Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise

Abstract

This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $H\in (0,1/2)$. We establish the Feynman–Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the $L^p(\Omega )$ moments of the solution.

Appendix

Lemma 4.4

For all $a, b, u,v,w >0$, if $u+v\le w+1/2$ and $w>1/2$, then

$$\begin{aligned} \sup _{n\in \mathbb {N}}\frac{\Gamma (an+u)\Gamma (bn+v)}{\Gamma ((a+b)n+w)}<\infty . \end{aligned}$$

Proof

We only need to show that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\Gamma (an+u)\Gamma (bn+v)}{\Gamma ((a+b)n+w)}<\infty . \end{aligned}$$

By Stirling’s formula (see [17, 5.11.3 or 5.11.7]), as n is large, we see that

$$\begin{aligned} \frac{\Gamma (an+u)\Gamma (bn+v)}{\Gamma ((a+b)n+w)}&\approx \sqrt{\pi }\exp \Big \{(an+u-1/2)\log (an)\\&\quad +(bn+v-1/2)\log (bn)\\&\quad -((a+b)n+w-1/2)\log ((a+b)n)\Big \}. \end{aligned}$$

Denote the right-hand side of the above quantity by $I_n$. By the supper-additivity of $f(x)=x\log x$, namely $f(x+y)\ge f(x)+f(y)$ for all $x,y\ge 0$, we see that

$$\begin{aligned} I_n\le \sqrt{\pi }\exp \Big \{ (u-1/2)\log (an)+ (v-1/2)\log (bn)-(w-1/2)\log ((a+b)n) \Big \}. \end{aligned}$$

Because $w>1/2$, we can apply the inequality $\log ((a+b)n)\ge \frac{1}{2}[\log (an)+\log (bn)]$ to obtain that

$$\begin{aligned} I_n\!\le&\,\sqrt{\pi }\exp \Big \{ (u\!-\!1/4\!-\!w/2)\log a\!+\! (v\!-\!1/4\!-\!w/2)\log b \!+\!(u\!+\!v\!-\!w\!-\!1/2)\!\log n \Big \}\\&= C n^{u+v-w-1/2} \!\le \!C,\quad \text {for all } n\in \mathbb {N}, \end{aligned}$$

where the last inequality is due to the assumption that $u+v-w-1/2\le 0$. $\square$

Let $E_{\alpha ,\beta }(z)$ be the Mittag–Leffler function

$$\begin{aligned} E_{\alpha ,\beta }(z)=\sum _{n=0}^\infty \frac{z^n}{\Gamma (\alpha n+\beta )}, \quad \mathfrak {R}\alpha >0,\, \beta \in \mathbb {C},\, z\in \mathbb {C}. \end{aligned}$$

Lemma 4.5

(Theorem 1.3 p. 32 in [19]) If $0<\alpha <2$, $\beta$ is an arbitrary complex number and $\mu$ is an arbitrary real number such that

$$\begin{aligned} \pi \alpha /2<\mu <\pi \wedge (\pi \alpha )\;, \end{aligned}$$

then for an arbitrary integer $p\ge 1$ the following expression holds:

$$\begin{aligned} E_{\alpha ,\beta }(z)= & {} \frac{1}{\alpha } z^{(1-\beta )/\alpha } \exp \left( z^{1/\alpha }\right) \\&-\sum _{k=1}^p \frac{z^{-k}}{\Gamma (\beta -\alpha k)} + O\left( |z|^{-1-p}\right) ,\, |z|\rightarrow \infty ,\, |\arg (z)|\le \mu \,. \end{aligned}$$

Lemma 4.6

For all $\alpha >0$ and $\beta \le 1$, there exists some constant $C=C_{\alpha ,\beta }\ge 1$ such that

$$\begin{aligned} E_{\alpha ,\beta }(z) \le C \exp \left\{ C z^{1/\alpha }\right\} ,\quad \text {for all } z\ge 0. \end{aligned}$$

Proof

By Lemma 4.5, we see that for some constants $C_{\alpha ,\beta }'>0$ and $C_{\alpha ,\beta }\ge 1$,

$$\begin{aligned} E_{\alpha ,\beta }(z)\le C_{\alpha ,\beta }'\left( 1+z^{(1-\beta )/\alpha } \exp \left( z^{1/\alpha }\right) \right) \le C_{\alpha ,\beta }\exp \left( C_{\alpha ,\beta } z^{1/\alpha }\right) , \end{aligned}$$

for all $z\ge 0$. $\square$