Semi-discrete semi-linear parabolic SPDEs

The main findings of this paper are:

- The \(k\)th moment Lyapunov exponent of \(u\) grows exactly as \(k^2\);
- The following random Radon--Nikodým theorem holds: \[ \lim_{\tau\downarrow 0}\frac{u_{t+\tau}(x)-u_t(x)}{ B_{t+\tau}(x)-B_t(x)}=\sigma(u_t(x))\quad\text{in probability;} \]
- 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
- 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 |