On semi-martingale characterizations of functionals of symmetric Markov processes. (English) Zbl 0936.60067

Fukushima, Masatoshi

Electron. J. Probab. 4, Paper No. 18, 32 p. (1999).

Summary: For a quasi-regular (symmetric) Dirichlet space \(( {\mathcal E}, {\mathcal F})\) and an associated symmetric standard process \((X_t, P_x)\), we show that, for \(u \in {\mathcal F}\), the additive functional \(u^*(X_t) - u^*(X_0)\) is a semimartingale if and only if there exist an \({\mathcal E}\)-nest \(\{F_n\}\) and positive constants \(C_n\) such that \( |{\mathcal E}(u,v)|\leq C_n \|v\|_\infty,\;v \in {\mathcal F}_{F_n,b}.\) In particular, a signed measure resulting from the inequality will be automatically smooth. One of the variants of this assertion is applied to the distorted Brownian motion on a closed subset of \(R^d\), giving stochastic characterizations of BV functions and Caccioppoli sets.

MSC:

60J45	Probabilistic potential theory
60J55	Local time and additive functionals
31C25	Dirichlet forms

Keywords:

Dirichlet form; strongly regular representation; additive functionals; smooth signed measure; BV function