×

zbMATH — the first resource for mathematics

A simple method for the existence of a density for stochastic evolutions with rough coefficients. (English) Zbl 1406.60089
Summary: We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [A. Debussche and the author, Probab. Theory Relat. Fields 158, No. 3–4, 575–596 (2014; Zbl 1452.76193)], by proving local estimates for the density, existence for the density with summable drift, and by improving the regularity of the density.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34K50 Stochastic functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] Randolf Altmayer, Estimating occupation time functionals, arXiv:1706.03418, 2017.
[2] Nachman Aronszajn and Kennan T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475. · Zbl 0102.32401
[3] David Baños and Paul Krühner, Optimal density bounds for marginals of Itô processes, Commun. Stoch. Anal. 10 (2016), no. 2, 131–150.
[4] David Baños and Paul Krühner, Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients, Stochastic Process. Appl. 127 (2017), no. 6, 1785–1799. · Zbl 1367.60064
[5] Vlad Bally and Lucia Caramellino, Regularity of probability laws by using an interpolation method, 2012, arXiv:1211.0052. · Zbl 1371.60096
[6] Vlad Bally and Lucia Caramellino, Integration by parts formulas, Malliavin calculus, and regularity of probability laws, Stochastic integration by parts and functional Itô calculus, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, [Cham], 2016, pp. 1–114. · Zbl 1372.60073
[7] Vlad Bally and Lucia Caramellino, Convergence and regularity of probability laws by using an interpolation method, Ann. Probab. 45 (2017), no. 2, 1110–1159. · Zbl 1377.60066
[8] Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246. · Zbl 0495.35024
[9] Giuseppe Cannizzaro, Peter K. Friz, and Paul Gassiat, Malliavin calculus for regularity structures: the case of gPAM, J. Funct. Anal. 272 (2017), no. 1, 363–419. · Zbl 1367.60062
[10] Rémi Catellier and Khalil Chouk, Paracontrolled distributions and the 3-dimensional stochastic quantization equation, arXiv:1310.6869, to appear on Annals of Probability, 2018. · Zbl 1433.60048
[11] Zhen-Qing Chen and Longmin Wang, Uniqueness of stable processes with drift, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2661–2675. · Zbl 1335.60085
[12] Giuseppe Da Prato and Arnaud Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl. (9) 82 (2003), no. 8, 877–947. · Zbl 1109.60047
[13] Stefano 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), no. 4, 1282–1321. · Zbl 1246.60082
[14] Arnaud Debussche and Nicolas Fournier, Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients, J. Funct. Anal. 264 (2013), no. 8, 1757–1778. · Zbl 1272.60032
[15] Arnaud Debussche and Marco Romito, Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise, Probab. Theory Related Fields 158 (2014), no. 3-4, 575–596. · Zbl 1452.76193
[16] Romain Duboscq and Anthony Réveillac, Stochastic regularization effects of semi-martingales on random functions, J. Math. Pures Appl. (9) 106 (2016), no. 6, 1141–1173. · Zbl 1351.35274
[17] Ennio Fedrizzi and Franco Flandoli, Hölder flow and differentiability for SDEs with nonregular drift, Stoch. Anal. Appl. 31 (2013), no. 4, 708–736. · Zbl 1281.60055
[18] Franco Flandoli, Massimiliano Gubinelli, and Enrico Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1–53. · Zbl 1200.35226
[19] Nicolas Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab. 25 (2015), no. 2, 860–897. · Zbl 1322.82013
[20] Nicolas Fournier and Jacques Printems, Absolute continuity for some one–dimensional processes, Bernoulli 16 (2010), no. 2, 343–360. · Zbl 1248.60062
[21] Martin Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. · Zbl 1332.60093
[22] Martin Hairer, Introduction to regularity structures, Braz. J. Probab. Stat. 29 (2015), no. 2, 175–210. · Zbl 1316.81061
[23] Martin Hairer, Regularity structures and the dynamical \(Φ ^4_3\) model, Current developments in mathematics 2014, Int. Press, Somerville, MA, 2016, pp. 1–49.
[24] Masafumi Hayashi, Arturo Kohatsu-Higa, and Gô Yûki, Local Hölder continuity property of the densities of solutions of SDEs with singular coefficients, J. Theoret. Probab. 26 (2013), no. 4, 1117–1134. · Zbl 1294.60081
[25] Masafumi Hayashi, Arturo Kohatsu-Higa, and Gô Yûki, Hölder continuity property of the densities of SDEs with singular drift coefficients, Electron. J. Probab. 19 (2014), no. 77, 22. · Zbl 1310.60081
[26] Lorick Huang, Density estimates for SDEs driven by tempered stable processes, arXiv:1504.04183, 2015. · Zbl 1336.60137
[27] Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992. · Zbl 0752.60043
[28] Victoria Knopova and Alexei Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by \(α \)-stable noise, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 100–140. · Zbl 1391.60191
[29] Arturo Kohatsu-Higa and Libo Li, Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients, Stoch. Anal. Appl. 34 (2016), no. 6, 979–1024. · Zbl 1350.60050
[30] Nicolai V. Krylov and Michael Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154–196.
[31] Seiichiro Kusuoka, Existence of densities of solutions of stochastic differential equations by Malliavin calculus, J. Funct. Anal. 258 (2010), no. 3, 758–784. · Zbl 1198.60025
[32] Claude Le Bris and Pierre-Louis Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations 33 (2008), no. 7-9, 1272–1317.
[33] Paul Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) (New York-Chichester-Brisbane), Wiley, 1978, pp. 195–263.
[34] Thilo Meyer-Brandis and Frank Proske, Construction of strong solutions of SDE’s via Malliavin calculus, J. Funct. Anal. 258 (2010), no. 11, 3922–3953. · Zbl 1195.60082
[35] Jean-Christophe Mourrat, Hendrik Weber, Global well-posedness of the dynamic \(Φ ^4\) model in the plane, Ann. Probab. 45 (2017), no. 4, 2398–2476. · Zbl 1381.60098
[36] Jean-Christophe Mourrat, Hendrik Weber, and Weijun Xu, Construction of \(Φ ^4_3\) diagrams for pedestrians, From particle systems to partial differential equations, Springer Proc. Math. Stat., vol. 209, Springer, Cham, 2017, pp. 1–46. · Zbl 1390.81266
[37] Jean-Christophe Mourrat and Hendrik Weber, The dynamic \(Φ ^4_3\) model comes down from infinity, Comm. Math. Phys. 356 (2017), no. 3, 673–753. · Zbl 1384.81068
[38] David Nualart, The Malliavin calculus and related topics, second ed., Probability and its Applications (New York), Springer-Verlag, Berlin, Berlin, 2006. · Zbl 1099.60003
[39] Marco Romito, Unconditional existence of densities for the Navier-Stokes equations with noise, Mathematical analysis of viscous incompressible fluid, RIMS Kôkyûroku, vol. 1905, Kyoto University, 2014, pp. 5–17.
[40] Marco Romito, Uniqueness and blow-up for a stochastic viscous dyadic model, Probab. Theory Related Fields 158 (2014), no. 3-4, 895–924. · Zbl 1387.35638
[41] Marco Romito, Hölder continuity of the densities for the Navier–Stokes equations with noise, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 3, 691–711. · Zbl 1352.60093
[42] Marco Romito, Some probabilistic topics in the Navier–Stokes equations, Recent Progress in the Theory of the Euler and Navier–Stokes Equations (James C. Robinson, José L. Rodrigo, Witold Sadowski, and Alejandro Vidal-López, eds.), London Math. Soc. Lecture Note Ser., vol. 430, Cambridge Univ. Press, Cambridge, 2016, pp. 175–232. · Zbl 1362.35215
[43] Marco Romito, Time regularity of the densities for the Navier–Stokes equations with noise, J. Evol. Equations 16 (2016), no. 3, 503–518. · Zbl 1360.76218
[44] Marta Sanz-Solé and André Süß, Absolute continuity for SPDEs with irregular fundamental solution, Electron. Commun. Probab. 20 (2015), no. 14, 11.
[45] Marta Sanz-Solé and André Süß, Non-elliptic SPDEs and ambit fields: existence of densities, Stochastics of environmental and financial economics—Centre of Advanced Study, Oslo, Norway, 2014–2015, Springer Proc. Math. Stat., vol. 138, Springer, Cham, 2016, pp. 121–144.
[46] René L. Schilling, Paweł Sztonyk, and Jian Wang, Coupling property and gradient estimates of Lévy processes via the symbol, Bernoulli 18 (2012), no. 4, 1128–1149. · Zbl 1263.60045
[47] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. · Zbl 0546.46028
[48] Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992. · Zbl 0763.46025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.