A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise

Abstract

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$du=(a^{ij}{u}_{x^i{x}^j}+b^i{u}_{x^i}+cu)dt+\xi |u{|}^{1+\lambda }dF,(t,x)\in (0,\infty )\times {\mathbb{R}}^d,$$ where $\lambda \ge 0$ and the coefficients depend on $(\omega ,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $\lambda$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$.

27 pages