Let D ⊂ R d be a bounded domain and let L = 1 2 ∇ · a∇ + b · ∇ be a second-order elliptic operator on D. Let ν be a probability measure on D. Denote by L the differential operator whose domain is specified by the following nonlocal boundary condition:and which coincides with L on its domain. Clearly 0 is an eigenvalue for L, with the corresponding eigenfunction being constant. It is known that L possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of L, indexed by ν, by γ 1 (ν) ≡ sup{Re λ: 0 = λ is an eigenvalue for L}.In this paper we investigate the eigenvalues of L in general and the spectral gap γ 1 (ν) in particular. 123 The operator L and its spectral gap γ 1 (ν) have probabilistic significance. The operator L is the generator of a diffusion process with random jumps from the boundary, and γ 1 (ν) measures the exponential rate of convergence of this process to its invariant measure.

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In this paper, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices.We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior.Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviations estimates are also obtained.Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

We study a model of species survival recently proposed by Michael and Volkov. We interpret it as a variant of empirical processes, in which the sample size is random and when decreasing, samples of smallest numerical values are removed. Micheal and Volkov proved that the empirical distributions converge to the sample distribution conditioned not to be below a certain threshold. We prove a functional central limit theorem for the fluctuations. There exists a threshold above which the limit process is

In this paper we provide some sharp asymptotic results for a stochastic model of species survival recently proposed by Guiol, Marchado, and Schinazi.1 Introduction and statement of results Recently, Guiol, Marchado, and Schinazi [7] proposed a new mathematical framework for modeling species survival which is closely related to the discrete Bak-Sneppen evolution model. In the original Bak-Sneppen model [3] a finite number of species are arranged in a circle, each species being characterized by its location and a parameter representing the fitness of the species and taking values between zero and one. The number of species and their location on the circle are fixed and remain unchanged throughout the evolution of the system. At discrete times n = 0, 1, . . . , the species with the lowest fitness and its two immediate neighbors update simultaneously their fitness values at random. The Bak-Sneppen evolution model is often referred to as an "ecosystem" because of the local interaction between different species. The distinguishing feature of the model is the emergence of self-organized criticality [1,2,6,8] regardless the simplicity and robustness of the underlying evolution mechanism. The Bak-Sneppen model has attracted significant attention over the past few decades, but it has also been proven to be difficult for analytical study. See for instance [6] for a relatively recent survey of the model.The asymptotic behavior of the Bak-Sneppen model, as the number of species gets arbitrarily large, was conjectured on the basis of computer simulations in [3]. It appears that the distribution of the fitness is asymptotically uniform on an interval (f crit , 1) for some critical parameter f crit , the value of which is close to 2/3 [1,8].Guiol, Marchado, and Schinazi [7] were able to prove a similar result for a related model with a stochastically growing number of species. Their analysis is based on a reduction to the study of a certain random walk, which allows them to build a proof using well-known results from the theory of random walks. The main result of [7] is *

Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.

Consider the partition function of a directed polymer in dimension d ≥ 1 in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is a well-known fact that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we provide a method to obtain estimates on the rate of decay of the lower tail of the large deviations, which are sharp up to multiplicative constants. As a consequence, we show that the lower tail of the large deviations exhibits three regimes, determined according to the tail of the negative part of the field. Our method is simple to apply and can be used to cover other oriented and non-oriented models including first/last-passage percolation and the parabolic Anderson model.