Lp-regularity theory for semilinear stochastic partial differential equations with multiplicative white noise

Abstract

We establish the $L_p$-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise: $$ du = (a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^{i}|u|^λu_{x^i})dt + σ^k(u)dw_t^k,\quad (t,x)\in(0,\infty)\times\bR^d; \quad u(0,\cdot) = u_0, $$ where $\lambda >0$, the set $\{w_t^k,k=1,2,\dots \}$ is a set of one-dimensional independent Wiener processes, and the function $u_0=u_0(\omega ,x)$ is a nonnegative random initial data. The coefficients $a^{ij},b^i,c$ depend on $(\omega ,t,x)$, and ${\bar{b}}^i$ depends on $(\omega ,t,x^1,\dots ,x^{i-1},x^{i+1},\dots ,x^d)$. The coefficients $a^{ij},b^i,c,{\bar{b}}^i$ are uniformly bounded and twice continuous differentiable. The leading coefficient $a$ satisfies ellipticity condition. Depending on the diffusion coefficient ${\sigma }^k(u)$, we consider two different cases; (i) $\lambda \in (0,\infty )$ and ${\sigma }^k(u)$ has Lipschitz continuity and linear growth in $u$, (ii) $\lambda ,{\lambda }_0\in (0,1/d)$ and ${\sigma }^k(u)={\mu }^k|u{|}^{1+{\lambda }_0}$ (${\sigma }^k(u)$ is super-linear). Each case has different regularity results. For example, in the case of $(i)$, for $\epsilon >0$ $$u \in C^{1/2 - \varepsilon,1 - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T<\infty, $$ almost surely. On the other hand, in the case of $(ii)$, if $\lambda ,{\lambda }_0\in (0,1/d)$, for $\epsilon >0$ $$ u \in C^{\frac{1-(λd) \vee (λ_0 d)}{2} - \varepsilon,1-(λd) \vee (λ_0 d) - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T<\infty $$ almost surely. It should be noted that $\lambda$ can be any positive number and the solution regularity is independent of nonlinear terms in case $(i)$. In case $(ii)$, however, $\lambda ,{\lambda }_0$ should satisfy $\lambda ,{\lambda }_0\in (0,1/d)$ and the regularities of the solution are affected by $\lambda ,{\lambda }_0$ and $d$.

arXiv comment28 pages

References in corpus (2)

• Super-Brownian motion as the unique strong solution to an SPDE
• Boundary behavior and interior Hölder regularity of solution to nonlinear stochastic partial differential equations driven by space-time white noise