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Metastability in reversible diffusion processes I. Sharp asymptotics for capcities and exit times
(2004)
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form +rF ( )r on R or subsets of , where F is a smooth function with finitely many local minima. In analogy to previous work in discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as # 0, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring-Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.
We continue the analysis of the problem of metastability for reversible diffusion processes,
initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator.
Recall that we are considering processes with generators of the form 1+rF()r on Rd or subsets
of Rd , where F is a smooth function with finitely many local minima. Here we consider only
the generic situation where the depths of all local minima are different. We show that in general
the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by
the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius
centered at the positions of the local minima of F. We also get very precise uniform control on the
corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision
with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven
that these mean times are given, again up to multiplicative errors that tend to one, by the classical
Eyring–Kramers formula.
We study a two-species interacting particle model on a subset of $\Z$
with open boundaries. The two species are injected with time
dependent rate on the left, resp.~right boundary.
Particles of different species annihilate when
they try to occupy the same site. This model has been proposed as a
simple model for the dynamics of an ``order book'' on a stock
market. We consider the hydrodynamic scaling limit for the empirical
process and prove a large deviation principle that implies
convergence to the solution of a non-linear parabolic equation.
We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
The randomized k-number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k-partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k = 2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k > 2 in the restricted problem and show that the vector of differences between the k sums converges to a k - 1-dimensional Poisson point process.