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On the long time behavior of the stochastic heat equation
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  • Published: June 1999

On the long time behavior of the stochastic heat equation

  • Lorenzo Bertini1 &
  • Giambattista Giacomin2 

Probability Theory and Related Fields volume 114, pages 279–289 (1999)Cite this article

  • 352 Accesses

  • 18 Citations

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma La Sapienza, P.le A. Moro 2, I-00185 Roma, Italy. e-mail: bertini@mat.uniroma1.it, , , , , , IT

    Lorenzo Bertini

  2. Institut für Angewandte Mathematik der Universität Zürich-Irchel, Winterthurerstr. 190, CH-8057 Zürich, Switzerland, , , , , , CH

    Giambattista Giacomin

Authors
  1. Lorenzo Bertini
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  2. Giambattista Giacomin
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Received: 11 November 1997 / Revised version: 31 July 1998

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Bertini, L., Giacomin, G. On the long time behavior of the stochastic heat equation. Probab Theory Relat Fields 114, 279–289 (1999). https://doi.org/10.1007/s004400050226

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  • Issue Date: June 1999

  • DOI: https://doi.org/10.1007/s004400050226

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  • Mathematics Subject Classification (1991): Primary 60H15, 60K35; secondary 82B44
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