On the pitchfork bifurcation for the Chafee-Infante equation with additive noise

• 成果类型: Article
• 署名作者: Blumenthal, Alex; Engel, Maximilian; Neamtu, Alexandra
• 署名单位: University System of Georgia; Georgia Institute of Technology; Free University of Berlin; University of Konstanz
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01235-3
• 发表日期: 2023
• 页码: 603-627
• 关键词: random dynamical-systems; invariant-measures; attractors; STABILITY

摘要

We investigate pitchfork bifurcations for a stochastic reaction diffusion equation perturbed by an infinite-dimensional Wiener process. It is well-known that the random attractor is a singleton, independently of the value of the bifurcation parameter; this phenomenon is often referred to as the destruction of the bifurcation by the noise. Analogous to the results of Callaway et al. (AIHP Prob Stat 53:1548-1574, 2017) for a 1D stochastic ODE, we show that some remnant of the bifurcation persists for this SPDE model in the form of a positive finite-time Lyapunov exponent. Additionally, we prove finite-time expansion of volumewith increasing dimension as the bifurcation parameter crosses further eigenvalues of the Laplacian.