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Register a new userMetastability of one-dimensional, non-reversible diffusions with periodic boundary conditions
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We consider small perturbations of a dynamical system on the one-dimensional torus. We derive sharp estimates for the pre-factor of the stationary state, we examine the asymptotic behavior of the solutions of the Hamilton-Jacobi equation for the pre-factor, we compute the capacities between disjoint sets, and we prove the metastable behavior of the process among the deepest wells following the martingale approach. We also present a bound for the probability that a Markov process hits a set before some fixed time in terms of the capacity of an enlarged process.
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This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the $epsilon$-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for our scheme. We combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also described in order to complete the theoretical results.
We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes.
We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.
We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of {it metastable points} as a subset of the state space $G_N$ such that (i) this set is reached from any point $xin G_N$ without return to x with probability at least $b_N$, while (ii) for any two point x,y in the metastable set, the probability $T^{-1}_{x,y}$ to reach y from x without return to x is smaller than $a_N^{-1}ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: item{(i)} To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. item{(ii)} To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is essentially equal to the inverse mean exit time from this state. The corresponding eigenfunctions are close to the indicator function of the support of the metastable state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.
We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $alpha$ at the left of the system and $beta$ at the right of the system. The strength of the reservoirs is ruled by $kappa$N --$theta$ > 0. Here N is the size of the system, $kappa$ > 0 and $theta$ $in$. Our results are valid for $theta$ $le$ 0. For $theta$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $kappa$ and with Dirichlet boundary conditions. Their solutions also depend on $kappa$. For $theta$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $theta$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $theta$ = 0 when we send the parameter $kappa$ to zero. Indeed, we conjecture that the limiting profile when $kappa$ $rightarrow$ 0 is the one that we should obtain when taking small values of $theta$ > 0.
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