Hölder estimates of mild solutions for nonlocal SPDEs

Abstract

Abstract We consider nonlocal PDEs driven by additive white noises on Rd ${\mathbb{R}}^{d}$. For Lq $L^{q}$ integrable coefficients, we derive the existence and uniqueness, as well as Hölder continuity, of mild solutions. Precisely speaking, the unique mild solution is almost surely Hölder continuous with Hölder index 0<θ<(1/2−d/(qα))(1∧α) $0<\theta <(1/2-d/(q \alpha))(1\wedge \alpha)$. Moreover, we show that any order γ(<q) $\gamma (< q)$ moment of Hölder normal for u on every bounded domain of R+×Rd ${\mathbb{R}}_{+}\times {\mathbb{R}}^{d}$ is finite.

Keywords

• Stochastic partial differential equation
• Hölder-estimates