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A Dirichlet process characterization of a class of reflected diffusions. (English) Zbl 1202.60059
The first main result of this paper (Theorem 3.1) states that multidimensional reflected diffusion that belong to a slight generalization of the family of reflected diffusions, obtained as approximations in [K. Ramanan and M. T. Reiman, Ann. Appl. Probab. 13, No. 1, 100–139 (2003; Zbl 1016.60083) and Ann. Appl. Probab. 18, No. 1, 22–58 (2008; Zbl 1144.60056)] fail to be semimartingales. It is the generalization to multidimensional case of similar statement known already in two dimensions and for the reflected Brownian motion.
The next main result (Th.3.5) states that for reflected diffusion belonging to certain class it is possible to decompose it as the sum of a local martingale and of a process of zero \(p\)-variation for some \(p>1\). This class consists of weak solutions of stochastic differential equations with reflection that are Markov processes, have locally bounded drift and dispersion coefficients and satisfy certain \(L^p\) continuity requirement.
Corollary 3.6 states that the nonsemimartingale reflected diffusions, considered in Theorem 3.1., are Dirichlet processes. In Corollary 3.7 Theorem 3.5 is applied to show that even in cupslike domains the associated reflected Brownian motions are Dirichlet processes.
The paper is organized as follows. The Introduction contains background and motivation and some notation used throughout the paper. In Section 2, the class of stochastic differential equations with reflaction under study is introduced, the main assumptions are presented and motivating examples are introduced. Section 3 contains rigorous statement of main results. The proof of Theorem 3.1 is presented in Section 4, while the proofs of Theorem 3.5 and Corollary 3.6 are given in Section 5. Some results needed in proofs are proved in the Appendix.
The volume of the paper is 44 pages. The list of references contains 32 positions.

MSC:
60G17 Sample path properties
60J55 Local time and additive functionals
60J65 Brownian motion
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