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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • MathSciNet review: 924775