## On reflecting diffusion processes and Skorokhod decompositions.(English)Zbl 0767.60074

Summary: Let $$G$$ be a $$d$$-dimensional bounded Euclidean domain, $$H^ 1(G)$$ the set of $$f$$ in $$L^ 2(G)$$ such that $$\nabla f$$ (defined in the distribution sense) is in $$L^ 2(G)$$. Reflecting diffusion processes associated with the Dirichlet spaces $$(H^ 1(G),{\mathcal E})$$ on $$L^ 2(G,\rho dx)$$ are considered, where ${\mathcal E}(f,g)={1\over 2}\int_ G\sum^ d_{i,j=1}a^{ij}{\partial f\over\partial x_ i}{\partial g\over\partial x_ j}\rho dx,\quad\text{for }f,g\in H^ 1(G),$ $${\mathcal A}=(a^{ij})$$ is a symmetric, bounded, uniformly elliptic $$d\times d$$ matrix-valued function such that $$a^{ij}\in H^ 1(G)$$ for each $$i$$, $$j$$, and $$\rho\in H^ 1(G)$$ is a positive bounded function on $$G$$ which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with $$(H^ 1(G),{\mathcal E})$$ having starting points in $$G$$ under a mild condition which is satisfied when $$\partial G$$ has finite $$(d-1)$$-dimensional lower Minkowski content.

### MSC:

 60J60 Diffusion processes 60J65 Brownian motion 60J55 Local time and additive functionals 60J35 Transition functions, generators and resolvents 31C25 Dirichlet forms
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### References:

 [1] Ahlfors, L.V.: Complex analysis, 3rd edn. New York: McGraw-Hill 1979 · Zbl 0395.30001 [2] Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab.19, 486-508 (1991). · Zbl 0732.60090 [3] Bass, R.F., Hsu, P.: The semimartingale structure of reflecting Brownian motion. Proc. Am. Math. Soc.108, 1007-1010 (1990). · Zbl 0694.60075 [4] Billingsley, P.: Convergence of probability measures. New York:Wiley 1969 · Zbl 0181.44303 [5] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York London: Academic Press 1968 · Zbl 0169.49204 [6] Chen, Z.Q.: On reflected Dirichlet spaces. Probab. Theory Relat. Fields (to appear) [7] Chen, Z.Q.: Pseudo Jordan domains and reflecting Brownian motions. Probab. Theory Relat. Fields (to appear) · Zbl 0767.60079 [8] Chen, Z.Q.. On reflecting diffusion processes. Ph. D. thesis. Washington University, St. Louis 1992 [9] Dellacherie, C., Meyer, P.A.: Probabilities and potential B (translated and prepared by J.P. Wilson). Amsterdam: North-Holland 1982 · Zbl 0494.60002 [10] Durrett, R.: Brownian motion and martingales in analysis. Belmont, CA: Wadsworth 1984 · Zbl 0554.60075 [11] Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969 · Zbl 0176.00801 [12] Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math4, 183-215 (1967) · Zbl 0317.60033 [13] Fukushima, M.: Regular representation of Dirichlet spaces. Trans. Am. Math. Soc.155, 455-473 (1971) · Zbl 0248.31007 [14] Fukushima, M.: Dirichlet forms and symmetric Markov processes. Amsterdam: North-Holland 1980 · Zbl 0422.31007 [15] 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 [16] 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) · Zbl 0661.60101 [17] Harrison, J.M., Williams, R.: Multidimensional reflecting Brownian motion having exponential stationary distribution. Ann. Probab.15, 115-137 (1987) · Zbl 0615.60072 [18] Hsu, P.: Reflecting Brownian motion, Boundary local time, and the Neumann boundary value problem. Ph.D. Dissertation, Stanford 1984 [19] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, 2nd edn., Amsterdam: North-Holland 1988 · Zbl 0495.60005 [20] Jerison, D., Kenig, C.: Boundary value problems on Lipschitz domains. Math. Assoc. Am. Stud.23, 1-68 (1982) · Zbl 0529.31007 [21] Jones, P.W.: Quasiconformal mappings and extendibility of functions in Sobolev spaces. Acta Math.147 (no. 1-2), 71-88 (1981) · Zbl 0489.30017 [22] Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math.37, 511-537 (1984) · Zbl 0598.60060 [23] 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) · Zbl 0116.30302 [24] 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 [25] Meyer, P.A.: Intégrales stochastiques. Séminaire de Probabilités 1. (Lect. Notes Math., vol. 39, pp. 72-162) Berlin Heidelberg New York: Springer 1967 [26] Meyer, P.A., Zheng, W.A.: Tightness criteria for laws of semimartingales. Ann. Inst. Henri Poincaré20 (no. 4), 357-372 (1984) · Zbl 0551.60046 [27] Milnor, J.W.: Topology from the differentiable viewpoint. Charlottesville: University Press of Virginia 1965 · Zbl 0136.20402 [28] Nehari, Z., Conformal mapping. New York: McGraw-Hill 1952 · Zbl 0048.31503 [29] Oshima, Y.: Lecture on Dirichlet spaces. (Preprint 1988) · Zbl 0729.22011 [30] Royden, H.L.: Real analysis, 3rd edn. New York: Macmillan 1988 · Zbl 0704.26006 [31] Rudin, W.: Real and complex analysis. New York: McGraw-Hill 1974 · Zbl 0278.26001 [32] Saisho, Y.: Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Theory Relat. Fields.74, 455-477 (1987) · Zbl 0591.60049 [33] Sato, K., Ueno, T.: Multidimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4-3, 529-605 (1965) · Zbl 0219.60057 [34] Silverstein, M.L.: Symmetric Markov processes. (Lect. Notes Math., vol. 426) Berlin Heidelberg New York: Springer 1974 · Zbl 0296.60038 [35] Silverstein, M.L.: The reflected Dirichlet space. Ill. J. Math.18 (2), 310-355 (1974) · Zbl 0303.60066 [36] Silverstein, M.L.: Boundary theory of symmetric Markov processes. (Lect. Notes Math., vol. 516) Berlin Heidelberg New York: Springer 1976 · Zbl 0331.60046 [37] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970 · Zbl 0207.13501 [38] Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure Appl. Math.24, 147-225 (1971). · Zbl 0227.76131 [39] Tanaka, H.: Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J.9, 163-177 (1979) · Zbl 0423.60055 [40] Williams, R.J., Zheng, W.A.: On reflecting Brownian motion?a weak convergence approach. Ann. Inst. Henri Poincaré26 (3), 461-488 (1990) · Zbl 0704.60081 [41] Wentzell, A.D.: On boundary conditions for multidimensional diffusion processes. Theory Probab. Appl.4, 164-177 (1959)
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