The Stochastic Heat Equation with Multiplicative Lévy Noise: Existence, Moments, and Intermittency

Abstract

We study the stochastic heat equation (SHE) $\partial _t u = \frac{1}{2} \Delta u + \beta u \xi$ driven by a multiplicative Lévy noise $\xi$ with positive jumps and coupling constant $\beta >0$, in arbitrary dimension $d\ge 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\ge 3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of u, we further show that the solution has finite pth moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order p of the solution, which are asymptotically sharp in the limit as $\beta \rightarrow 0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity ${\beta }>0$, in any dimension $d\ge 1$. This behavior contrasts with that observed for the SHE on $\mathbb {Z}^d$ and for the SHE on $\mathbb {R}^d$ with Gaussian noise, for which intermittency does not occur in high dimensions for small $\beta$.

Appendix A: Some Technical Results

1.1 A.1. Proof of Theorem 4.3

The result is contained in [75], but only implicitly, so we give a short proof. Let us first check that (4.3) ensures that all integrals in (4.4) are well defined and finite. Applying the BDG inequality and using the subadditivity (5.4) of the function $x\mapsto x^{p/2}$ for $x>0$ iteratively, we deduce that

$$\begin{aligned}{} & {} \mathbb {E}\Biggl [ \biggl |\int _{ \mathbb {X}^N} f({\varvec{w}}) \prod _{j=1}^N M^{\omega }(\textrm{d}w_j) \biggr |^p \Biggr ]\nonumber \\{} & {} \quad \le C_p\,\mathbb {E}\Biggl [\Biggr (\int _\mathbb {X}\Biggl (\int _{\mathbb {X}^{N-1}} f({\varvec{w}}) \prod _{j=1}^{N-1} M^{\omega }(\textrm{d}w_j)\Biggr )^2\,\delta _{\omega }(\textrm{d}w_N)\Biggr )^{\frac{p}{2}} \Biggr ]\nonumber \\{} & {} \quad \le C_p\,\int _\mathbb {X}\mathbb {E}\Biggl [ \biggl |\int _{\mathbb {X}^{N-1}} f({\varvec{w}})\prod _{j=1}^N M^{\omega }(\textrm{d}w_j)\biggr |^p \Biggr ]\,\nu (\textrm{d}w_N)\nonumber \\{} & {} \quad \le \cdots \le C_p^N \int _{ \mathbb {X}^N} |f({\varvec{w}})|^p \,\nu (\textrm{d}w_{1})\cdots \nu (\textrm{d}w_N)<\infty . \end{aligned}$$

(A.1)

These inequalities remain unchanged if ${\omega }$ is replaced by ${\omega }_1,\ldots ,{\omega }_N$ and $\mathbb {E}$ is replaced by $\mathbb {E}^{\otimes N}$.

Now we move to the proof of (4.4), for which we shall prove the second inequality. To obtain the reverse inequality it is sufficient follow the same proof and observe that all estimates are two-sided (i.e., one can always substitute “$\ge$” for “$\le$” if one also replaces 1/C by C). We now consider a state space on which ${\omega }$ is jointly defined with our i.i.d. copies ${\omega }_1,\ldots ,{\omega }_N$ (and ${\omega }$ is independent of $({\omega }_1,\ldots ,{\omega }_N)$). We let ${\bar{\mathbb {P}}}:= \mathbb {P}\otimes \mathbb {P}^{\otimes N}$ denote the associated probability. In analogy with (1.5), we consider a filtration $({\bar{\mathcal {F}}}_t)_{t\ge 0}$ on this state space defined by ${\bar{\mathcal {F}}}_t:= \mathcal {F}_t \otimes \mathcal {F}^{(1)}_t \otimes \cdots \otimes \mathcal {F}^{(N)}_t$, where $\mathcal {F}^{(i)}_t:= \sigma ( {\omega }\cap ([0,t]\cap \mathbb {R}^d \times (0,\infty )) )$ for $i=1,\ldots ,N$. Our result follows from applying the following inequality, valid for $i=1,\ldots , N$, iteratively:

$$\begin{aligned}{} & {} {\bar{\mathbb {E}}}\bigg [ \int _{ \mathbb {X}^N} f({\varvec{w}})\prod _{j=1}^i M^{\omega }(\textrm{d}w_j)\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j)\bigg ]\nonumber \\{} & {} \quad \le \frac{C}{1-p} {\bar{\mathbb {E}}}\bigg [ \int _{ \mathbb {X}^N} f({\varvec{w}})\prod _{j=1}^{i-1} M^{\omega }(\textrm{d}w_j)\prod _{j=i}^N M^{{\omega }_j}(\textrm{d}w_j)\bigg ]. \end{aligned}$$

(A.2)

Let us first spend some time on the first step $i=N$. By elementary properties of Itô integrals, $X_t:=\int _{ \mathbb {X}^N} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_N)\prod _{j=1}^N M^{\omega }(\textrm{d}w_j)$ and $Y_t:=\int _{ \mathbb {X}^N} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_N)(\prod _{j=1}^{N-1} M^{\omega }(\textrm{d}w_j))\, M^{{\omega }_N}(\textrm{d}w_N)$ define $(\bar{{\mathcal {F}}}_t)_{t\ge 0}$-martingales, with $[X]_t= \int _\mathbb {X}(\int _{ \mathbb {X}^{N-1}} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_N)\prod _{j=1}^{N-1} M^{\omega }(\textrm{d}w_j) )^2\,\delta _{{\omega }}(\textrm{d}w_N)$ and $[Y]_t= \int _\mathbb {X}(\int _{ \mathbb {X}^{N-1}} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_N)\prod _{j=1}^{N-1} M^{\omega }(\textrm{d}w_j) )^2\,\delta _{{\omega }_{N}}(\textrm{d}w_N)$ being the corresponding quadratic variation processes. Since ${\omega }$ and ${\omega }_N$ are Poisson random measures with the same $({\bar{{{\mathcal {F}}}}}_t)_{t\ge 0}$-intensity measures (namely $\textrm{d}t\otimes \textrm{d}x\otimes {\lambda }(\textrm{d}z)$), the jump measures associated to [X] and [Y] have the same predictable compensator in $({\bar{{{\mathcal {F}}}}}_t)_{t\ge 0}$. As a result, X and Y are weakly tangential martingales in the sense of [75]. Thus, by [75, Thm. 4.1] and Doob’s inequality, there is $C_p>0$ such that

$$\begin{aligned} {\bar{\mathbb {E}}} [ |X_{\infty }|^p ]\le {\bar{\mathbb {E}}}\bigg [ \sup _{t\ge 0} {|X_t|^p} \bigg ] \le C_p {\bar{\mathbb {E}}}\bigg [ \sup _{t\ge 0} {|Y_t|^p} \bigg ] \le C_p\bigg (\frac{ p}{p-1}\bigg )^p {\bar{\mathbb {E}}}[|Y_\infty |^p].\quad \end{aligned}$$

(A.3)

The reader can check that for the implicit constant $C_p$ from [75, Thm. 4.2], one may take $C_p:=(24p^2)^p [ 2\times 3^{p/2} ( 2^{p/2+1}+14\times 3^{p/2}(28\times 3^{p/2})^{p/2} ) ]$. The expression in brackets comes from the last equation in [75, p. 38] (with ${\varphi }(x)=|x|^{p/2}$ and $c=(14\times 3^{p/2})^{-1}$), while the computation in [75, p. 39], combined with the BDG inequality as in [53, Ch. VII, Thm. 92], entails an additional factor of $(24p^2)^p$. Most importantly, $C_{p}$ is bounded uniformly in $p\in (1,2]$, so that there exists a universal constant C such that for $p\in (1,2]$, ${\bar{\mathbb {E}}} [ |X_{\infty }|^p ]\ \le {C}(p-1)^{-1} {\bar{\mathbb {E}}} [ |Y_{\infty }|^p ],$ which finishes the proof of (A.2) for $i=N$.

In order to iterate and prove (A.2) for $i \le N-1$, we wish to interchange $M^{\omega }(\textrm{d}w_1)\cdots M^{\omega }(\textrm{d}w_{i})$ with $M^{{\omega }_{i+1}}(\textrm{d}w_{i+1})\cdots M^{{\omega }_{N}}(\textrm{d}w_{N})$ as follows:

$$\begin{aligned}{} & {} \int _{ \mathbb {X}^N} f({\varvec{w}})\prod _{j=1}^i M^{\omega }(\textrm{d}w_i)\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j) \nonumber \\{} & {} \quad =\int _{\mathbb {X}^{i}} \Biggl (\int _{\mathbb {X}^{N-i}} f({\varvec{w}})\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j)\Biggr ) \prod _{j=1}^i M^{\omega }(\textrm{d}w_i). \end{aligned}$$

(A.4)

Even though the integral on the r.h.s. is anticipative when considering the filtration $({\bar{\mathcal {F}}}_t)_{t\ge 0}$, we can recover an integral in Itô’s sense by constructing the inner integrals $\int _{\mathbb {X}^{N-i}} f({\varvec{w}})\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j)$ using the filtration $({\bar{\mathcal {F}}}_t)_{t\ge 0}$ and the outer integrals using the filtration ${\bar{\mathcal {F}}}^{(i)}$ defined by ${\bar{\mathcal {F}}}^{(i)}_t:= \mathcal {F}_t \otimes \bigotimes _{j=1}^i \mathcal {F}^{(j)}_t \otimes \bigotimes _{j=i+1}^N \mathcal {F}^{(j)}_\infty$. Note that the inner integrals are ${\bar{{{\mathcal {F}}}}}^{(i)}_0$-measurable. With this convention, we can justify (A.4) as follows: It certainly holds if f is a step function, that is, if f only assumes finitely many values (in that case, the integrals are simply finite sums). For general f, take a sequence of step functions $(f_n)_{n\in \mathbb {N}}$ such that $|f_n|\le |f|$ for all n and $f_n\rightarrow f$ pointwise as $n\rightarrow \infty$. Equation (A.4) holds for $f_n$, and arguing similarly to (A.1) and using dominated convergence, we deduce (A.4). The fact that we are able to interpret the latter integral in Itô’s sense is crucial for the BDG inequality, which was needed in (A.1), to apply. Once (A.4) is established, we can prove (A.2) by considering the weakly tangential martingales (for the filtration $({\bar{\mathcal {F}}}^{(i)}_t)_{t\ge 0}$)

$$\begin{aligned} X^{(i)}_t&:=\int _{\mathbb {X}^{i}} \Biggl (\int _{\mathbb {X}^{N-i}} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_i)\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j)\Biggr )\prod _{j=1}^i M^{{\omega }}(\textrm{d}w_j) \\ Y^{(i)}_t&:=\int _{\mathbb {X}^{i}} \Biggl (\int _{\mathbb {X}^{N-i}} f({\varvec{w}}){\textbf{1}}_{(0,t]}(t_i)\prod _{j=i+1}^N M^{{\omega }_j}(\textrm{d}w_j)\Biggr ) \prod _{j=1}^{i-1} M^{{\omega }}(\textrm{d}w_j) \,M^{{\omega }_i}(\textrm{d}w_{i}), \end{aligned}$$

applying (A.3) and re-arranging the integrals similarly to (A.4).

1.2 A.2. Proof of Lemma 4.4

For $A, u,{\varepsilon }>0$ and $H\in {{\mathcal {S}}}$ that satisfies $|H|\le u|K|$, we have that

$$\begin{aligned}{} & {} \mathbb {P}\Biggl ( \Biggl |\int _\mathbb {X}H({\omega },t,x)\,\xi _{{\omega }_<}(\textrm{d}t,\textrm{d}x) \Biggr |>{\varepsilon }\Biggr ) \le \mathbb {P}_\ge \Bigl ( A u \Vert K\Vert _{\xi _{{\omega }_<},p; \mathbb {P}_{<}}>{\varepsilon }\Bigr )\\{} & {} \quad + \mathbb {P}\Biggl ( \Biggl |\int _\mathbb {X}H({\omega },t,x)\,\xi _{{\omega }_<}(\textrm{d}t,\textrm{d}x) \Biggr |>uA\Vert K\Vert _{\xi _{{\omega }_<},p; \mathbb {P}_{<}}\Biggr ). \end{aligned}$$

The second probability is bounded by

$$\begin{aligned} \mathbb {E}_{\ge }\Biggl [ \frac{1}{u^pA^p\Vert K\Vert ^p_{\xi _{{\omega }_<},p; \mathbb {P}_{<}}} \mathbb {E}_<\left[ \left|\int _\mathbb {X}H({\omega },t,x)\,\xi _{{\omega }_<}(\textrm{d}t,\textrm{d}x) \right|^p \right] \Biggr ]\le A^{-p}. \end{aligned}$$

Therefore, using dominated convergence for the first term, we get

$$\begin{aligned}{} & {} \lim _{u\rightarrow 0}\sup _{H\in {{\mathcal {S}}},|H|\le u|K|}\mathbb {P}\Biggl ( \Biggl |\int _\mathbb {X}H({\omega },t,x)\,\xi _{{\omega }_<}(\textrm{d}t,\textrm{d}x) \Biggr |>{\varepsilon }\Biggr ) \\{} & {} \quad \le \lim _{u\rightarrow 0} \mathbb {E}_\ge [{\textbf{1}}_{\{ A u \Vert K\Vert _{\xi _{{\omega }_<},p; \mathbb {P}_{<}}>{\varepsilon }\}}]+A^{-p}\le A^{-p}. \end{aligned}$$

Sending $A\rightarrow \infty$ shows that the left-hand side is 0, which is equivalent to $\Vert uK\Vert _{\xi _<,0}\rightarrow 0$ as $u\rightarrow 0$. By (4.6), K is integrable with respect to $\xi _{{\omega }_<}$. $\square$