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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Random perturbations of reaction-diffusion equations: the quasideterministic approximation
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by Mark I. Freidlin PDF
Trans. Amer. Math. Soc. 305 (1988), 665-697 Request permission

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. \[ \frac {{\partial u_k^\varepsilon }} {{\partial t}} = {L_k}u_k^\varepsilon + {f_k}(x; u_1^\varepsilon , \ldots ,u_n^\varepsilon ) + \varepsilon {\zeta _k}(t, x)\] 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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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 305 (1988), 665-697
  • MSC: Primary 35K57; Secondary 35R60, 60H15, 60J60
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0924775-7
  • MathSciNet review: 924775