Geodesic
On some functional inequalities for skew Brownian motion
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 9-20

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We study the Poincaré and logarithmic Sobolev inequalities. We provide several constructions of skew Brownian motion; this is an example of diffusion with singular drift interesting from different points of view. We obtain inequalities for skew Brownian motion that naturally generalize the Gaussian case. It turns out that for skew Brownian motion the estimates depend on the local time of the process. 
@article{TM_2014_287_a1,
     author = {A. T. Abakirova},
     title = {On some functional inequalities for skew {Brownian} motion},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {9--20},
     publisher = {mathdoc},
     volume = {287},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_287_a1/}
}
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A. T. Abakirova. On some functional inequalities for skew Brownian motion. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 9-20. http://geodesic.mathdoc.fr/item/TM_2014_287_a1/