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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
DOI: https://doi.org/10.1090/S0002-9947-1988-0924775-7
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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  • [1] M. C. Agranovich and M. I. Višik, Elliptic problems with parameter and parabolic problems of general form, Uspehi Mat. Nauk 19 (1964), 53-161. (Russian) MR 0192188 (33:415)
  • [2] R. Azencott, Ecole d'eté de probabilités de Saint-Flour VIII-1978 (R. Azencott, Y. Guivarch and R. Gundy, eds.), Lecture Notes in Math., vol. 774, Springer, Berlin, 1980.
  • [3] S. K. Christensen and G. Kallianpur, Stochastic differential equations for neuronal behavior, Center for Stochastic Processes, Univ. of North Carolina, Technical report No. 103, 1985. MR 898262 (89c:60070)
  • [4] W. Faris and G. Jona-Lasinio, Large deviations for a nonlinear heat equation with noise, J. Phys. A 15 (1982), 3025-3055. MR 684578 (84j:81073)
  • [5] X. M. Fernique, J. P. Conze and J.Gani, Ecole d'eté de Saint-Flour IV-1974 (P.-L. Hennequin, ed.), Lecture Notes in Math., vol. 480, Springer, Berlin, 1975.
  • [6] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Springer-Verlag, New York and Berlin, 1984. MR 722136 (85a:60064)
  • [7] M. Freidlin, Random perturbations of infinite dimensional dynamical systems. Report in the Maimonides Conf., Moscow, 1985.
  • [8] -, Random perturbations of infinite dimensional dynamical systems, Abstracts of reports in the 1st World Congress of Bernoulli Society, Vol. II, "Nauka", Moscow, 1986.
  • [9] A. N. Kolmogorov, Zur umkehrbarkeit der statistischen Naturgesetze, Math. Ann. 113 (1937), 766-772. MR 1513121
  • [10] S. Kozlov, Some problems concerning stochastic partial differential equations, Trudy Sem. Petrovsk. 4 (1970), 147-172. (Russian)
  • [11] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York and Berlin, 1983. MR 688146 (84d:35002)
  • [12] L. Volevich and B. Panejah, Some spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk 20 (1965), 3-74. (Russian) MR 0174970 (30:5160)
  • [13] J. B. Walsh, A stochastic model of neutral response, Adv. in Appl. Probab. 13 (1981). MR 612203 (82f:92020)
  • [14] -, An introduction to stochastic partial differential equations, Preprint.
  • [15] A. Wentzell and M. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys 25 (1970), 1-55. MR 0267221 (42:2123)

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Additional Information

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

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