Abstract
Consider a family of probability measures {ν ξ } on a bounded open region D ⊂ ℝd with a smooth boundary and a positive parameter set {β ξ }, all indexed by ξ ∈ ∂D. For any starting point inside D, we run a diffusion until it first exits D, at which time it stays at the exit point ξ for an independent exponential holding time with rate β ξ and then leaves ξ by a jump into D according to the distribution ν ξ . Once the process jumps inside, it starts the diffusion afresh. The same evolution is repeated independently each time the process jumped into the domain. The resulting Markov process is called diffusion with holding and jumping boundary (DHJ), which is not reversible due to the jumping. In this paper we provide a study of DHJ on its generator, stationary distribution and the speed of convergence.
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References
Ben-Ari I, Pinsky R G. Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure. J Funct Anal, 2007, 251: 122–140
Ben-Ari I, Pinsky R G. Ergodic behavior of diffusion with random jumps from the boundary. Stoch Proc Appl, 2009, 119: 864–881
Bernt O. Stochastic Differential Equations, the sixth edition. New York: Springer-Verlag, 2002
Burzy K, Holyst R, March P. A Fleming-Viot particle representation of the Dirichlet Laplacian. Comm Math Phys, 2000, 214: 679–703
Bieniek M, Burdzy K, Finch S. Non-extinction of a Fleming-Viot particle model. Preprint
Chen Z Q. Symmetric jump processes and their heat kernel estimates. Sci China Ser A, 2009, 52: 1423–1445
Chung K L, Zhao Z. From Brownian Motion to Schrodinger’s Equation. Berlin: Springer-Verlag Berlin Heidelberg, 1995
Down D, Meyn P, Tweedie R L. Exponential and uniform ergodicity of Markov processes. Ann Probab, 1995, 23: 1671–1691
Ethier S, Kurtz T. Markov Processes: Characterization and Convergence. In: Wiley Series in Probability and Mathematical Statistics. New York: John Wiley, Sons Inc, 1986
Ethier S N, Kurtz T G. Coupling and ergodic theorems for Fleming-Viot processes. Ann of Probab, 1998, 26: 533–561
Feller W. The parabolic differential equations and the associated semi-groups of transformations. Ann of Math, 1952, 55: 468–519
Feller W. Diffusion processes in one dimension. Trans Amer Math Soc, 1954, 17: 1–31
Feller W. Generalized second order differential operators and their lateral conditions. Illinois J Math, 1957, 1: 459–504
Grigorescu I, Kang M. Brownian motion on the figure eight. J Theory Probab, 2002, 15: 817–844
Grigorescu I, Kang M. Ergodic ptoperty of multidimensional Brownian motion with rebirth. Electron J Probab, 2007, 12: 1299–1322
Hou Z, Liu G. Markov Skeleton Processes and Their Applications. Beijing/Hong Kong: Science Press and International Press, 2005
Ito K, Mckean H P. Diffusion Processes and Their Sample Paths. Berlin: Springer-Verlag, 1965
Karlin S, Taylor H M. A Second Course in Stochastic Processes. New York: Academic Press, 1981
Karatzas I, Shreve S E. Brownian Motion and Stochastic Calculus. New York: Springer-Verlag, 1991
Kolb M, Wubker A. On the spectral gap of Brownian motion with jump boundary. Electron J Probab, 2011, 43: 1214–1237
Kosygina E. Brownian flow on a finite interval with jump boundary conditions. Disc Cont Dyn Syst Ser B, 2006, 6: 867–880
Leung Y, Li W V, Rakesh. Spectral analysis of Brownian motion with jump boundary. Proc Amer Math Soc, 2008, 136: 4427–4436
Leung Y, Li W V. Fastest rate of convergence for Brownian motion with jump boundary. Preprint, 2011
Liggett T M. Interacting Particle Systems. New York: Springer-Verlag, 1985
Mandl P. Analytical Treatment of One-dimensional Markov Processes. New York: Springer-Verlag, 1968
Meyn P, Tweedie R L. Markov Chains and Stochastic Stability. New York: Springer-Verlag, 1993
Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
Pinsky R G. Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1995
Rudin W. Real and complex analysis, second edition, McGraw-Hill Series in Higher Mathematics. New York: McGraw-Hill Book Co., 1974
Taira K. Diffusion Processes and Partial Differential Equations. New York: Academic Press, 1988
Venttsel A D. On boundary conditions for multidimensional diffusion processes. Theory Probab Appl, 1959, 4: 164–177
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Peng, J., Li, W.V. Diffusions with holding and jumping boundary. Sci. China Math. 56, 161–176 (2013). https://doi.org/10.1007/s11425-012-4416-9
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DOI: https://doi.org/10.1007/s11425-012-4416-9