On the long time behavior of the stochastic heat equation |
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Authors: | Lorenzo Bertini Giambattista Giacomin |
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Institution: | Dipartimento di Matematica, Università di Roma La Sapienza, P.le A. Moro 2, I-00185 Roma, Italy. e-mail: bertini@mat.uniroma1.it, IT Institut für Angewandte Mathematik der Universit?t Zürich-Irchel, Winterthurerstr. 190, CH-8057 Zürich, Switzerland, CH
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Abstract: | We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum
– the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds
to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain
a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature
via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion
process.
Received: 11 November 1997 / Revised version: 31 July 1998 |
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Keywords: | Mathematics Subject Classification (1991): Primary 60H15 60K35 secondary 82B44 |
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