Reflection Brownian motions: Quasimartingales and strong Caccioppoli sets. (English) Zbl 0812.60065

The paper deals mainly with the following problem: When is a reflecting Brownian function in a bounded domain a quasi-martingale? The answer depends on the smoothness of the topological boundary of \(D \subset R^ d\) and it is well-known that when the boundary of \(D\) is \(C^ 2\)-smooth, then for every given Dirichlet space \(\{H^ 1(D), \xi)\) there is a continuous Markov process \(X\) associated with \(\xi\) which admits the Skorokhod semi-martingale decomposition starting at any point in \(\overline D:=D\cup\partial D\).
The authors introduce the notion of strong Caccioppoli sets which are nothing else than bounded domains \(D \subset R^ d\) such that \[ \int_ D {\partial g \over \partial x_ i} dm \leq C | g |_ \infty, \qquad i = 1,2, \dots, \;\forall g \in H^ 1(D) \cap C_ b(D), \] where \(|\;|_ \infty\) denotes the supremum norm, \(C_ b\) the class of continuous bounded functions and \(H^ 1\) the class of \(g \in L^ 2(D,dm)\) whose first distributional derivatives \(\partial g/ \partial x_ i\) belong also to \(L^ 2\), \(C\) a fixed constant. Let \(\{H^ 1, \xi, dm\}\) be a symmetric Dirichlet space (in the sense given by Fukushima) on a given strong Caccioppoli set \(D\subset R^ d\) and let \(X = (X_ t)\) be the stationary continuous Markov process on \(\overline D\) associated with \(\xi\) (called usually the stationary reflecting Brownian motion on \(\overline D)\), then the main result is the following
Theorem 1.1. The stationary reflecting Brownian motion on \(\overline D\) is a quasi-martingale iff \(D\) is a strong Caccioppoli set.
It is noted that the definition of a quasi-martingale given by the author here is slightly different from the known one introduced by Dellacherie- Meyer. In the last Sections 4 and 5 the authors extend the above mentioned result to the case of symmetric reflecting diffusion processes associated with some class of strongly elliptic operators with coefficients in \(H^ 1(D)\).
Reviewer: X.L.Nguyen (Hanoi)


60J65 Brownian motion
31C25 Dirichlet forms
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
60J60 Diffusion processes
60J55 Local time and additive functionals
Full Text: DOI


[1] Adams, R. A.:Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030
[2] Bass, R. F. and Hsu, P.: ?The Semimartingale Structure of Reflecting Brownian Motion?,Proc. Amer. Math. Soc. 108 (1990), 1007-1010. · Zbl 0694.60075
[3] Bass, R. F. and Hsu, P.: ?Some Potential Theory for Reflecting Brownian Motion in Hölder and Lipschitz Domains?,Ann. Prob. 19 (1991), 486-508. · Zbl 0732.60090
[4] Blumenthal, R. M. and Getoor, R. K.:Markov Processes and Potential Theory, Academic Press, New York, 1968. · Zbl 0169.49204
[5] Chen Z. Q.: ?On Reflecting Diffusion Processes and Skorokhod Decompositions?,Probability Theory and Related Fields 94 (1993), 281-315. · Zbl 0767.60074
[6] Chen, Z. Q.: ?Pseudo Jordan Domains and Reflecting Brownian Motions?,Probability Theory and Related Fields 94 (1993), 271-280. · Zbl 0767.60079
[7] Dellacherie, C. and Meyer, P.-A.:Probabilités et Potentiel, Théorie des Martingales, Hermann, Paris, 1980. · Zbl 0464.60001
[8] Evans, L. C. and Gariepy, R. F.:Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. · Zbl 0804.28001
[9] Federer, H.:Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969. · Zbl 0176.00801
[10] Fitzsimmons, P. J. and Getoor, R. K.: ?On the Potential Theory of Symmetric Markov Processes?,Math. Ann. 281 (1988), 495-512. · Zbl 0627.60067
[11] Fukushima, M.: ?A Construction of Reflecting Barrier Brownian Motions for Bounded Domains?,Osaka J. Math. 4 (1967), 183-215. · Zbl 0317.60033
[12] Fukushima, M.:Dirichlet Forms and Symmetric Markov Processes, North-Holland, Amsterdam, 1980. · Zbl 0422.31007
[13] Giusti, E.:Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. · Zbl 0545.49018
[14] Jones, P. W.: ?Quasiconformal Mappings and Extendability of Functions in Sobolev Spaces?,Acta. Math. 147 (1981), 71-88. · Zbl 0489.30017
[15] Pardoux, E. and Williams, R. J.: ?Symmetric Reflected Diffusions?, to appear inAnn. Inst. Henri Poincaré. · Zbl 0794.60078
[16] Stein, E.:Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[17] Williams, R. J.: ?Reflected Brownian Motion: Hunt Process and Semimartingale Representation?, to appear inProceedings of the Barcelona Seminar on Stochastic Analysis, September 1991, Progress in Probability, Birkhäuser, Boston.
[18] Williams, R. J. and Zheng, W. A.: ?On Reflecting Brownian Motion ? a Weak Convergence Approach?,Ann. Inst. Henri Poincaré 26 (1990), 461-488. · Zbl 0704.60081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.