Absolute continuity of symmetric diffusions

• 成果类型: Article
• 署名作者: Fitzsimmons, PJ
• 署名单位: University of California System; University of California San Diego
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 230-258
• 关键词: generalized schrodinger-operators; reversible markov-processes; uniqueness; PRINCIPLE; forms

摘要

Let X and Y be symmetric diffusion processes with a common state space, and let P-m (resp. Q(mu)) be the law of X (resp. Y) with its symmetry measure m (resp. mu) as initial distribution. We study the consequences of the absolute continuity condition Q(mu) much less than loc P-m. We show that under this condition there is a smooth version rho of the Radon-Nikodym derivative d(mu)/dm such that 1/2[log rho(X-t) - log rho(X-0)] = M-t + N-t, t < sigma, where M is a continuous local martingale additive functional, N is a zero-energy continuous additive functional and a is an explosion time. The Girsanov density L-t := dQ(mu)\(Ft)/dP(m)\(Ft) then admits the representation L-t = exp(M-t - 1/2[M](t))1((t