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Random perturbations of reaction-diffusion equations: the quasideterministic approximation


Author: Mark I. Freidlin
Journal: Trans. Amer. Math. Soc. 305 (1988), 665-697
MSC: Primary 35K57; Secondary 35R60, 60H15, 60J60
MathSciNet review: 924775
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Abstract: Random fields $ {u^\varepsilon }(t,\,x) = (u_1^\varepsilon (t,\,x), \ldots ,u_n^\varepsilon (t,\,x))$, defined as the solutions of a system of the PDE due.

$\displaystyle \frac{{\partial u_k^\varepsilon }} {{\partial t}} = {L_k}u_k^\var... ...x;\,u_1^\varepsilon , \ldots ,u_n^\varepsilon ) + \varepsilon {\zeta _k}(t,\,x)$

are considered. Here $ {L_k}$ are second-order linear elliptic operators, $ {\zeta _k}$ are Gaussian white-noise fields, independent for different $ k$, and $ \varepsilon $ is a small parameter. The most attention is given to the problem of determining the behavior of the invariant measure $ {\mu ^\varepsilon }$ of the Markov process $ u_t^\varepsilon = (u_1^\varepsilon (t,\, \cdot ), \ldots ,u_n^\varepsilon (t,\, \cdot ))$ in the space of continuous functions as $ \varepsilon \to 0$, and also of describing transitions of $ u_t^\varepsilon $ between stable stationary solutions of nonperturbed systems of PDE. The behavior of $ {\mu ^\varepsilon }$ and the transitions are defined by large deviations for the field $ {u^\varepsilon }(t,\,x)$.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0924775-7
Article copyright: © Copyright 1988 American Mathematical Society