Nonlinear Parabolic Equations with Cc Adll Ag Noise Stochastic Parabolic Equations

We dispense with semimartingale (and Dirichlet process) assumptions while investigating arbitrary-order stochastic semi-linear parabolic equations. The emergence of fractional L evy processes in pressing applications like communication networks and mathematical nance highlights the need for studying stochastic evolutionary equations under general noise conditions. Our principle result states that there exists a unique mild CN -valued solution tody(t; x) = 24 X jmj 2pAm(t; x)@m x y(t; x) + (t; y(t))(x)35 dt + d t(x)u(t; x) on [0;1) Rd that is Holder continuous on average, where f t; t 0g and fut; t 0g are any given processes such that t ! tut and t ! t1 1 2p t@tut are DH1 [0;1)-valued respectivelyDH2 [0;1)-valued processes. H1;H2 are Hilbert spaces of functions de ned within. Naturally, due to the full generality allowed for , we will have to specify how to interpret our stochastic integrals and mild solutions. In fact, our purely analytical methods are general enough to allow to be a higher variation process like an iterated Brownian motion. Some novel and technical results on bounds for fundamental solutions to parabolic equations as well as for approximations in some extended Sobolev-like spaces are also given.