L^p-Theory of semi-linear SPDEs on general measure spaces and applications

In general settings, applying evolutional semigroup arguments, we prove the existence and uniqueness of L^p-solutions to semi-linear SPDEs of the typed u (t, x) = [L u (t, x) + f (t, x, u (t))] d t + under(∑, k) g^k (t, x, u (t)) d w_t^k, u (0, x) = u₀ (x), x ∈ E, where L is an unbounded linear negative operator on L^p (E, B, μ), {w_t^k ; t ≥ 0, k = 1, 2, ...} is a sequence of independent Brownian motions, and (E, B, μ) is a general measure space. We also discuss the regularities of solutions in Sobolev spaces. Moreover, a time discretized approximation for above equation is proved to convergence in Hölder spaces. As applications, we study several classes of solutions for different types SPDEs on abstract Wiener space and Riemannian manifold.