Summary
LetG be ad-dimensional bounded Euclidean domain, H1 (G) the set off in L2(G) such that ∇f (defined in the distribution sense) is in L2(G). Reflecting diffusion processes associated with the Dirichlet spaces (H1(G), ℰ) on L2(G, σdx) are considered in this paper, where
A=(aij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ij∈H1(G) for eachi,j, and σ∈H1(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H1(G), ℰ) having starting points inG under a mild condition which is satisfied when ϖG has finite (d−1)-dimensional lower Minkowski content.
Article PDF
Similar content being viewed by others
References
Ahlfors, L.V.: Complex analysis, 3rd edn. New York: McGraw-Hill 1979
Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab.19, 486–508 (1991).
Bass, R.F., Hsu, P.: The semimartingale structure of reflecting Brownian motion. Proc. Am. Math. Soc.108, 1007–1010 (1990).
Billingsley, P.: Convergence of probability measures. New York:Wiley 1969
Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York London: Academic Press 1968
Chen, Z.Q.: On reflected Dirichlet spaces. Probab. Theory Relat. Fields (to appear)
Chen, Z.Q.: Pseudo Jordan domains and reflecting Brownian motions. Probab. Theory Relat. Fields (to appear)
Chen, Z.Q.. On reflecting diffusion processes. Ph. D. thesis. Washington University, St. Louis 1992
Dellacherie, C., Meyer, P.A.: Probabilities and potential B (translated and prepared by J.P. Wilson). Amsterdam: North-Holland 1982
Durrett, R.: Brownian motion and martingales in analysis. Belmont, CA: Wadsworth 1984
Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969
Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math4, 183–215 (1967)
Fukushima, M.: Regular representation of Dirichlet spaces. Trans. Am. Math. Soc.155, 455–473 (1971)
Fukushima, M.: Dirichlet forms and symmetric Markov processes. Amsterdam: North-Holland 1980
Fukushima, M.: On absolute continuity of multidimensional symmetrizable diffusions. In: Fukushima, M. (ed.). Functional analysis in Markov processes. (Lect. Notes Math., vol. 923, pp. 146–176) Berlin Heidelberg New York: Springer 1982
Gong, G.L., Qian, M.P., Silverstein, M.L.: Normal derivative for bounded domains with general boundary. Trans. Am. Math. Soc.308, 785–809 (1988)
Harrison, J.M., Williams, R.: Multidimensional reflecting Brownian motion having exponential stationary distribution. Ann. Probab.15, 115–137 (1987)
Hsu, P.: Reflecting Brownian motion, Boundary local time, and the Neumann boundary value problem. Ph.D. Dissertation, Stanford 1984
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn., Amsterdam: North-Holland 1988
Jerison, D., Kenig, C.: Boundary value problems on Lipschitz domains. Math. Assoc. Am. Stud.23, 1–68 (1982)
Jones, P.W.: Quasiconformal mappings and extendibility of functions in Sobolev spaces. Acta Math.147 (no. 1-2), 71–88 (1981)
Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math.37, 511–537 (1984)
Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equation with discontinuous coefficients. Ann. Sci. Norm Super. Pisa, III. Ser.17, 43–77 (1963)
Lyons, T.J., Zheng, W.A.: A crossing estimate for the canonical process on a Dirichlet space and a tightness result. In: Colloque Paul Lévy sur lés processus stochatiques. (Asterisque, vols. 157–158, pp. 249–271 Paris: Soc. Math. Fr. 1988
Meyer, P.A.: Intégrales stochastiques. Séminaire de Probabilités 1. (Lect. Notes Math., vol. 39, pp. 72–162) Berlin Heidelberg New York: Springer 1967
Meyer, P.A., Zheng, W.A.: Tightness criteria for laws of semimartingales. Ann. Inst. Henri Poincaré20 (no. 4), 357–372 (1984)
Milnor, J.W.: Topology from the differentiable viewpoint. Charlottesville: University Press of Virginia 1965
Nehari, Z., Conformal mapping. New York: McGraw-Hill 1952
Oshima, Y.: Lecture on Dirichlet spaces. (Preprint 1988)
Royden, H.L.: Real analysis, 3rd edn. New York: Macmillan 1988
Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1974
Saisho, Y.: Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Relat. Fields.74, 455–477 (1987)
Sato, K., Ueno, T.: Multidimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4-3, 529–605 (1965)
Silverstein, M.L.: Symmetric Markov processes. (Lect. Notes Math., vol. 426) Berlin Heidelberg New York: Springer 1974
Silverstein, M.L.: The reflected Dirichlet space. Ill. J. Math.18 (2), 310–355 (1974)
Silverstein, M.L.: Boundary theory of symmetric Markov processes. (Lect. Notes Math., vol. 516) Berlin Heidelberg New York: Springer 1976
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970
Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure Appl. Math.24, 147–225 (1971).
Tanaka, H.: Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J.9, 163–177 (1979)
Williams, R.J., Zheng, W.A.: On reflecting Brownian motion—a weak convergence approach. Ann. Inst. Henri Poincaré26 (3), 461–488 (1990)
Wentzell, A.D.: On boundary conditions for multidimensional diffusion processes. Theory Probab. Appl.4, 164–177 (1959)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen, ZQ. On reflecting diffusion processes and Skorokhod decompositions. Probab. Th. Rel. Fields 94, 281–315 (1993). https://doi.org/10.1007/BF01199246
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01199246