Stochastic heat equation with general rough noise

Abstract

We study the well-posedness of a nonlinear one dimensional stochastic heat equation driven by Gaussian noise: $\frac{\partial \mathit{u}}{\partial \mathit{t}}=\frac{{\partial }^2\mathit{u}}{\partial {\mathit{x}}^2}+\mathit{\sigma }(\mathit{u})\dot{\mathit{W}}$, where $\dot{\mathit{W}}$ is white in time and fractional in space with Hurst parameter $\mathit{H}\in (\frac{1}{4},\frac{1}{2})$. In a recent paper (Ann. Probab. 45 (2017) 4561–616) by Hu, Huang, Lê, Nualart and Tindel a technical and unusual condition of $\mathit{\sigma }(0)=0$ was assumed which is critical in their approach. The main effort of this paper is to remove this condition. The idea is to work on a weighted space ${\mathcal{Z}}_{\mathit{\lambda },\mathit{T}}^{\mathit{p}}$ for some power decay weight $\mathit{\lambda }(\mathit{x})={\mathit{c}}_{\mathit{H}}{(1+|\mathit{x}{|}^2)}^{\mathit{H}-1}$. In addition, when $\mathit{\sigma }(\mathit{u})=1$ we obtain the exact asympotics of the solution ${\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})$ as t and x go to infinity. In particular, we find the exact growth of ${sup}_{|\mathit{x}|\le \mathit{L}}|{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|$ and the sharp growth rate for the Hölder coefficients, namely, ${sup}_{|\mathit{x}|\le \mathit{L}}\frac{|{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x}+\mathit{h})-{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|}{|\mathit{h}{|}^{\mathit{\beta }}}$ and ${sup}_{|\mathit{x}|\le \mathit{L}}\frac{|{\mathit{u}}_{\mathrm{add}}(\mathit{t}+\mathit{\tau },\mathit{x})-{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|}{{\mathit{\tau }}^{\mathit{\alpha }}}$.

Nous étudions une équation de chaleur stochastique ā une dimension spatiale non linéaire dirigée par le bruit gaussien : $\frac{\partial \mathit{u}}{\partial \mathit{t}}=\frac{{\partial }^2\mathit{u}}{\partial {\mathit{x}}^2}+\mathit{\sigma }(\mathit{u})\dot{\mathit{W}}$, où $\dot{\mathit{W}}$ est blanc dans le temps et fractionnaire dans le espace avec le paramètre Hurst $\mathit{H}\in (\frac{1}{4},\frac{1}{2})$. Dans un article récent (Ann. Probab. 45 (2017) 4561–616) par Hu, Huang, Lê, Nualart et Tindel une condition technique et inhabituelle $\mathit{\sigma }(0)=0$ a été supposée, ce qui est critique dans leur approche. Le principal effort de ce document est de supprimer cette condition. L’idée est de travailler sur un espace pondéré ${\mathcal{Z}}_{\mathit{\lambda },\mathit{T}}^{\mathit{p}}$ pour un certain poids de décroissance polynomiale $\mathit{\lambda }(\mathit{x})={\mathit{c}}_{\mathit{H}}{(1+|\mathit{x}{|}^2)}^{\mathit{H}-1}$. Lorsque $\mathit{\sigma }(\mathit{u})=1$ nous obtenons les asympotiques exactes de la solution ${\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})$ quand t et x tendent vers l’infini. En particulier, nous trouvons la croissance exacte de ${sup}_{|\mathit{x}|\le \mathit{L}}|{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|$ et la croissance exacte des coefficients de Hölder, c’est-à-dire, ${sup}_{|\mathit{x}|\le \mathit{L}}\frac{|{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x}+\mathit{h})-{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|}{|\mathit{h}{|}^{\mathit{\beta }}}$ et ${sup}_{|\mathit{x}|\le \mathit{L}}\frac{|{\mathit{u}}_{\mathrm{add}}(\mathit{t}+\mathit{\tau },\mathit{x})-{\mathit{u}}_{\mathrm{add}}(\mathit{t},\mathit{x})|}{{\mathit{\tau }}^{\mathit{\alpha }}}$.