非线性扩散大系统的小噪声谱间隙渐近性
Small noise spectral gap asymptotics for a large system of nonlinear diffusions
摘要：
本文研究了双势阱下无界状态空间上强耦合扩散大系统的$L^2$谱隙。该系统可以看作是一维环面上随机Allen-Cahn方程的空间离散逼近。我们证明了上，下界的首项的光谱间隙在小的温度制度，统一控制系统的大小。上界由Eyring-Kramers型公式给出。证明了该下界对对数Sobolev常数也成立。我们建立了一个充分条件的上界的渐近最优性，并表明，这个条件是满足适当的假设下的系统规模的增长。我们的结果可以在大维数下用半经典Witten-Laplacian来表示。
英文摘要：
We study the $L^2$ spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen-Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring-Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fulfilled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.
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