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Stochastic formulations of the parametrix method. (English) Zbl 1403.60047
Summary: In this manuscript, we consider stochastic expressions of the parametrix method for solutions of $$d$$-dimensional stochastic differential equations (SDEs) with drift coefficients which belong to $$L^p(\mathbb R^d)$$, $$p > d$$. We prove the existence and Hölder continuity of probability density functions for distributions of solutions at fixed points and obtain an explicit expansion via (stochastic) parametrix methods. We also obtain Gaussian type upper and lower bounds for these probability density functions.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes
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##### References:
 [1] W. Arendt, G. Metafune and D. Pallara, Gaussian estimates for elliptic operators with unbounded drift. J. Math. Anal. Appl. 338 (2008) 505-517. · Zbl 1138.47038 [2] V. Bally and A. Kohatsu-Higa, A probabilistic interpretation of the parametrix method. Ann. Appl. Probab. 25 (2015) 3095-3138. · Zbl 1329.35164 [3] D. Baños and P. Krühner, Optimal density bounds for marginals of Itô processes. Commun. Stoch. Anal. 10 (2016) 1. [4] D. Baños and P. Krühner, Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients. Stoch. Proc. Appl. 127 (2017) 1785-1799. · Zbl 1367.60064 [5] A.S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, Vol. 1858. Springer Science & Business Media, Berlin (2005). · Zbl 1071.60003 [6] F. Corielli, P. Foschi and A. Pascucci. Parametrix approximation of diffusion transition densities. SIAM J. Finan. Math. 1 (2010) 833-867. [7] G. Da Prato, F. Flandoli, E. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations in hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 (2013) 3306-3344. · Zbl 1291.35455 [8] G. Da Prato, F. Flandoli, E. Priola and M. Röckner, Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theor. Probab. 28 (2015) 1571-1600. · Zbl 1332.60091 [9] S. De Marco, Smoothness and asymptotic estimates of densities for sdes with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 (2011) 1282-1321. · Zbl 1246.60082 [10] M.D. Francesco and A. Pascucci, On a class of degenerate parabolic equations of kolmogorov type. AMRX Appl. Math. Res. Express3 (2005) 77-116. · Zbl 1085.35086 [11] A. Friedman, Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs, NJ (1964). · Zbl 0144.34903 [12] M. Hayashi, A. Kohatsu-Higa and G. Yûki, Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients. J. Theoret. Probab. 26 (2013) 1117-1134. · Zbl 1294.60081 [13] M. Hayashi, A. Kohatsu-Higa and G. Yûki, Hölder continuity property of the densities of SDEs with singular drift coefficients. Electron. J. Probab. 19 (2014) 1-22. · Zbl 1310.60081 [14] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd Edn. Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1991). · Zbl 0734.60060 [15] P. Kim and R. Song, Two-sided estimates on the density of brownian motion with singular drift. Illinois J. Math. 50 (2006) 635-688. · Zbl 1110.60071 [16] A. Kohatsu-Higa and A. Makhlouf, Estimates for the density of functionals of SDEs with irregular drift. Stoch. Process. Appl. 123 (2013) 1716-1728. · Zbl 1279.60070 [17] A. Kohatsu-Higa and A. Tanaka, A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts. Ann.Inst. Henri Poincaré Probab. Stat. 48 (2012) 871-883. · Zbl 1248.60058 [18] N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields131 (2005) 154-196. · Zbl 1072.60050 [19] S. Kusuoka, Continuity and gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation. Stoch. Process. Appl. (2016) 359-384. · Zbl 1354.60059 [20] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1-76. · Zbl 0568.60059 [21] A. Makhlouf, Existence and Gaussian bounds for the density of Brownian motion with random drift. Commun. Stoch. Anal. 10 (2016) 151-162. [22] D. Nualart, The Malliavin Calculus and Related Topics, 2nd Edn. Probability and its Applications (New York). Springer-Verlag, Berlin (2006). · Zbl 1099.60003 [23] A. Pascucci and A. Pesce, The Parametrix Method for Parabolic Spdes. Preprint (2018). [24] N. Portenko, Generalized Diffusion Processes. Translations of Mathematical Monographs. American Mathematical Society, Providence,RI (1990). [25] Z. Qian and W. Zheng, A representation formula for transition probability densities of diffusions and applications. Stoch. Process. Appl. 111 (2004) 57-76. · Zbl 1070.60072 [26] Z. Qian, F. Russo and W. Zheng, Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Relat. Fields127 (2003) 388-406. · Zbl 1050.60061 [27] M. Romito, A Simple Method for the Existence of a Density for Stochastic Evolutions with Rough Coefficients. Preprint (2017). [28] J. Shin and G. Trutnau, On singular stochastic differential equations and dirichlet forms. Rev. Roumaine Math. Pures Appl. 62 (2017a) 217-258. · Zbl 1399.31012 [29] J. Shin and G. Trutnau, On the stochastic regularity of distorted brownian motions. Trans. Am. Math. Soc. 369 (2017b) 7883-7915. · Zbl 1379.60092 [30] A.J. Veretennikov and N.V. Krylov, Explicit formulae for the solutions of stochastic equations. Mat. Sb. (N.S.)100 (1976) 266-284. · Zbl 0353.60059
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