November  2016, 21(9): 3219-3237. doi: 10.3934/dcdsb.2016095

Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results

1. 

Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse 25, 33615 Bielefeld, Germany

2. 

School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02445, South Korea

3. 

Department of Mathematical Sciences and Research Institute of Mathematics of Seoul National University, 1, Gwanak-Ro, Gwanak-Gu, Seoul 08826, South Korea

Received  September 2015 Revised  March 2016 Published  October 2016

Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. This non-symmetric distorted Brownian motion is also proved to be strong Feller. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] that the constructed weak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet forms to prove further properties of the solutions. For example, we obtain new non-explosion criteria for them. We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion related to a class of Muckenhoupt weights and corresponding divergence free perturbations.
Citation: Michael Röckner, Jiyong Shin, Gerald Trutnau. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3219-3237. doi: 10.3934/dcdsb.2016095
References:
[1]

S. Albeverio, Yu. G. Kondratiev and M. Röckner, Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interactions,, Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, (2001), 15. doi: 10.1090/conm/317/05517. Google Scholar

[2]

W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Form Methods for Evolution Equations and Application,, The 18th internet seminar, (2014). Google Scholar

[3]

B. Baur, Elliptic Boundary Value Problems and Constructions of $L^p$-strong Feller Processes with Singular Drift and Reflection,, Dissertation, (2013). doi: 10.1007/978-3-658-05829-6. Google Scholar

[4]

V. Bogachev, N. Krylov and M. Röckner, Elliptic regularity and essential self-adjointness of Dirichlet operators on $R^d$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 451. Google Scholar

[5]

V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. Partial Differential Equations, 26 (2001), 2037. doi: 10.1081/PDE-100107815. Google Scholar

[6]

K. L. Chung and J. B. Walsh, Markov Processes, Brownian Motion, and Time Symmetry,, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2005). doi: 10.1007/0-387-28696-9. Google Scholar

[7]

E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218. Google Scholar

[8]

T. Fattler and M. Grothaus, 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 (2007), 217. doi: 10.1016/j.jfa.2007.01.014. Google Scholar

[9]

E. Fedrizzi and F. Flandoli, Hölder flow and differentiability for SDEs with nonregular drift,, Stoch. Anal. Appl., 31 (2013), 708. doi: 10.1080/07362994.2012.628908. Google Scholar

[10]

M. Fukushima, Energy forms and diffusion processes,, Mathematics physics, 1 (1985), 65. doi: 10.1142/9789814415125_0002. Google Scholar

[11]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes,, Second revised and extended edition. de Gruyter Studies in Mathematics, (2011). Google Scholar

[12]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus,, (second edition), (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[13]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Prob. Th. Rel. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z. Google Scholar

[14]

Z. M. Ma and M. Röckner, Introduction to the Theory of (Non-symmetric) Dirichlet Forms,, Universitext. Springer-Verlag, (1992). doi: 10.1007/978-3-642-77739-4. Google Scholar

[15]

Y. Oshima, Semi-Dirichlet Forms and Markov Processes,, De Gruyter Studies in Mathematics, (2013). doi: 10.1515/9783110302066. Google Scholar

[16]

L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities,, Internat. Math. Res. Notices, (1992), 27. doi: 10.1155/S1073792892000047. Google Scholar

[17]

J. Shin and G. Trutnau, On the stochastic regularity of distorted Brownian motions,, to appear in Trans. Amer. Math. Soc., (). doi: 10.1090/tran/6887. Google Scholar

[18]

W. Stannat, (Nonsymmetric) Dirichlet operators on $L^1$: Existence, uniqueness and associated Markov processes,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 99. Google Scholar

[19]

G. Trutnau, On a class of non-symmetric diffusions containing fully non-symmetric distorted Brownian motions,, Forum Math., 15 (2003), 409. doi: 10.1515/form.2003.022. Google Scholar

[20]

G. Trutnau, On Hunt processes and strict capacities associated with generalized Dirichlet forms,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8 (2005), 357. doi: 10.1142/S0219025705002013. Google Scholar

[21]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces,, Lecture notes in mathematics;1736. Springer, (2000). doi: 10.1007/BFb0103908. Google Scholar

[22]

X. Zhang, Strong solutions of SDES with singular drift and Sobolev diffusion coefficients,, Stochastic Process. Appl., 115 (2005), 1805. doi: 10.1016/j.spa.2005.06.003. Google Scholar

show all references

References:
[1]

S. Albeverio, Yu. G. Kondratiev and M. Röckner, Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interactions,, Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, (2001), 15. doi: 10.1090/conm/317/05517. Google Scholar

[2]

W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Form Methods for Evolution Equations and Application,, The 18th internet seminar, (2014). Google Scholar

[3]

B. Baur, Elliptic Boundary Value Problems and Constructions of $L^p$-strong Feller Processes with Singular Drift and Reflection,, Dissertation, (2013). doi: 10.1007/978-3-658-05829-6. Google Scholar

[4]

V. Bogachev, N. Krylov and M. Röckner, Elliptic regularity and essential self-adjointness of Dirichlet operators on $R^d$,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 451. Google Scholar

[5]

V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. Partial Differential Equations, 26 (2001), 2037. doi: 10.1081/PDE-100107815. Google Scholar

[6]

K. L. Chung and J. B. Walsh, Markov Processes, Brownian Motion, and Time Symmetry,, Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2005). doi: 10.1007/0-387-28696-9. Google Scholar

[7]

E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations,, Comm. Partial Differential Equations, 7 (1982), 77. doi: 10.1080/03605308208820218. Google Scholar

[8]

T. Fattler and M. Grothaus, 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 (2007), 217. doi: 10.1016/j.jfa.2007.01.014. Google Scholar

[9]

E. Fedrizzi and F. Flandoli, Hölder flow and differentiability for SDEs with nonregular drift,, Stoch. Anal. Appl., 31 (2013), 708. doi: 10.1080/07362994.2012.628908. Google Scholar

[10]

M. Fukushima, Energy forms and diffusion processes,, Mathematics physics, 1 (1985), 65. doi: 10.1142/9789814415125_0002. Google Scholar

[11]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes,, Second revised and extended edition. de Gruyter Studies in Mathematics, (2011). Google Scholar

[12]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus,, (second edition), (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[13]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Prob. Th. Rel. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z. Google Scholar

[14]

Z. M. Ma and M. Röckner, Introduction to the Theory of (Non-symmetric) Dirichlet Forms,, Universitext. Springer-Verlag, (1992). doi: 10.1007/978-3-642-77739-4. Google Scholar

[15]

Y. Oshima, Semi-Dirichlet Forms and Markov Processes,, De Gruyter Studies in Mathematics, (2013). doi: 10.1515/9783110302066. Google Scholar

[16]

L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities,, Internat. Math. Res. Notices, (1992), 27. doi: 10.1155/S1073792892000047. Google Scholar

[17]

J. Shin and G. Trutnau, On the stochastic regularity of distorted Brownian motions,, to appear in Trans. Amer. Math. Soc., (). doi: 10.1090/tran/6887. Google Scholar

[18]

W. Stannat, (Nonsymmetric) Dirichlet operators on $L^1$: Existence, uniqueness and associated Markov processes,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 99. Google Scholar

[19]

G. Trutnau, On a class of non-symmetric diffusions containing fully non-symmetric distorted Brownian motions,, Forum Math., 15 (2003), 409. doi: 10.1515/form.2003.022. Google Scholar

[20]

G. Trutnau, On Hunt processes and strict capacities associated with generalized Dirichlet forms,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8 (2005), 357. doi: 10.1142/S0219025705002013. Google Scholar

[21]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces,, Lecture notes in mathematics;1736. Springer, (2000). doi: 10.1007/BFb0103908. Google Scholar

[22]

X. Zhang, Strong solutions of SDES with singular drift and Sobolev diffusion coefficients,, Stochastic Process. Appl., 115 (2005), 1805. doi: 10.1016/j.spa.2005.06.003. Google Scholar

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