Self-interacting diffusions: a simulated annealing version

• 成果类型: Article
• 署名作者: Raimond, Olivier
• 署名单位: Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0147-9
• 发表日期: 2009
• 页码: 247-279
• 关键词: convergence

摘要

We study asymptotic properties of processes X, living in a Riemannian compact manifold M, solution of the stochastic differential equation (SDE) dX(t) = dW(t)(X-t) - beta(t)del V mu(t)(X-t)dt with W a Brownian vector field, beta(t) = a log(t + 1), mu(t) = 1/t integral(t)(0) delta(Xs)ds and V mu(t)(x) = 1/t integral(t)(0) V(x, X-s)ds, V being a smooth function. We show that the asymptotic behavior of mu(t) can be described by a non-autonomous differential equation. This class of processes extends simulated annealing processes for which V(x, y) is only a function of x. In particular we study the case M = S-n, the n-dimensional sphere, and V(x, y) = - cos(d(x, y)), where d(x, y) is the distance on S-n, which corresponds to a process attracted by its past trajectory. In this case, it is proved that mu(t) converges almost surely towards a Dirac measure.