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 Electronic Journal of Probability > Vol. 9 (2004) > Paper 23 open journal systems 


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.


Full text: PDF

Pages: 674-709

Published on: October 11, 2004


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Electronic Journal of Probability. ISSN: 1083-6489