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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.

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Acknowledgements

The authors wish to thank the referees for their useful suggestions to improve the paper.

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Authors and Affiliations

Authors

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Correspondence to David Nualart.

Additional information

Hu is partially supported by a Grant from the Simons Foundation #209206; Nualart is partially supported by the NSF Grant DMS1512891 and the ARO Grant FED0070445.

Kalbasi is supported by a fellowship from the Swiss National Science Foundation.

Appendix

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 \(z\ge 0\). \(\square \)

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Chen, L., Hu, Y., Kalbasi, K. et al. Intermittency for the stochastic heat equation driven by a rough time fractional Gaussian noise. Probab. Theory Relat. Fields 171, 431–457 (2018). https://doi.org/10.1007/s00440-017-0783-z

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  • DOI: https://doi.org/10.1007/s00440-017-0783-z

Keywords

  • Stochastic heat equation
  • Feynman–Kac integral
  • Feynman–Kac formula
  • Time fractional Gaussian noise
  • Fractional calculus
  • Moment bounds
  • Lyapunov exponents
  • Intermittency

Mathematics Subject Classification

  • Primary 60H15
  • Secondary 60G60
  • 35R60