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

Röckner M, Shin J, Trutnau G (2016)
Discrete and Continuous Dynamical Systems. Series B 21(9): 3219-3237.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
Abstract / Bemerkung
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 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.
Erscheinungsjahr
Zeitschriftentitel
Discrete and Continuous Dynamical Systems. Series B
Band
21
Ausgabe
9
Seite(n)
3219-3237
ISSN
eISSN
PUB-ID

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Röckner M, Shin J, Trutnau G. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems. Series B. 2016;21(9):3219-3237.
Röckner, M., Shin, J., & Trutnau, G. (2016). Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems. Series B, 21(9), 3219-3237. doi:10.3934/dcdsb.2016095
Röckner, M., Shin, J., and Trutnau, G. (2016). Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems. Series B 21, 3219-3237.
Röckner, M., Shin, J., & Trutnau, G., 2016. Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems. Series B, 21(9), p 3219-3237.
M. Röckner, J. Shin, and G. Trutnau, “Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results”, Discrete and Continuous Dynamical Systems. Series B, vol. 21, 2016, pp. 3219-3237.
Röckner, M., Shin, J., Trutnau, G.: Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results. Discrete and Continuous Dynamical Systems. Series B. 21, 3219-3237 (2016).
Röckner, Michael, Shin, Jiyong, and Trutnau, Gerald. “Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results”. Discrete and Continuous Dynamical Systems. Series B 21.9 (2016): 3219-3237.