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

Abstract

We establish the Lp-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise:du=(aijuxixj+biuxi+cu+b¯i|u|λuxi)dt+σk(u)dwtk,(ω,t,x)∈Ω×(0,∞)×Rd;u(0,⋅)=u0, where λ>0, the set {wtk,k=1,2,…} is a set of one-dimensional independent Wiener processes, and the function u0=u0(ω,x) is nonnegative random initial data. The coefficients aij,bi,c depend on (ω,t,x), and b¯i depends on (ω,t,x1,…,xi−1,xi+1,…,xd). The coefficients aij,bi,c,b¯i are uniformly bounded and twice continuously differentiable. The leading coefficient a satisfies the ellipticity condition. Depending on the diffusion coefficient σk(u)=σk(ω,t,x,u), we consider two cases:(i)λ∈(0,∞) and σk(u) has a Lipschitz continuity and linear growth in u.(ii)λ,λ0∈(0,1/d) and σk(ω,t,x,u)=μk(ω,t,x)|u|1+λ0 with |∑k|μk(ω,t,⋅)|2|C(Rd)<∞ for all (ω,t)∈Ω×[0,∞). We obtain the existence, uniqueness, Lp regularity, and Hölder regularity of the solution. It should be noted that each case has different regularity results. For example, in the case of (i), for ε>0u∈Ct,x1/2−ε,1−ε([0,T]×Rd)∀T<∞, almost surely. On the other hand, in the case of (ii), if λ,λ0∈(0,1/d), for ε>0u∈Ct,x1−(λd)∨(λ0d)2−ε,1−(λd)∨(λ0d)−ε([0,T]×Rd)∀T<∞ almost surely. It should be noted that λ can be any positive number and that the solution regularity is independent of the nonlinear terms in case (i). In case (ii), however, λ and λ0 should satisfy λ,λ0∈(0,1/d), and the regularities of the solution are affected by λ,λ0 and d. This difference is due to the need to use a different proof method for the super-linear diffusion coefficient σk(u).