Open Access
1999 On Semi-Martingale Characterizations of Functionals of Symmetric Markov Processes
Masatoshi Fukushima
Author Affiliations +
Electron. J. Probab. 4: 1-32 (1999). DOI: 10.1214/EJP.v4-55


For a quasi-regular (symmetric) Dirichlet space $( {\cal E}, {\cal F})$ and an associated symmetric standard process $(X_t, P_x)$, we show that, for $u in {\cal F}$, the additive functional $u^*(X_t) - u^*(X_0)$ is a semimartingale if and only if there exists an ${\cal E}$-nest $\{F_n\}$ and positive constants $C_n$ such that $ \vert {\cal E}(u,v)\vert \leq C_n \Vert v\Vert_\infty, v \in {\cal 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.


Download Citation

Masatoshi Fukushima. "On Semi-Martingale Characterizations of Functionals of Symmetric Markov Processes." Electron. J. Probab. 4 1 - 32, 1999.


Accepted: 6 October 1999; Published: 1999
First available in Project Euclid: 4 March 2016

zbMATH: 0936.60067
MathSciNet: MR1741537
Digital Object Identifier: 10.1214/EJP.v4-55

Primary: 60J45
Secondary: 31C25 , 60J55

Keywords: Additive functionals , BV function , quasi-regular Dirichlet form , Semimartingale , smooth signed measure , strongly regular representation

Vol.4 • 1999
Back to Top