Generators of some non-commutative stochastic processes

• 成果类型: Article
• 署名作者: Anshelevich, Michael
• 署名单位: Texas A&M University System; Texas A&M University College Station
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0470-z
• 发表日期: 2013
• 页码: 777-815
• 关键词: free brownian-motion; Respect; convolution; POLYNOMIALS; EVOLUTION; monotone

摘要

A fundamental result of Biane (Math Z 227:143-174, 1998) states that a process with freely independent increments has the Markov property, but that there are two kinds of free L,vy processes: the first kind has stationary increments, while the second kind has stationary transition operators. We show that a process of the first kind (with mean zero and finite variance) has the same transition operators as the free Brownian motion with appropriate initial conditions, while a process of the second kind has the same transition operators as a monotone L,vy process. We compute an explicit formula for the generators of these families of transition operators, in terms of singular integral operators, and prove that this formula holds on a fairly large domain. We also compute the generators for the -Brownian motion, and for the two-state free Brownian motions.