Abstract
We study spectral Galerkin approximations of an Allen--Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon}$. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring-Kramers formula for the limiting renormalised stochastic PDE. The effect of the "infinite renormalisation" is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring-Kramers law by a renormalised Carleman-Fredholm determinant.
Highlights
Metastability is a common physical phenomenon in which a system spends a long time in metastable states before reaching its equilibrium
The mean transition time between minima is governed by the Eyring–Kramers law [12, 22]: If τ denotes the expected hitting time of a neighbourhood of a local minimum y of the solution of (1.1) started in another minimum x, and under suitable assumptions on the potential V, the Eyring–Kramers law gives the asymptotic expression
A crucial idea is to change point of view with respect to the usual finite dimensional setting as presented in [7, 8] and to regard capacities and partition functions as expectations of random variables under Gaussian measures, which are well-defined in infinite dimension
Summary
Metastability is a common physical phenomenon in which a system spends a long time in metastable states before reaching its equilibrium. As we work in finite dimensions throughout, we can avoid making any use of the analytic tools developed in recent years to deal with singular SPDEs. A crucial idea is to change point of view with respect to the usual finite dimensional setting as presented in [7, 8] and to regard capacities and partition functions as expectations of random variables under Gaussian measures, which are well-defined in infinite dimension. A crucial idea is to change point of view with respect to the usual finite dimensional setting as presented in [7, 8] and to regard capacities and partition functions as expectations of random variables under Gaussian measures, which are well-defined in infinite dimension This idea already appeared in [11] for the analysis of metastability of the Allen-Cahn equation in space dimension d = 1. Some well-known facts about Hermite polynomials and Wick powers are collected in Appendix A
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