Sharp upper bound on the almost-sure exponential behavior of a stochastic parabolic partial differential equation

Rene Carmona; Frederi G. Viens; S. A. Molchanov

doi:10.1515/rose.1996.4.1.43

Sharp upper bound on the almost-sure exponential behavior of a stochastic parabolic partial differential equation

Rene Carmona, Frederi G. Viens, S. A. Molchanov

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Abstract

Using a Gaussian estimate, we prove an upper bound on the almost-sure large time exponential growth of the solution to the parabolic Anderson model {equation presented} on the lattice Z_d when the potential V(t} x) is a mean zero Gaussian field that is white in time and homogeneous in space. As the diffusivity parameter K tends to 0, our upper bound is of the same order as the lower bound given by Carmona and Molchanov in [2].

Original language	English (US)
Pages (from-to)	43-49
Number of pages	7
Journal	Random Operators and Stochastic Equations
Volume	4
Issue number	1
DOIs	https://doi.org/10.1515/rose.1996.4.1.43
State	Published - Jan 1996

All Science Journal Classification (ASJC) codes

Analysis
Statistics and Probability

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10.1515/rose.1996.4.1.43

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abstract = "Using a Gaussian estimate, we prove an upper bound on the almost-sure large time exponential growth of the solution to the parabolic Anderson model {equation presented} on the lattice Zd when the potential V(t} x) is a mean zero Gaussian field that is white in time and homogeneous in space. As the diffusivity parameter K tends to 0, our upper bound is of the same order as the lower bound given by Carmona and Molchanov in [2].",

author = "Rene Carmona and Viens, {Frederi G.} and Molchanov, {S. A.}",

year = "1996",

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AU - Viens, Frederi G.

AU - Molchanov, S. A.

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N2 - Using a Gaussian estimate, we prove an upper bound on the almost-sure large time exponential growth of the solution to the parabolic Anderson model {equation presented} on the lattice Zd when the potential V(t} x) is a mean zero Gaussian field that is white in time and homogeneous in space. As the diffusivity parameter K tends to 0, our upper bound is of the same order as the lower bound given by Carmona and Molchanov in [2].

AB - Using a Gaussian estimate, we prove an upper bound on the almost-sure large time exponential growth of the solution to the parabolic Anderson model {equation presented} on the lattice Zd when the potential V(t} x) is a mean zero Gaussian field that is white in time and homogeneous in space. As the diffusivity parameter K tends to 0, our upper bound is of the same order as the lower bound given by Carmona and Molchanov in [2].

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Sharp upper bound on the almost-sure exponential behavior of a stochastic parabolic partial differential equation

Abstract

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