Self-interacting diffusions. III. Symmetric interactions
Self-interacting diffusions. III. Symmetric interactions
复制标题
自相互作用扩散。
DOI：
10.1214/009117905000000251
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发表时间：
2005
影响因子：
2.3
通讯作者：
Olivier Raimond
中科院分区：
文献类型：
--
作者：
M. Benaïm;Olivier Raimond
设M是紧致黎曼流形. M上的自相互作用扩散是dXt = dWt（Xt）-1t（t0 $> VXs（Xt）ds）dt的随机过程解，其中{Wt}是M上的布朗向量场，Vx（y）= V（x，y）是光滑函数.设μ t = 1 tt 0 δ Xsds表示Xt的归一化占有测度.证明了当V是对称的时，μ t几乎必然收敛于某个非线性自由能泛函J的临界集，并且J具有一般的n个临界点，μ t几乎必然收敛于J的一个局部极小值，每个局部极小值都有一个正概率被选择.
Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to dX t =dW t (X t )-1 t( t 0 ⊇V Xs (X t )ds)dt, where {W t } is a Brownian vector field on M and V x (y) = V(x, y) a smooth function. Let μ t = 1 t t 0 δ Xs ds denote the normalized occupation measure of X t . We prove that, when V is symmetric, μ t converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μ t converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.