An L-p-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

Abstract

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: du = ((a) over tilde (ij) (omega, t)u(xi xj) + f) dt + g(k)d w(t)(k), t is an element of (0, T); u(0, .) = 0, (1) where T is an element of(0, infinity), w(k) (k = 1, 2, ...) are independent Wiener processes, ((a) over bar (ij) (omega, t)) is a (predictable) nonnegative sysmeric matrix valued stochastic process such that kappa vertical bar xi vertical bar(2) <= (a) over bar (ij) (omega,t)xi(i)xi(j) <= K vertical bar xi vertical bar(2) for all (omega, t, xi) is an element of Omega x (0, T) x R-d for some kappa, K is an element of(0, infinity), f is an element of L-p ((0, T) x R-d, dt x dx; L-r(Omega, F, dP) and g, g(x). L-p ((0, T) x R-d, dt x dx; L-r(Omega, F, dP; l(2)) with 2 <= r <= p < infinity and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate integral(T)(0) integral(d)(R) (E[vertical bar u(t, x)(r)](p/r) dxdt + integral(T)(0) integral(d)(R) (E[vertical bar u(xx) (t, x)vertical bar(r)](p/r) dxdt <= N(integral(T)(0) integral(d)(R) (E[vertical bar f(t, x)vertical bar(r)])(p/r) dxdt + integral(T)(0) integral(d)(R) (E[vertical bar g(t, x)vertical bar(r)(l2))(p/r) dxdt where N is a positive constant depending only on d, p, r, kappa, K, and T. As an application, for the solution u to (1), the r th moment mr (t, x) := E vertical bar u(t, x)vertical bar(r) is in the parabolic Sobolev space W-p/r(1,2) ((0, T) x R-d ).