Electronic Journal of Probability

Stochastic differential equations with sticky reflection and boundary diffusion

Martin Grothaus and Robert Voßhall

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We construct diffusion processes in bounded domains $\Omega $ with sticky reflection at the boundary $\Gamma $ in use of Dirichlet forms. In particular, the occupation time on the boundary is positive. The construction covers a static boundary behavior and an optional diffusion along $\Gamma $. The process is a solution to a given SDE for q.e. starting point. Using regularity results for elliptic PDE with Wentzell boundary conditions we show strong Feller properties and characterize the constructed process even for every starting point in $\overline{\Omega } \backslash \Xi $, where $\Xi $ is given explicitly by the involved densities. By a time change we obtain pointwise solutions to SDEs with immediate reflection under weak assumptions on $\Gamma $ and the drift. A non-trivial extension of the construction yields N-particle systems with the stated boundary behavior and singular drifts. Finally, the setting is applied to a model for particles diffusing in a chromatography tube with repulsive interactions.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 7, 37 pp.

Received: 23 March 2016
Accepted: 11 January 2017
First available in Project Euclid: 27 January 2017

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Zentralblatt MATH identifier

Primary: 60J50: Boundary theory 60J60: Diffusion processes [See also 58J65] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 31C25: Dirichlet spaces 35J15: Second-order elliptic equations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

sticky reflection boundary diffusions Wentzell boundary conditions strong Feller properties interacting particle systems singular interactions

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Grothaus, Martin; Voßhall, Robert. Stochastic differential equations with sticky reflection and boundary diffusion. Electron. J. Probab. 22 (2017), paper no. 7, 37 pp. doi:10.1214/17-EJP27. https://projecteuclid.org/euclid.ejp/1485486107

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