Abstract
We provide a general construction scheme for \(\mathcal L^p\)-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvent of kernels having the \(\mathcal L^p\)-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.
Similar content being viewed by others
References
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 Leonard Gross, Contemporary Mathematics, vol. 317. Amer. Math. Soc. Providence, RI (2003)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Bauer, H.: Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie. 3., neubearb. Aufl. de Gruyter Lehrbuch. Walter de Gruyter, Berlin-New York (1978)
Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. Pure and Applied Mathematics, vol. 29. A Series of Monographs and Textbooks. Academic Press, X, New York-London (1968)
Bogachev, V.I., Krylov, N.V., Röckner, M.: Elliptic regularity and essential self-adjointness of Dirichlet operators on ℝn. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24(3), 451–461 (1997)
Bogachev, V.I., Krylov, N.V., Röckner, M.: On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Commun. Partial Differ. Equ. 26(11–12), 2037–2080 (2001)
Dohmann, J.M.: Feller-type properties and path regularities of Markov processes. Forum Math. 17(3), 343–359 (2005)
Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
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(2), 217–241 (2007)
Fattler, T., Grothaus, M.: Construction of elliptic diffusions with reflecting boundary condition and an application to continuous N-particle systems with singular interactions. Proc. Edinb. Math. Soc., II. Ser. 51(2), 337–362 (2008)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. 2nd Revised and Extended ed. de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter, Berlin (1994)
Fukushima, M.: Energy forms and diffusion processes. In: Streit, L. (ed.) Mathematics and Physics, vol. 1, pp. viii+338, World Scientific Publishing Co., Singapore (1985)
Henning, S.-O.: Strong Feller properties for elliptic diffusions and applications to interacting particle systems. Diploma thesis, University of Kaiserslautern (2008)
Liskevich, V.A., Semenov, Y.A.: Some problems on Markov semigroups. In: Demuth, M., et al. (eds.) Schrödinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras. Math. Top., vol. 11, pp. 163–217. Akademie Verlag, Berlin (1996)
Ma, Z.-M., Röckner. M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer, VIII, Berlin (1992)
Morrey, C.B.: Multiple integrals in the calculus of variations. Reprint of the 1966 Original. Classics in Mathematics. Springer, IX, Berlin (2008)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, VIII, New York (1983)
Shaposhnikov, S.V.: On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations. Math. Notes 79(3), 413–430 (2006)
Simmons, G.F.: Introduction to topology and modern analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Company, XV, New York etc. (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baur, B., Grothaus, M. & Stilgenbauer, P. Construction of \({\mathcal L}^{{p}}\)-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions. Potential Anal 38, 1233–1258 (2013). https://doi.org/10.1007/s11118-012-9314-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-012-9314-9