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