A General Analytical Result for Non-linear SPDE's and Applications
Laurent Denis, Université du Maine, France
L. Stoica, University of Bucharest, Romania
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.
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