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=(aijuxixj+biuxi+cu)dt+ξ|u|1+λdF,(t,x)∈(0,∞)×Rd, where λ≥0 and the coefficients depend on (ω,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 λ, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of F and the spatial dimension d.