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 Electronic Journal of Probability > Vol. 4 (1999) > Paper 18 open journal systems 


On Semi-Martingale Characterizations of Functionals of Symmetric Markov Processes

Masatoshi Fukushima, Kansai University


Abstract
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 vVert_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.


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Pages: 1-32

Published on: October 6, 1999
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Electronic Journal of Probability. ISSN: 1083-6489