Path properties of the solution to the stochastic heat equation with Lévy noise

Abstract

We consider sample path properties of the solution to the stochastic heat equation, in ${\mathbb {R}}^d$ or bounded domains of ${\mathbb {R}}^d$, driven by a Lévy space–time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than $-\frac{d}{2}$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal–Getoor index of the Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Lévy noises, and to light- as well as heavy-tailed jumps.

Appendix A: Integration with respect to random measures

Just as Lévy processes are special instances of semimartingales, a Lévy noise as in (1.6) is a random measure as introduced in [4]. We give a short introduction into the integration theory associated to it, hereby concentrating on the main ideas and results that we need for our purposes. All details not mentioned or explained can be found in [4, 13, 25, 26]. Given a filtered probability space $(\Omega ,{\mathcal {F}},({\mathcal {F}}_t)_{t\in [0,T]},{\mathbb {P}})$ satisfying the usual conditions, consider a Polish space E (e.g., $E=D$, the spatial domain in (1.1)), and denote by ${\mathcal {P}}$ the $\sigma$-field ${\mathcal {P}}_0\otimes {\mathcal {B}}(E)$, where ${\mathcal {P}}_0$ is the usual predictable $\sigma$-field. With an abuse of terminology, ${\mathcal {P}}$-measurable mappings from ${\tilde{\Omega }} :=\Omega \times [0,T]\times E$ to ${\mathbb {R}}$ are again called predictable and their collection again denoted by ${\mathcal {P}}$.

Given a sequence $({\tilde{\Omega }}_k)_{k\geqslant 1}$ in ${\mathcal {P}}$ satisfying ${\tilde{\Omega }}_k \uparrow {\tilde{\Omega }}$, a mapping $M:{\mathcal {P}}_M:=\bigcup _{k\geqslant 1} {\mathcal {P}}|_{{\tilde{\Omega }}_k} \rightarrow L^p$ where $p\in [0,+\infty )$ is called an $L^p$-random measure if for every sequence $(A_i)_{i\geqslant 1}$ of pairwise disjoint sets in ${\mathcal {P}}_M$ with $\bigcup _{i\geqslant 1} A_i\in {\mathcal {P}}_M$, we have $M(\bigcup _{i\geqslant 1} A_i) = \sum _{i\geqslant 1} M(A_i)$ in $L^p$, and some additional “$({\mathcal {F}}_t)_{t\in [0,T]}$-adaptedness” conditions are satisfied. In our example of a Lévy noise on $[0,T]\times D$, we can take ${\tilde{\Omega }}_k = D$ if D is bounded, and ${\tilde{\Omega }}_k = [-k,k]^d$ if $D={\mathbb {R}}^d$.

The stochastic integral of a simple integrand of the form $S=\sum _{i=1}^r a_i \mathbb {1}_{A_i}$, where $r\in {\mathbb {N}}$, $a_i\in {\mathbb {R}}$ and $A_i\in {\mathcal {P}}_M$, is defined in the canonical way by

$$\begin{aligned} \int _0^T\int _E S(t,x)\,M(\mathrm {d}t,\mathrm {d}x) := \sum _{i=1}^r a_i M(A_i). \end{aligned}$$

Denoting by ${\mathcal {S}}_M$ the collection of such simple integrands, the extension of the integral to a larger subset of ${\mathcal {P}}$ is carried out using the Daniell mean

$$\begin{aligned} \Vert H\Vert _{M,p} :=\Vert H\Vert _{M,p,{\mathbb {P}}}:= \sup _{S\in {\mathcal {S}}_M, |S|\leqslant |H|} \left\| \int _0^T\int _E S(t,x)\,M(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p}\, ,\quad H\in {\mathcal {P}}.\nonumber \\ \end{aligned}$$

(A.1)

A predictable process H is called p-integrable with respect to M if there exists a sequence $(S_n)_{n\geqslant 1}$ of simple integrands with $\Vert H-S_n \Vert _{M,p} \rightarrow 0$ as $n\rightarrow +\infty$. The collection of p-integrable processes is denoted by $L^{1,p}(M)$ (or $L^{1,p}(M,{\mathbb {P}})$ if we want to emphasize the probability measure). The stochastic integral of H with respect to M is then defined as the $L^p$-limit of $\int _0^T\int _E S_n(t,x)\,M(\mathrm {d}t,\mathrm {d}x)$ (which exists and does not depend on the choice of $S_n$). In all notions introduced, the prefix p is suppressed if $p=0$. The constructed integral obeys the dominated convergence theorem, see [4, (2.6)].

Theorem A.1

If $(H_n)_{n\geqslant 1}$ are predictable and converge pointwise to H, and $|H_n|\leqslant H_0$ for all $n\geqslant 1$ and some $H_0\in L^{1,p}(M)$, then $H_n, H \in L^{1,p}(M)$ and $\Vert H-H_n\Vert _{M,p}\rightarrow 0$ as $n\rightarrow +\infty$.

In this paper, we are particularly interested in the case where M is a linear combination of random measures of one of the following forms:

• (a) M is a predictable strict random measure, that is, almost every realization of M is a measure on $[0,T]\times E$ and $t\mapsto \int _0^T \int _E \mathbb {1}_A(s,y) \mathbb {1}_{[0,t]}(s)\,M(\mathrm {d}s,\mathrm {d}y)$ is a predictable process for all $A\in {\mathcal {P}}_M$.

• (b) $M(\mathrm {d}t,\mathrm {d}x) = \int _{z\in {\mathbb {R}}} W(t,x,z)\,{\tilde{J}}(\mathrm {d}t,\mathrm {d}x,\mathrm {d}z)$, where J is an $({\mathcal {F}}_t)_{t\in [0,T]}$-Poisson random measure with intensity measure $\nu (\mathrm {d}t,\mathrm {d}x,\mathrm {d}z)$, ${\tilde{J}} = J-\nu$, and $W\mathbb {1}_{{\tilde{\Omega }}_k}$ is 1-integrable with respect to ${\tilde{J}}$ (a random measure on $E\times {\mathbb {R}}$) in the sense above for every $k\geqslant 1$.

• (c) M is a strict random measure of the form $M(\mathrm {d}t,\mathrm {d}x) = \int _{z\in {\mathbb {R}}} W(t,x,z)\,J(\mathrm {d}t,\mathrm {d}x, \mathrm {d}z)$ where $W\mathbb {1}_{{\tilde{\Omega }}_k}$ is integrable with respect to J for every $k\geqslant 1$.

In these cases, the Daniell mean can be computed (or estimated) explicitly.

Lemma A.2

Let $\Vert X\Vert _{L^p} = {\mathbb {E}}[|X|^p]$ for $0<p<1$ and $\Vert X\Vert _{L^p} = ({\mathbb {E}}[|X|^p])^{\frac{1}{p}}$ be the usual $L^p$-norm for $p\geqslant 1$.

• 1. In the case (a) above, we have for every $0<p<+\infty$ and $H\in {\mathcal {P}}$,

  $$\begin{aligned} \Vert H\Vert _{M,p} = \left\| \int _0^T \int _E |H(t,x)|\,|M|(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p}, \end{aligned}$$

  (A.2)

  where |M| is the total variation measure of M.

• 2. In the case (b) above, there exist for every $p\geqslant 1$ constants $c_p, C_p>0$ such that for all $H\in {\mathcal {P}}$,

  $$\begin{aligned}&c_p \left\| \left( \int _0^T \int _E H^2(t,x)\,[M](\mathrm {d}t,\mathrm {d}x) \right) ^{\frac{1}{2}} \right\| _{L^p} \\&\quad \leqslant \Vert H\Vert _{M,p}\leqslant C_p \left\| \left( \int _0^T \int _E H^2(t,x)\,[M](\mathrm {d}t,\mathrm {d}x) \right) ^{\frac{1}{2}} \right\| _{L^p}, \end{aligned}$$

  where $[M](\mathrm {d}t,\mathrm {d}x) = \int _{z\in {\mathbb {R}}} W^2(t,x,z)\,J(\mathrm {d}t,\mathrm {d}x,\mathrm {d}z)$ is the quadratic variation measure of M. In particular,

  $$\begin{aligned} \Vert H\Vert _{M,p}\leqslant C_p \left( \int _0^T\int _E\int _{\mathbb {R}}\Vert H(t,x) W(t,x,z)\Vert _{L^p}^p\,\nu (\mathrm {d}t,\mathrm {d}x,\mathrm {d}z)\right) ^{\frac{1}{p}}. \end{aligned}$$

  (A.3)

• 3. In the case (c) above, we have for $0<p\leqslant 1$ and $H\in {\mathcal {P}}$,

  $$\begin{aligned} \Vert H\Vert _{M,p} \leqslant \int _0^T \int _E \int _{\mathbb {R}}\Vert H(t,x)W(t,x,z)\Vert _{L^p}\, \nu (\mathrm {d}t,\mathrm {d}x,\mathrm {d}z). \end{aligned}$$

Proof

For the first statement, the “$\leqslant$”-part follows from

$$\begin{aligned} \left| \int _0^T\int _E S(t,x)\,M(\mathrm {d}t,\mathrm {d}x)\right|&\leqslant \int _0^T\int _E |S(t,x)|\,|M|(\mathrm {d}t,\mathrm {d}x)\\&\leqslant \int _0^T\int _E |H(t,x)|\,|M|(\mathrm {d}t,\mathrm {d}x) \end{aligned}$$

for all S with $|S|\leqslant |H|$. For the “$\geqslant$”-part, observe that the right-hand side of (A.2) equals $\Vert H\Vert _{|M|,p}$ by dominated convergence. Next, consider the measure $\mu (\mathrm {d}\omega ,\mathrm {d}t,\mathrm {d}x) = M(\omega ,\mathrm {d}t,\mathrm {d}x)\,{\mathbb {P}}(\mathrm {d}\omega )$ on ${\mathcal {P}}$ and let $D(\omega ,t,x)$ be its Radon–Nikodym derivative with respect to $|\mu |(\mathrm {d}\omega ,\mathrm {d}t,\mathrm {d}x)$. Then D is predictable, $|D|\equiv 1$ and $|M|(\omega ,\mathrm {d}t,\mathrm {d}x)=D(\omega ,t,x)\,M(\omega ,\mathrm {d}t,\mathrm {d}x)$. Hence,

$$\begin{aligned}&\sup _{S\in {\mathcal {S}}_M, |S|\leqslant |H|} \left\| \int _0^T\int _E S(t,x)\,|M|(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p}= \sup _{S\in {\mathcal {S}}_M, 0\leqslant S\leqslant |H|} \left\| \int _0^T\int _E S(t,x)\,|M|(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p} \\&\qquad = \sup _{S\in {\mathcal {S}}_M, 0\leqslant S\leqslant |H|} \left\| \int _0^T\int _E S(t,x)D(t,x)\,M(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p}\\&\qquad \leqslant \sup _{S\in {\mathcal {S}}_M, 0\leqslant S\leqslant |H|} \left\| \int _0^T\int _E S(t,x)\,M(\mathrm {d}t,\mathrm {d}x) \right\| _{L^p} \leqslant \Vert H\Vert _{M,p}, \end{aligned}$$

and (A.2) is proved

For the second statement, we observe that $t\mapsto \int _0^T\int _E S(s,y)\mathbb {1}_{[0,t]}(s)\,M(\mathrm {d}s,\mathrm {d}y)$ is a local martingale for all $S\in {\mathcal {S}}_M$, see [4, Proposition 4.9(b)]. So the statement for $S\in {\mathcal {S}}_M$ follows from the Burkholder–Davis–Gundy inequalities. The general case is again a consequence of the dominated convergence theorem. Inequality (A.3) can be proved along the lines of the third statement, which we now establish.

Let $(T_i,X_i,Z_i)_{i\geqslant 1}$ be the points of J in $[0,T]\times E\times {\mathbb {R}}$. Then, using $(x+y)^p \leqslant x^p+y^p$ for $x,y>0$ and $0<p\leqslant 1$, we get

$$\begin{aligned}&{\mathbb {E}}\left[ \left| \int _0^T \int _E S(t,x)\,M(\mathrm {d}t,\mathrm {d}x) \right| ^p \right] = {\mathbb {E}}\left[ \left| \sum _{i\geqslant 1} S(T_i,X_i)W(T_i,X_i,Z_i) \right| ^p \right] \\&\qquad \leqslant {\mathbb {E}}\left[ \sum _{i\geqslant 1} | S(T_i,X_i)W(T_i,X_i,Z_i) |^p \right] \\&\qquad \leqslant {\mathbb {E}}\left[ \int _0^T \int _E \int _{\mathbb {R}}|H(t,x)W(t,x,z)|^p\,J(\mathrm {d}t,\mathrm {d}x,\mathrm {d}z) \right] \\&\qquad ={\mathbb {E}}\left[ \int _0^T \int _E \int _{\mathbb {R}}|H(t,x)W(t,x,z)|^p\,\nu (\mathrm {d}t,\mathrm {d}x,\mathrm {d}z) \right] , \end{aligned}$$

and the proof is complete. $\square$

With the help of the Daniell mean, one can obtain the following stochastic Fubini theorem.

Theorem A.3

Let $(A,{\mathcal {A}},\mu )$ be a $\sigma$-finite measure space, M be an $L^p$-random measure for some $p>0$, and H be a ${\mathcal {P}}\otimes {\mathcal {A}}$-measurable function.

• 1. If $p\geqslant 1$ and $\int _A \Vert H(\cdot ,\cdot ,a)\Vert _{M,p}\,\mu (\mathrm {d}a)<+\infty$, then

  $$\begin{aligned} \int _A \left( \int _0^T \int _E H(t,x,a)\, M(\mathrm {d}t,\mathrm {d}x)\right) \,\mu (\mathrm {d}a) \end{aligned}$$

  and

  $$\begin{aligned} \int _0^T \int _E \left( \int _A H(t,x,a)\,\mu (\mathrm {d}a) \right) \,M(\mathrm {d}t,\mathrm {d}x) \end{aligned}$$

  are equal almost surely, and all integrals involved are well defined.

• 2. If $0<p\leqslant 1$ and M is a random measure as in (c) above, the conclusion of the first part continues to hold if

  $$\begin{aligned} \int _0^T\int _E \int _{\mathbb {R}}\left\| \int _A |H(t,x,a)|\,\mu (\mathrm {d}a) |W(t,x,z)| \right\| _{L^p} \,\nu (\mathrm {d}t,\mathrm {d}x,\mathrm {d}z)<+\infty .\qquad \end{aligned}$$

  (A.4)

The first part has been proved in [26, Theorem 2] for $p=1$ (see also [5] for processes indexed only by time), but is obviously also valid for $p>1$ by the monotonicity of $L^p$-norms. In particular, Lemma A.2 can be used to verify the integrability assumption. The second part follows from the ordinary Fubini theorem (M is a strict random measure here) together with an argument as in the proof of third part of Lemma A.2.

A last result that we need relates to the possibility of recovering $L^2$-integrability from $L^p$-integrability, $0\leqslant p<2$, upon an equivalent change of probability measure. For semimartingales, this result is well known, see [33, Chapter IV, Theorem 34], for example. For random measures, it is proved in [25, Corollary of Theorem 2].

Theorem A.4

If M is an $L^p$-random measure and $H\in L^{1,p}(M)$ for some $0\leqslant p<2$, then there exists a probability measure ${\mathbb {Q}}$ that is equivalent to ${\mathbb {P}}$ on ${\mathcal {F}}$ such that $\frac{\mathrm {d}{\mathbb {Q}}}{\mathrm {d}{\mathbb {P}}}$ is bounded, $\frac{\mathrm {d}{\mathbb {P}}}{\mathrm {d}{\mathbb {Q}}} \in L^{\frac{p}{2-p}}({\mathbb {P}})$, M is an $L^2$-random measure under ${\mathbb {Q}}$, and $H\in L^{1,2}(M,{\mathbb {Q}})$.