## The Annals of Applied Probability

### Semi-discrete semi-linear parabolic SPDEs

#### Abstract

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

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

Dates
Revised: September 2014
First available in Project Euclid: 30 July 2015

https://projecteuclid.org/euclid.aoap/1438261057

Digital Object Identifier
doi:10.1214/14-AAP1065

Mathematical Reviews number (MathSciNet)
MR3375892

Zentralblatt MATH identifier
1325.60108

#### Citation

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. https://projecteuclid.org/euclid.aoap/1438261057

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