Mathematics – Probability
Scientific paper
2005-03-25
Annals of Probability 2005, Vol. 33, No. 1, 244-299
Mathematics
Probability
Published at http://dx.doi.org/10.1214/009117904000000991 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009117904000000991
We investigate the close connection between metastability of the reversible diffusion process X defined by the stochastic differential equation dX_t=-\nabla F(X_t) dt+\sqrt2\epsilon dW_t,\qquad \epsilon >0, and the spectrum near zero of its generator -L_{\epsilon}\equiv \epsilon \Delta -\nabla F\cdot\nabla, where F:R^d\to R and W denotes Brownian motion on R^d. For generic F to each local minimum of F there corresponds a metastable state. We prove that the distribution of its rescaled relaxation time converges to the exponential distribution as \epsilon \downarrow 0 with optimal and uniform error estimates. Each metastable state can be viewed as an eigenstate of L_{\epsilon} with eigenvalue which converges to zero exponentially fast in 1/\epsilon. Modulo errors of exponentially small order in 1/\epsilon this eigenvalue is given as the inverse of the expected metastable relaxation time. The eigenstate is highly concentrated in the basin of attraction of the corresponding trap.
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