# Preprint: Semi-discrete semi-linear parabolic SPDEs

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

Abstract. Consider the semi-discrete semi-linear Itô stochastic heat equation, $\partial_t u_t(x) = (\mathscr{L} u_t)(x) + \sigma(u_t(x))\, \partial_t B_t(x),$ started at a non-random bounded initial profile $$u_0:{\bf Z}^d\to{\bf R}_+$$. Here: $$\{B(x)\}_{x\in{\bf Z}^d}$$ is an field of i.i.d. Brownian motions; $$\mathscr{L}$$ denotes the generator of a continuous-time random walk on $${\bf Z}^d$$; and $$\sigma:{\bf R}\to{\bf R}$$ is Lipschitz continuous and non-random with $$\sigma(0)=0$$.

The main findings of this paper are:

1. The $$k$$th moment Lyapunov exponent of $$u$$ grows exactly as $$k^2$$;
2. The following random Radon--Nikodým theorem holds: $\lim_{\tau\downarrow 0}\frac{u_{t+\tau}(x)-u_t(x)}{ B_{t+\tau}(x)-B_t(x)}=\sigma(u_t(x))\quad\text{in probability;}$
3. Under some non-degeneracy conditions, there often exists a "scale function" $$S:{\bf R}\to(0\,,\infty)$$, such that the finite-dimensional distributions of $$x\mapsto\{S(u_{t+\tau}(x))-S(u_t(x))\}/\sqrt\tau$$ converge to those of white noise as $$\tau\downarrow 0$$; and
4. When the underlying walk is transient and the "noise level is sufficiently low," the solution can be a.s. uniformly dissipative provided that $$u_0\in\ell^1({\bf Z}^d)$$.

Keywords. The stochastic heat equation; interacting diffusions.

AMS Classification (2000) Primary: 60J60, 60K35, 60K37; Secondary: 47B80, 60H25

Support. Research supported in part by the NSF grants DMS-0747758 (N.G.; M.J.) and DMS-1006903 (M.J.; D.K.), the NSC grant 101-2115-M-008-10-MY2 (S.-Y.S.), and the NCU grant 102G607-3 (S.-Y.S.).

Pre/E-Prints. This paper is available in

 Nicos Georgiou Dept. Mathematics Univ. of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090 georgiou@math.utah.edu Mathrew Joseph Dept. Probab. Statist. Univ. Sheffield Sheffield, S3 7RH, UK m.joseph@shef.ac.uk Davar Khoshnevisan Dept. Mathematics Univ. of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090 davar@math.utah.edu Shang-Yuan Shiu Dept. Mathematics National Central University Jhongli City Taoyuan County, 32001, Taiwan shiu@math.ncu.edu.tw Last Update: November 9, 2013 © 2013 - Nicos Georgiou, Mathew Joseph, D. Khoshnevisan, and Shang-Yuan Shiu