$ for $h=0$) and obtain
The product of (9) raised to the power λ and (10) raised to the power $1-\lambda $, $0<\lambda <1$, now satisfies
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big|q(z,s+h)-q(z,s)\big|& \displaystyle \le C|h{|}^{\lambda \beta (\sigma )}{(t-s)}^{-\beta (\sigma )\lambda /2}|x_{1}-x_{2}{|}^{(1-\lambda )\beta (\sigma )},\\{} \displaystyle w_{2}\big(q(z,\cdot ),r\big)& \displaystyle \le C{r}^{\lambda \beta (\sigma )}|x_{1}-x_{2}{|}^{(1-\lambda )\beta (\sigma )}.\end{array}\]
If $\lambda \beta (\sigma )>1/2$, then the integral from (2) is finite for some $\alpha >1/2$. In this case, the integral does not exceed $C|x_{1}-x_{2}{|}^{(1-\lambda )\beta (\sigma )}$.From the estimate of $E_{1}$ for $h=0$ we obtain
\[\big|q(z,0)\big|\le C|x_{1}-x_{2}{|}^{\beta (\sigma )},\hspace{2em}\big\| q(z,\cdot )\big\| _{\mathsf{L}_{2}([0,t])}\le C|x_{1}-x_{2}{|}^{\beta (\sigma )}.\]
Therefore, we have
\[\big|\vartheta (x_{1})-\vartheta (x_{2})\big|\le C(\omega )|x_{1}-x_{2}{|}^{\gamma _{1}},\hspace{1em}\gamma _{1}=(1-\lambda )\beta (\sigma ).\]
Under the restriction $\lambda \beta (\sigma )>1/2$, we can get any $\gamma _{1}<\beta (\sigma )-1/2$. □5 Hölder continuity in t
Lemma 3.
Assume that Assumptions A5, A6, and A7 hold. Then, if $\gamma _{2}\le \beta (\mu )$ and $\gamma _{2}<\beta (\sigma )-1/2$, then for any fixed $x\in \mathbb{R}$, the stochastic process
\[\bar{\vartheta }(t)=\int _{(0,t]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}dy,\hspace{1em}t\in [0,T],\]
has a Hölder continuous version with exponent $\gamma _{2}$.
Proof.
For $t_{1}<t_{2}$, we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \bar{\vartheta }(t_{2})-\bar{\vartheta }(t_{1})& \displaystyle =\int _{(t_{1},t_{2}]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}p(t_{2}-s,x-y)\sigma (s,y)\hspace{0.1667em}dy\\{} & \displaystyle \hspace{1em}+\int _{(0,t_{1}]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}\big(p(t_{2}-s,x-y)-p(t_{1}-s,x-y)\big)\sigma (s,y)\hspace{0.1667em}dy\\{} & \displaystyle :=F_{1}+F_{2}.\end{array}\]
Step 1. Estimation of $\boldsymbol{F}_{\mathbf{1}}$. Consider segments $[0,T]$, ${\varDelta _{kn}^{(T)}}=((k-1){2}^{-n}T,k{2}^{-n}T]$, and the function
\[\bar{q}(z,s)=\int _{\mathbb{R}}p(t_{2}-s,x-y)\sigma (s,y)\hspace{0.1667em}dy,\hspace{1em}s\in [t_{1},t_{2}],\hspace{2.5pt}z=(x,t_{2}).\]
From the estimates of $D_{1}$ in (7) and (8) it follows that
Let $k_{n1}$ and $k_{n2}$ be such that $t_{1}\in {\varDelta _{k_{n1}n}^{(T)}}$, $t_{2}\in {\varDelta _{k_{n2}n}^{(T)}}$. For the functions
Here $n_{0}$ is such that
We have
\[\bar{q}_{n}(z,s)=\sum \limits_{k=1}^{{2}^{n}}\bar{q}\big(z,(k-1){2}^{-n}T\vee t_{1}\wedge t_{2}\big)\mathbf{1}_{{\varDelta _{kn}^{(T)}}}(s),\]
the analogue of the Lebesgue theorem [3, Proposition 7.1.1] implies that
(12)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{(t_{1},t_{2}]}\bar{q}(z,s)\hspace{0.1667em}d\mu (s)\Bigg|=\Bigg|\mathrm{p}\underset{n\to \infty }{\lim }\int _{(t_{1},t_{2}]}\bar{q}_{n}(z,s)\hspace{0.1667em}d\mu (s)\Bigg|\\{} & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{1em}=\Bigg|\int _{(t_{1},t_{2}]}\bar{q}_{0}(z,s)\hspace{0.1667em}d\mu (s)+\sum \limits_{n=1}^{\infty }\bigg(\int _{(t_{1},t_{2}]}\bar{q}_{n}(z,s)\hspace{0.1667em}d\mu (s)-\int _{(t_{1},t_{2}]}\bar{q}_{n-1}(z,s)\hspace{0.1667em}d\mu (s)\bigg)\Bigg|\\{} & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{1em}\le \big|\bar{q}(z,t_{1})\mu \big((t_{1},t_{2}]\big)\big|+\sum \limits_{n=n_{0}}^{\infty }\big|\big(\bar{q}\big(z,k_{n1}{2}^{-n}T\big)-\bar{q}(z,t_{1})\big)\mu \big({\varDelta _{(k_{n1}+1)n}^{(T)}}\big)\big|\\{} & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{2em}+\sum \limits_{n=n_{0}}^{\infty }\sum \limits_{k:k_{n1}\le 2k-2<k_{n2}-2}\big|\big(\bar{q}\big(z,(2k-1){2}^{-n}T\big)-\bar{q}\big(z,(2k-2){2}^{-n}T\big)\big)\mu \big({\varDelta _{(2k)n}^{(T)}}\big)\big|\\{} & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{2em}+\sum \limits_{n=n_{0}}^{\infty }\big|\big(\bar{q}\big(z,(k_{n2}-1){2}^{-n}T\big)-\bar{q}\big(z,(k_{n2}-2){2}^{-n}T\big)\big)\mu \big(\big((k_{n2}-1){2}^{-n}T,t_{2}\big]\big)\big|.\end{array}\]Applying Assumptions A5, A7, (11), and the Cauchy inequality from (12) for $0<\varepsilon <2\beta (\sigma )-1$, we obtain
Also, analogously to (7) and (8), we have
From (15) and (16) for $0<\lambda <1$ and $0\le s\le t_{1}-h$, we obtain
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{(t_{1},t_{2}]}\bar{q}(z,s)\hspace{0.1667em}d\mu (s)\Bigg|\\{} & \displaystyle \hspace{1em}\le C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}+C(\omega )\sum \limits_{n=n_{0}}^{\infty }{2}^{-n\beta (\mu )}\\{} & \displaystyle \hspace{2em}+C\sum \limits_{n=n_{0}}^{\infty }\sum \limits_{k:k_{n1}\le 2k-2<k_{n2}-2}{2}^{-n\beta (\sigma )}{\big(t_{2}-(2k-2){2}^{-n}T\big)}^{-\beta (\sigma )/2}\big|\mu \big({\varDelta _{(2k)n}^{(T)}}\big)\big|\\{} & \displaystyle \hspace{2em}+C(\omega )\sum \limits_{n=n_{0}}^{\infty }{2}^{-n\beta (\mu )}\\{} & \displaystyle \hspace{1em}\le C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}+C(\omega ){2}^{-n_{0}\beta (\mu )}\\{} & \displaystyle \hspace{2em}+C{\Bigg(\sum \limits_{n=n_{0}}^{\infty }{2}^{\varepsilon n}{2}^{-2n\beta (\sigma )}\sum \limits_{k:k_{n1}\le 2k-2<k_{n2}-2}{\big(t_{2}-(2k-2){2}^{-n}T\big)}^{-\beta (\sigma )}\Bigg)}^{1/2}\\{} & \displaystyle \hspace{2em}\times {\Bigg(\sum \limits_{n=0}^{\infty }{2}^{-\varepsilon n}\sum \limits_{k=1}^{{2}^{n}}{\mu }^{2}\big({\varDelta _{kn}^{(T)}}\big)\Bigg)}^{1/2}\\{} & \displaystyle \stackrel{\text{(4)},\text{(13)}}{\le }C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}\\{} & \displaystyle \hspace{2em}+C(\omega ){\Bigg(\sum \limits_{n=n_{0}}^{\infty }{2}^{\varepsilon n}{2}^{-2n\beta (\sigma )}\sum \limits_{1\le i<(k_{n2}-k_{n1})/2}{\big(i{2}^{-n}T\big)}^{-\beta (\sigma )}\Bigg)}^{1/2}\\{} & \displaystyle \hspace{1em}\le C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}+C(\omega ){\Bigg(\sum \limits_{n=n_{0}}^{\infty }{2}^{n(\varepsilon -\beta (\sigma ))}{(k_{n2}-k_{n1})}^{1-\beta (\sigma )}\Bigg)}^{1/2}\\{} & \displaystyle \hspace{1em}\stackrel{\text{(14)}}{\le }C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}+C(\omega ){(t_{2}-t_{1})}^{(1-\beta (\sigma ))/2}{2}^{-n_{0}(2\beta (\sigma )-\varepsilon -1)/2}\\{} & \displaystyle \hspace{1em}\stackrel{\text{(13)}}{\le }C(\omega ){(t_{2}-t_{1})}^{\beta (\mu )}+C(\omega ){(t_{2}-t_{1})}^{(\beta (\sigma )-\varepsilon )/2}\le C(\omega ){(t_{2}-t_{1})}^{\gamma _{2}}.\end{array}\]
Step 2. Estimation of $\boldsymbol{F}_{\mathbf{2}}$. Now denote
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \tilde{q}(z,s)=\int _{\mathbb{R}}\big(p(t_{2}-s,x-y)-p(t_{1}-s,x-y)\big)\sigma (s,y)\hspace{0.1667em}dy,\\{} & \displaystyle \hspace{1em}s\in [0,t_{1}],\hspace{2.5pt}z=(x,t_{1},t_{2}).\end{array}\]
Using the change of variables
we get
(15)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \big|\tilde{q}(z,s)\big|=& \displaystyle C\Bigg|\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s,x-2av\sqrt{t_{2}-s})\hspace{0.1667em}dv\\{} & \displaystyle -\int _{\mathbb{R}}{e}^{-{v}^{2}}\sigma (s,x-2av\sqrt{t_{1}-s})\hspace{0.1667em}dv\Bigg|\\{} \displaystyle \stackrel{\mathrm{A}\mathrm{6}}{\le }& \displaystyle C\int _{\mathbb{R}}{e}^{-{v}^{2}}{\big|v(\sqrt{t_{2}-s}-\sqrt{t_{1}-s})\big|}^{\beta (\sigma )}\hspace{0.1667em}dv\\{} \displaystyle \stackrel{\text{(8)}}{\le }& \displaystyle C{(t_{2}-t_{1})}^{\beta (\sigma )}{(t_{2}-s)}^{-\beta (\sigma )/2}.\end{array}\](16)
\[\big|\tilde{q}(z,s+h)-\tilde{q}(z,s)\big|\le C|h{|}^{\beta (\sigma )}{(t_{1}-s)}^{-\beta (\sigma )/2}.\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \big|\tilde{q}(z,s+h)-\tilde{q}(z,s)\big|\\{} & \displaystyle \hspace{1em}\le C|h{|}^{\lambda \beta (\sigma )}{(t_{1}-s)}^{-\beta (\sigma )\lambda /2}{(t_{2}-t_{1})}^{(1-\lambda )\beta (\sigma )}{(t_{2}-s-h)}^{-(1-\lambda )\beta (\sigma )/2},\\{} & \displaystyle w_{2}\big(\tilde{q}(z,\cdot ),r\big)\le C{r}^{\lambda \beta (\sigma )}{(t_{2}-t_{1})}^{(1-\lambda )\beta (\sigma )}.\end{array}\]
If $\lambda \beta (\sigma )>1/2\Leftrightarrow (1-\lambda )\beta (\sigma )<\beta (\sigma )-1/2$, then the integral from (2) is finite for some $\alpha >1/2$. In this case, the integral does not exceed $C{(t_{2}-t_{1})}^{(1-\lambda )\beta (\sigma )}$.From (15) we get
\[\big|\tilde{q}(z,0)\big|\le C{(t_{2}-t_{1})}^{\beta (\sigma )/2},\hspace{2em}\big\| \tilde{q}(z,\cdot )\big\| _{\mathsf{L}_{2}([0,t_{1}])}\le C{(t_{2}-t_{1})}^{\beta (\sigma )/2}.\]
Therefore, from (3) we have $|F_{2}|\le C(\omega ){(t_{2}-t_{1})}^{\gamma _{2}}$, which finishes the proof. □6 Solution to the equation
Theorem 1.
Suppose that Assumptions A1–A6 hold.
-
2. For any fixed $t\in [0,T]$ and $\gamma _{1}<\beta (\sigma )-1/2$, the stochastic function $u(t,x)$, $x\in \mathbb{R}$, has a Hölder continuous version with exponent $\gamma _{1}$.
-
3. In addition, let Assumption A7 hold. Then for any fixed $\delta >0$ and $\gamma _{1}$, $\gamma _{2}$ such that $\gamma _{1}<\beta (\sigma )-1/2$, $\gamma _{2}\le \beta (\mu )$, and $\gamma _{2}<\beta (\sigma )-1/2$, the stochastic function $u(t,x)$ has a version $\tilde{u}(t,x)$ such that
Proof.
Consider the standard iteration process. Take ${u}^{(0)}(t,x)=0$ and set
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {u}^{(n+1)}(t,x)& \displaystyle =\int _{\mathbb{R}}p(t,x-y)u_{0}(y)\hspace{0.1667em}dy\\{} & \displaystyle \hspace{1em}+{\int _{0}^{t}}\hspace{0.1667em}ds\int _{\mathbb{R}}p(t-s,x-y)f\big(s,y,{u}^{(n)}(s,y)\big)\hspace{0.1667em}dy\\{} & \displaystyle \hspace{1em}+\int _{(0,t]}\hspace{0.1667em}d\mu (s)\int _{\mathbb{R}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}dy.\end{array}\]
Remark 1.
For u, we obtained less regularity than for elements of equation (5). However, a solution to a heat equation usually has the same regularity or even more regular than the coefficients. One may expect that using other methods gives (17) with exponents $\gamma _{2}\le \beta (\mu )\wedge \gamma _{2}<\beta (\sigma )$ and $\gamma _{1}<\beta (\sigma )$.
