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

Using analytical methods, we prove existence uniqueness and estimates for s.p.d.e. of the type duₜ+Auₜdt+f ( t,uₜ ) dt+R g(t, uₜ ) dt=h(t,x,uₜ) dBₜ, 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 √Au, $Rg$ 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.