SEMI-DISCRETE SEMI-LINEAR PARABOLIC SPDES

• 成果类型: Article
• 署名作者: Georgiou, Nicos; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan
• 署名单位: University of Sussex; University of Sheffield; Utah System of Higher Education; University of Utah; National Central University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1065
• 发表日期: 2015
• 页码: 2959-3006
• 关键词: stochastic heat-equation; ergodic-theorems; BEHAVIOR

摘要

Consider an infinite system partial derivative(t)u(t) (x) = (Lu-t) (x) + sigma (u(t) (x)) partial derivative(t) B-t (x) of interacting Ito 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 kth moment Lyapunov exponent is frequently of sharp order k(2), in contrast to the continuous-space stochastic heat equation whose kth 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 l(1)(z(d)).