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Strong Feller Property of Sticky Reflected Distorted Brownian Motion

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Abstract

Using Girsanov transformations we construct from sticky reflected Brownian motion on \([0,\infty )\) a conservative diffusion on \(E:=[0,\infty )^n\), \(n \in \mathbb {N}\), and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies of the mathematical model are motivated by its applications to the dynamical wetting model with \(\delta \)-pinning and repulsion.

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References

  1. Albeverio, S., Kondratiev, Y., Röckner, M.: Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interactions. In: Finite and Infinite Dimensional Analysis in Honor of Lenorad Gross, Volume 317 of Contemporary Mathematics. American Mathematical Society, Providence (2003)

  2. Baur, B., Grothaus, M.: Construction and strong Feller property of distorted elliptic diffusion with reflecting boundary. Potential Anal. 40(4), 391–425 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baur, B., Grothaus, M., Stilgenbauer, P.: Construction of \(\cal{L}^p\)-strong Feller processes via Dirichlet forms and applications to elliptic diffusions. Potential Anal. 38(4), 1233–1258 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. Walter de Gruyter & Co., Berlin (1991)

    Book  MATH  Google Scholar 

  5. Chen, Z., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory, Volume 35 of London Mathematical Society Monographs, vol. 35. Princeton University Press, Princeton (2012)

    Google Scholar 

  6. Chen, Z., Kuwae, K.: On doubly Feller property. Osaka J. Math. 46, 909–930 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Chung, K.L.: Doubly-Feller process with multiplicative functional. In: Seminar on Stochastic Processes, 1985, Volume 12 of Progress in Probability and Statistics, pp. 63–78. Birkhäuser, Boston (1986)

  8. Eberle, A.: Girsanov-type transformations of local Dirichlet forms: an analytic approach. Osaka J. Math. 33, 497–531 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Engelbert, H.-J., Peskir, G.: Stochastic differential equations for sticky Brownian motion. Stochastics 86(6), 993–1021 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fattler, T., Grothaus, M.: Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous \(N\)-particle systems with singular interactions. J. Funct. Anal. 246, 217–241 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fitzsimmons, P.J.: Absolute continuity of symmetric diffusions. Ann. Probab. 25(1), 230–258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fitzsimmons, P.J.: The Dirichlet form of a gradient-type drift transformation of a symmetric diffusion. Acta Math. Sinica 24(7), 1057–1066 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fattler, T., Grothaus, M., Voßhall, R.: Construction and analysis of a sticky reflected distorted Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 52(2), 735–762 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Volume 19 of de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin (2011)

    Google Scholar 

  15. Funaki, T.: Stochastic interface models. In: Lectures on Probability Theory and Statistics, Volume 1869 of Lecture Notes in Mathematics, pp. 103–274. Springer, Berlin (2005)

  16. Fukushima, M., Tomisaki, M.: Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab. Theory Relat. Fields 106, 521–557 (1996)

    Article  MATH  Google Scholar 

  17. Giacomin, G.: Limit theorems for random interface models of Ginzburg–Landau \(\nabla \phi \) type. In: Stochastic Partial Differential Equations and Applications (Trento, 2002), Volume 227 of Lecture Notes in Pure and Applied Mathematics, pp. 235–253. Dekker, New York (2002)

  18. Graham, C.: The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary. Ann. Inst. Henri Poincaré 24(1), 45–72 (1988)

    MathSciNet  MATH  Google Scholar 

  19. Ikeda, N., Watanabe, Sh: Stochastic Differential Equations and Diffusion Processes, Volume 24 of North-Holland Mathematical Library, 2nd edn. North-Holland Publishing Co., Amsterdam (1989)

    Google Scholar 

  20. Kallenberg, O.: Foundations of modern probability. Probability and its Applications. Springer, New York (1997)

    MATH  Google Scholar 

  21. Knight, F.B.: Essentials of Brownian motion and diffusion, Volume 18 of Mathematical Surveys, vol. 18. American Mathematical Society, Providence (1981)

    Book  Google Scholar 

  22. Kostrykin, V., Potthoff, J., Schrader, R.: Brownian motion on metric graphs: Feller Brownian motions on intervals revisited. arXiv:1008.3761 (2010)

  23. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, Volume 113 of Graduate Texts in Mathematics, vol. 113. Springer, New York (1998)

    Book  Google Scholar 

  24. Revuz, D.: Markov Chains, 2nd edn. North-Holland Mathematical Library, North-Holland (1984)

    MATH  Google Scholar 

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Acknowledgements

We thank Torben Fattler for helpful comments and discussions. R. Voßhall gratefully acknowledges financial support in the form of a fellowship of the German state Rhineland-Palatine. Moreover, we thank an anonymous referee for helpful comments improving the readability of the paper.

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  1. University of Kaiserslautern, P.O. Box 3049, 67653, Kaiserslautern, Germany

    Martin Grothaus & Robert Voßhall

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  1. Martin Grothaus
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  2. Robert Voßhall
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Correspondence to Martin Grothaus.

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Grothaus, M., Voßhall, R. Strong Feller Property of Sticky Reflected Distorted Brownian Motion. J Theor Probab 31, 827–852 (2018). https://doi.org/10.1007/s10959-016-0735-z

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  • DOI: https://doi.org/10.1007/s10959-016-0735-z

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