The Annals of Applied Probability

Semi-discrete semi-linear parabolic SPDEs

Nicos Georgiou, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan Shiu

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Consider an infinite system

\[\partial_{t}u_{t}(x)=(\mathscr{L}u_{t})(x)+\sigma(u_{t}(x))\partial_{t}B_{t}(x)\] of interacting Itô diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solution under standard regularity assumptions on the nonlinearity $\sigma$. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the $k$th moment Lyapunov exponent is frequently of sharp order $k^{2}$, in contrast to the continuous-space stochastic heat equation whose $k$th moment Lyapunov exponent can be of sharp order $k^{3}$. When the underlying walk is transient and the noise level is sufficiently low, we prove also that the solution is a.s. uniformly dissipative provided that the initial profile is in $\ell^{1}(\mathbf{Z}^{d})$.

Article information

Ann. Appl. Probab., Volume 25, Number 5 (2015), 2959-3006.

Received: November 2013
Revised: September 2014
First available in Project Euclid: 30 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments
Secondary: 47B80: Random operators [See also 47H40, 60H25] 60H25: Random operators and equations [See also 47B80]

The stochastic heat equation semi-discrete stochastic heat equation discrete space parabolic Anderson model Lyapunov exponents dissipative behavior comparison principle interacting diffusions BDG inequality


Georgiou, Nicos; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan. Semi-discrete semi-linear parabolic SPDEs. Ann. Appl. Probab. 25 (2015), no. 5, 2959--3006. doi:10.1214/14-AAP1065.

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