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Principal values of Brownian local times and their related topics. (English) Zbl 0878.60049
Ikeda, N. (ed.) et al., Itô’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Itô on the occasion of his 80th birthday. Tokyo: Springer. 413-422 (1996).
It is reviewed an interesting branch of stochastic analysis of continuous additive functionals of the symmetric Hunt processes, which cannot be treated in the frame of the theory of semimartingales. Main attention is paid to the properties of the functionals $C^a_t= 2(B_t- a)\log|B_t-a|- 2(B_0- a)\log|B_0-a|-$
$-2(B_t- a)+2(B_0- a)-2\int^t_0\log|B_s- a|dB_s,\quad t\geq 0,\;a\in\mathbb{R}^1,$
$H^a_t(1-\alpha)= 2{(B_t- a)^{1-\alpha}_+\over(-\alpha)(1- \alpha)}- 2 {(B_0- a)^{1-\alpha}_+\over(- \alpha)(1-\alpha)}-2\int^t_0 {(B_s- a)^{-\alpha}_+\over (-\alpha)} dB_s,\quad t\geq 0,\;0<\alpha<1/2,$ and their relationship to the local time $$L^a_t$$ at a linear Brownian motion $$B$$. Representations of Brownian continuous additive functionals locally of zero energy are also discussed.
For the entire collection see [Zbl 0852.00016].

MSC:
 60J55 Local time and additive functionals 60J65 Brownian motion 60J40 Right processes