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On the stochastic regularity of distorted Brownian motions

Authors: Jiyong Shin and Gerald Trutnau
Journal: Trans. Amer. Math. Soc. 369 (2017), 7883-7915
MSC (2010): Primary 60J60, 60J35, 31C25, 31C15; Secondary 60J55, 35J25
Published electronically: April 15, 2016
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Abstract: We systematically develop general tools to apply Fukushima's absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density w.r.t. the reference measure and methods to estimate drift potentials comfortably. We then apply our results to distorted Brownian motions and construct weak solutions to singular stochastic differential equations, i.e., equations with possibly unbounded and discontinuous drift and reflection terms which may be the sum of countably many local times. The solutions can start from any point of the explicitly specified state space. We consider different kinds of weights, like Muckenhoupt $ A_2$ weights and weights with moderate growth at singularities as well as different kinds of (multiple) boundary conditions. Our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected Brownian motions with singular drift in bounded and unbounded multi-dimensional domains.

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Additional Information

Jiyong Shin
Affiliation: School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdae- mungu, Seoul 02445, South Korea

Gerald Trutnau
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics of Seoul National University, San56-1 Shinrim-dong Kwanak-gu, Seoul 151-747, South Korea

Keywords: Transition functions, singular diffusions, skew Brownian motion, reflected Brownian motion, Feller processes.
Received by editor(s): August 17, 2014
Received by editor(s) in revised form: November 30, 2015
Published electronically: April 15, 2016
Additional Notes: This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2012R1A1A2006987).
Article copyright: © Copyright 2016 American Mathematical Society

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