We consider two-dimensional stochastic differential equations, describing the motion of a slowly and periodically forced overdamped particle in a double-well potential, subjected to weak additive noise. We give sharp asymptotics of Eyring-Kramers type for the expected transition time from one potential well to the other one. Our results cover a range of forcing frequencies that are large with respect to the maximal transition rate between potential wells of the unforced system. The main difficulty of the analysis is that the forced system is non-reversible, so that standard methods from potential theory used to obtain Eyring-Kramers laws for reversible diffusions do not apply. Instead, we use results by Landim, Mariani and Seo that extend the potential-theoretic approach to non-reversible systems.
We prove an Eyring-Kramers law for the small eigenvalues and mean first-passage times of a metastable Markovian jump process which is invariant under a group of symmetries. Our results show that the usual Eyring-Kramers law for asymmetric processes has to be corrected by a factor computable in terms of stabilisers of group orbits. Furthermore, the symmetry can produce additional Arrhenius exponents and modify the spectral gap. The results are based on representation theory of finite groups.
We consider non-reversible random walks evolving on a potential field in a bounded domain of $\mathbb{R}^d$. We describe the complete metastable behavior of the random walk among the landscape of valleys, and we derive the Eyring-Kramers formula for the mean transition time from a metastable set to a stable set.
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 formu…
In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $α$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $α$ vanishes on T is also considered.
In this article, we prove the Eyring-Kramers formula for non-reversible metastable diffusion processes that have a Gibbs invariant measure. Our result indicates that non-reversible processes exhibit faster metastable transitions between neighborhoods of local minima, compared to the reversible process considered in [Bovier, Eckhoff, Gayrard, and Klein, J. Eur. Math. Soc. 6: 399-424, 2004]. Therefore, by adding non-reversibility to the model, we can indeed accelerate the metastable transition. Our proof is based on the potential theoretic approach to metastability through accurate estimation of…
We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions.
In this paper we consider the mean transition time of an over-damped Brownian particle between local minima of a smooth potential. When the minima and saddles are non-degenerate this is in the low noise regime exactly characterized by the so called Eyring-Kramers law and gives the mean transition time as a quantity depending on the curvature of the minima and the saddle. In this paper we find an extension of the Eyring-Kramers law giving an upper bound on the mean transition time when both the minima/saddles are degenerate (flat) while at the same time covering multiple saddles at the same hei…
The Eyring-Kramers law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. In the weak-noise limit, the transition time is to leading order exponential in the potential difference to overcome. This exponential is corrected by a prefactor which depends on the principal curvatures of the potential at the starting minimum and at the highest saddle crossed by an optimal transition path. The Eyring-Kramers law, however, does not hold whenever one of these principal curvatures vanishes, since it would predict a vanishing or infinite tr…
We study the convergence rate of continuous-time simulated annealing $(X_t; \, t \ge 0)$ and its discretization $(x_k; \, k =0,1, \ldots)$ for approximating the global optimum of a given function $f$. We prove that the tail probability $\mathbb{P}(f(X_t) > \min f +δ)$ (resp. $\mathbb{P}(f(x_k) > \min f +δ)$) decays polynomial in time (resp. in cumulative step size), and provide an explicit rate as a function of the model parameters. Our argument applies the recent development on functional inequalities for the Gibbs measure at low temperatures -- the Eyring-Kramers law. In the discrete setting…
Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $Ω\subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $Ω$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $Ω$ through each of the saddle points of $f$ on $\partial Ω$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the …
We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs…
We consider a class of parabolic semi-linear stochastic partial differential equations driven by space-time white noise on a compact space interval. Our aim is to obtain precise asymptotics of the transition times between metastable states. A version of the so-called Eyring-Kramers Formula is proven in an infinite dimensional setting. The proof is based on a spatial finite difference discretization of the stochastic partial differential equation. The expected transition time is computed for the finite dimensional approximation and controlled uniformly in the dimension.
Given an energy potential on the Euclidian space, a piecewise deterministic Markov process is designed to sample the corresponding Gibbs measure. In dimension one an Eyring-Kramers formula is obtained for the exit time of the domain of a local minimum at low temperature, and a necessary and sufficient condition is given on the cooling schedule in a simulated annealing algorithm to ensure the process converges to the set of global minima. This condition is similar to the classical one for diffusions and involves the critical depth of the potential. In higher dimension a non optimal sufficient c…
This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof …
We study the metastable dynamics of a discretised version of the mass-conserving stochastic Allen-Cahn equation. Consider a periodic one-dimensional lattice with $N$ sites, and attach to each site a real-valued variable, which can be interpreted as a spin, as the concentration of one type of metal in an alloy, or as a particle density. Each of these variables is subjected to a local force deriving from a symmetric double-well potential, to a weak ferromagnetic coupling with its nearest neighbours, and to independent white noise. In addition, the dynamics is constrained to have constant total m…
The main contribution of the current study is two-fold. First, we investigate the energy landscape of the Ising and Potts models on finite two-dimensional lattices without external fields in the low temperature regime. The complete analysis of the energy landscape of these models was unknown because of its complicated plateau saddle structure between the ground states. We characterize this structure completely in terms of a random walk on the set of sub-trees of a ladder graph. Second, we provide a considerable simplification of the well-known potential-theoretic approach to metastability. In …
We consider a system of $ N \in \mathbb{N} $ mean-field interacting stochastic differential equations that are driven by a single-site potential of double-well form and by Brownian noise. The strength of the noise is measured by a small parameter $ \varepsilon >0$ (which we interpret as the \emph{temperature}), and we suppose that the strength of the interaction is given by $ J>0 $. Choosing the \emph{empirical mean} ($ P:\mathbb{R}^N \rightarrow \mathbb{R} $, $ Px =1/N \sum_i x_i $) as the macroscopic order parameter for the system, we show that the resulting macroscopic Hamiltonian has two g…
