2018/08/13 by Pavlos Tsatsoulis, Tsatsoulis, Pavlos, Hendrik Weber +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #35K57 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1808.04171
openalex publication_date 2018/08/13 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
We prove an asymptotic coupling theorem for the 2-dimensional Allen--Cahn\nequation perturbed by a small space-time white noise. We show that with\noverwhelming probability two profiles that start close to the minimisers of the\npotential of the deterministic system contract exponentially fast in a suitable\ntopology. In the 1-dimensional case a similar result was shown in\n citeMS88,MOS89.\n It is well-known that in more than one dimension solutions of this equation\nare distribution-valued, and the equation has to be interpreted in a\nrenormalised sense. Formally, this renormalisation corresponds to moving the\nminima of the potential infinitely far apart and making them infinitely deep.\nWe show that despite this renormalisation, solutions behave like perturbations\nof the deterministic system without renormalisation: they spend large stretches\nof time close to the minimisers of the (un-renormalised) potential and the\nexponential contraction rate of different profiles is given by the second\nderivative of the potential in these points.\n As an application we prove an Eyring--Kramers law for the transition times\nbetween the stable solutions of the deterministic system for fixed initial\nconditions.\n