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October 2015 Semi-discrete semi-linear parabolic SPDEs
Nicos Georgiou, Mathew Joseph, Davar Khoshnevisan, Shang-Yuan Shiu
Ann. Appl. Probab. 25(5): 2959-3006 (October 2015). DOI: 10.1214/14-AAP1065

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 $\ell^{1}(\mathbf{Z}^{d})$.

Citation

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Nicos Georgiou. Mathew Joseph. Davar Khoshnevisan. Shang-Yuan Shiu. "Semi-discrete semi-linear parabolic SPDEs." Ann. Appl. Probab. 25 (5) 2959 - 3006, October 2015. https://doi.org/10.1214/14-AAP1065

Information

Received: 1 November 2013; Revised: 1 September 2014; Published: October 2015
First available in Project Euclid: 30 July 2015

zbMATH: 1325.60108
MathSciNet: MR3375892
Digital Object Identifier: 10.1214/14-AAP1065

Subjects:
Primary: 60J60, 60K35, 60K37
Secondary: 47B80, 60H25

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 5 • October 2015
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