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DISCRETE APPROXIMATIONS TO REFLECTED BROWNIAN MOTION 1
"... In this paper we investigate three discrete or semidiscrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in R n that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete ..."
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Cited by 8 (4 self)
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In this paper we investigate three discrete or semidiscrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in R n that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D ∩ 2 −k Z n moving at the rate 2 −2k with stationary initial distribution converge weakly in the space D([0, 1], R n), equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on D. We further show that the following “myopic conditioning ” algorithm generates, in the limit, a reflected Brownian motion on any bounded domain D. For every integer k ≥ 1, let {X k j2 −k,j = 0, 1, 2,...} be a discrete time Markov chain with onestep transition probabilities being the same as those for the Brownian motion in D conditioned not to exit D before time 2 −k. We prove that the laws of X k converge to that of the reflected Brownian motion on D. These approximation
Perturbation of symmetric Markov processes
 Probab. Theory Related Fields 140 (2008), no. 12, 239–275, DOI 10.1007/s0044000700652. MR2357677 (2008m:60151
"... Abstract We present a pathspace integral representation of the semigroup associated with the quadratic form obtained by a lowerorder perturbation of the L 2 infinitesimal generator L of a general symmetric Markov process. An illuminating concrete example for Crucial to the development is the use ..."
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Cited by 7 (5 self)
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Abstract We present a pathspace integral representation of the semigroup associated with the quadratic form obtained by a lowerorder perturbation of the L 2 infinitesimal generator L of a general symmetric Markov process. An illuminating concrete example for Crucial to the development is the use of an extension of Nakao's stochastic integral for zeroenergy additive functionals and the associated Itô formula, both of which were recently developed in
A Dirichlet process characterization of a class of reflected diffusions
 Ann. Probab
, 2010
"... Abstract. For a general class of stochastic differential equations with reflection that admit a Markov weak solution and satisfy a certain L p continuity condition, p > 1, it is shown that the associated reflected diffusion can be decomposed as the sum of a local martingale and a continuous, ada ..."
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Cited by 4 (3 self)
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Abstract. For a general class of stochastic differential equations with reflection that admit a Markov weak solution and satisfy a certain L p continuity condition, p > 1, it is shown that the associated reflected diffusion can be decomposed as the sum of a local martingale and a continuous, adapted process of zero pvariation. In particular, when p = 2, this implies that the associated reflected diffusion is a Dirichlet processes in the sense of Föllmer. As motivation for such a characterization, it is also shown that reflected diffusions belonging to a specific family within this class are not semimartingales, but are Dirichlet processes. This family of diffusions arise naturally as approximations of certain stochastic networks that use the socalled generalized processor sharing scheduling policy.