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
Mathrew Joseph
Dept. Probab. Statist.
Univ. Sheffield
Sheffield, S3
Davar Khoshnevisan
Dept. Mathematics
Univ. of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090
Shang-Yuan Shiu
Dept. Mathematics
National Central University
Jhongli City
Taoyuan County, 32001, Taiwan
Last Update: November 9, 2013
© 2013 - Nicos Georgiou, Mathew Joseph, D. Khoshnevisan, and Shang-Yuan Shiu