Abstract
We consider the properties of a random set ϕ t (ℝ d+ ), where ϕ t (x) is a solution of a stochastic differential equation in ℝ d+ with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ t (ℝ d+ ) and prove that the Hausdorff dimension of the boundary ∂ϕ t (ℝ d+ ) does not exceed d − 1.
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REFERENCES
I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1982).
A. Yu. Pilipenko, “Flows generated by stochastic equations with reflection,” Random Oper. Stochast. Equat., 12, No.4, 389–396 (2004).
A. P. Calderon, “On the differentiability of absolutely continuous functions,” Riv. Mat. Univ. Parma, 2, 203–213 (1951).
N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de Gruyter, Berlin (1991).
P. L. Lions, J.-L. Menaldi, and A.-S. Sznitman, “Construction de processus de diffusion reflechis par penalisation du domaine,” C. R. Acad. Sci. Math., 292, No.11, 559–562 (1981).
J.-L. Menaldi, “Stochastic variational inequality for reflected diffusion,” Indiana Univ. Math. J., 2, No.5, 733–744 (1983).
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
O. Kallenberg, Foundations of Modern Probability, Springer, New York (2002).
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge (1990).
N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience, New York (1958).
M. Cranston and Y. Le Jan, “Noncoalescence for the Skorokhod equation in a convex domain of R 2,” Probab. Theory Relat. Fields, 87, No.2, 241–252 (1990).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.
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Pilipenko, A.Y. Properties of the Flows Generated by Stochastic Equations with Reflection. Ukr Math J 57, 1262–1274 (2005). https://doi.org/10.1007/s11253-005-0260-1
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DOI: https://doi.org/10.1007/s11253-005-0260-1