On self-attracting d-dimensional random walks

• 成果类型: Article
• 署名作者: Bolthausen, E; Schmock, U
• 署名单位: University of Zurich; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 531-572
• 关键词: markov process expectations; reinforced random-walk; asymptotic evaluation; large time; phase

摘要

Let {X-t}(t greater than or equal to 0) be a symmetric, nearest-neighbor random walk on Z(d) with exponential holding times of expectation 1/d, starting at the origin. For a potential V: Z(d) --> [0, infinity) with finite and nonempty support, define transformed path measures by d (P) over cap(T) = exp(T(-1)integral(0)(T)integral(0)(T)V(X-s - X-t)ds/dt)dP/Z(T) for T > 0, where ZT is the normalizing constant. If d = 1 or if the self-attraction is sufficiently strong, then \\X-t\\(infinity) has an exponential moment under (P) over cap(T) which is uniformly bounded for T > 0 and t is an element of [0, T]. We also prove that {X-t}(t greater than or equal to 0) under suitable subsequences of {(P) over cap(T)}(T>0) behaves for large T asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of X-T as well as of the law of the empirical measure L-T under {(P) over cap(T)}(T>0).