A General Analytical Result for Non-linear SPDE's and Applications

Abstract

Using analytical methods, we prove existence uniqueness and estimates for s.p.d.e. of the type $$ du_t+Au_tdt+f ( t,u_t ) dt+R g(t, u_t ) dt=h(t,x,u_t) dB_t, $$ where $A$ is a linear non-negative self-adjoint (unbounded) operator, $f$ is a nonlinear function which depends on $u$ and its derivatives controlled by $\sqrt{A}u$, $Rg$ corresponds to a nonlinearity involving $u$ and its derivatives of the same order as $Au$ but of smaller magnitude, and the right term contains a noise involving a $d$-dimensional Brownian motion multiplied by a non-linear function. We give a neat condition concerning the magnitude of these nonlinear perturbations. We also mention a few examples and, in the case of a diffusion generator, we give a double stochastic interpretation.