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Density for solutions to stochastic differential equations with unbounded drift. (English) Zbl 1427.60109
Summary: Via a special transform and by using the techniques of the Malliavin calculus, we analyze the density of the solution to a stochastic differential equation with unbounded drift.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
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[1] Bally, V. and Caramellino, L. (2017). Regularity of probability laws by using an interpolation method. Annals of Probability45, 1110-1159. · Zbl 1377.60066
[2] Bally, V., Caramellino, L. and Cont, R. (2016). Stochastic integration by parts and functional Itô calculus. In Lecture Notes of the Barcelona Summer School on Stochastic Analysis Held in Barcelona, July 23-27, 2012. Advanced Courses in Mathematics. CRM Barcelona Cham: Birkhäuser/Springer.
[3] Baños, D. and Krühner, P. (2017). Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients. Stochastic Processes and Their Applications127, 1785-1799.
[4] De Marco, S. (2011). Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. The Annals of Applied Probability21, 1282-1321. · Zbl 1246.60082
[5] Debussche, A. and Fournier, N. (2013). Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. Journal of Functional Analysis264, 1757-1778. · Zbl 1272.60032
[6] Flandoli, F., Gubinelli, M. and Priola, E. (2010a). Well-posedness of the transport equation by stochastic perturbation. Inventiones Mathematicae180, 1-53. · Zbl 1200.35226
[7] Flandoli, F., Gubinelli, M. and Priola, E. (2010b). Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bulletin Des Sciences Mathématiques134, 405-422. · Zbl 1198.60023
[8] Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimension al processes. Bernoulli16, 343-360. · Zbl 1248.60062
[9] Hayashi, M., Kohatsu-Higa, A. and Yuki, G. (2013). Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients. Journal of Theoretical Probability26, 1117-1134. · Zbl 1294.60081
[10] Kohatsu-Higa, A. (2003). Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probability Theory and Related Fields126, 421-457. · Zbl 1022.60056
[11] Kohatsu-Higa, A. and Makhlouf, A. (2013). Estimates for the density of functionals of sdes with irregular drift. Stochastic Processes and Their Applications123, 1716-1728. · Zbl 1279.60070
[12] Kohatsu-Higa, A. and Tanaka, A. (2012). Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts. Annales de L’IHP Probabilités et Statistiques48, 871-883. · Zbl 1248.60058
[13] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In Ecole D’été de Probabilités de Saint-Flour, XII-1982. Lecture Notes in Math.1097, 143-303. Berlin: Springer.
[14] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus from Stein’s Method to Universality. Cambridge: Cambridge University Press. · Zbl 1266.60001
[15] Nourdin, I. and Viens, F. (2009). Density formula and concentration inequalities with Malliavin calculus. Electronic Journal of Probability14, 2287-2309. · Zbl 1192.60066
[16] Nualart, D. (2006). Malliavin Calculus and Related Topics, 2nd ed. New York: Springer. · Zbl 1099.60003
[17] Nualart, D. and Quer-Sardanyons, L. (2009). Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations. Stochastic Processes and Their Applications119, 3914-3938. · Zbl 1190.60041
[18] Romito, M. (2016). Time regularity of the densities for the Navier-Stokes equations with noise. Journal of Evolution Equations16, 503-518. · Zbl 1360.76218
[19] Romito, M. (2017). A simple method for the existence of a density for stochastic evolutions with rough coefficients. Preprint. Available at arXiv:1707.05042. · Zbl 1406.60089
[20] Sanz-Solé, M. (1995). Malliavin Calculus. With Applications to Stochastic Partial Differential Equations. Lausanne: Fundamental Sciences, EPFL Press.
[21] Shigekawa, I. (1980). Derivatives of Wiener functionals and absolute continuity of induced measures. Journal of Mathematics of Kyoto University20, 263-289. · Zbl 0476.28008
[22] Zhang, X. (2014). Stochastic differential equations with Sobolev diffusion and singular drift. The Annals of Applied Probability26, 2697-2732. · Zbl 1353.60056
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