Electronic Journal of Probability

A simple method for the existence of a density for stochastic evolutions with rough coefficients

Marco Romito

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We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [15], by proving local estimates for the density, existence for the density with summable drift, and by improving the regularity of the density.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 113, 43 pp.

Received: 13 October 2017
Accepted: 3 November 2018
First available in Project Euclid: 23 November 2018

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H15: Stochastic partial differential equations [See also 35R60] 34K50: Stochastic functional-differential equations [See also , 60Hxx]

density of laws stochastic differential equations irregular coefficients Besov spaces

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Romito, Marco. A simple method for the existence of a density for stochastic evolutions with rough coefficients. Electron. J. Probab. 23 (2018), paper no. 113, 43 pp. doi:10.1214/18-EJP242. https://projecteuclid.org/euclid.ejp/1542942364

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  • [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.
  • [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.
  • [5] Vlad Bally and Lucia Caramellino, Regularity of probability laws by using an interpolation method, 2012, arXiv:1211.0052.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [11] Zhen-Qing Chen and Longmin Wang, Uniqueness of stable processes with drift, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2661–2675.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [17] Ennio Fedrizzi and Franco Flandoli, Hölder flow and differentiability for SDEs with nonregular drift, Stoch. Anal. Appl. 31 (2013), no. 4, 708–736.
  • [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.
  • [19] Nicolas Fournier, Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition, Ann. Appl. Probab. 25 (2015), no. 2, 860–897.
  • [20] Nicolas Fournier and Jacques Printems, Absolute continuity for some one–dimensional processes, Bernoulli 16 (2010), no. 2, 343–360.
  • [21] Martin Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504.
  • [22] Martin Hairer, Introduction to regularity structures, Braz. J. Probab. Stat. 29 (2015), no. 2, 175–210.
  • [23] Martin Hairer, Regularity structures and the dynamical $\Phi ^4_3$ 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.
  • [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.
  • [26] Lorick Huang, Density estimates for SDEs driven by tempered stable processes, arXiv:1504.04183, 2015.
  • [27] Peter E. Kloeden and Eckhard Platen, Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23, Springer-Verlag, Berlin, 1992.
  • [28] Victoria Knopova and Alexei Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha $-stable noise, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 100–140.
  • [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.
  • [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.
  • [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.
  • [35] Jean-Christophe Mourrat, Hendrik Weber, Global well-posedness of the dynamic $\Phi ^4$ model in the plane, Ann. Probab. 45 (2017), no. 4, 2398–2476.
  • [36] Jean-Christophe Mourrat, Hendrik Weber, and Weijun Xu, Construction of $\Phi ^4_3$ diagrams for pedestrians, From particle systems to partial differential equations, Springer Proc. Math. Stat., vol. 209, Springer, Cham, 2017, pp. 1–46.
  • [37] Jean-Christophe Mourrat and Hendrik Weber, The dynamic $\Phi ^4_3$ model comes down from infinity, Comm. Math. Phys. 356 (2017), no. 3, 673–753.
  • [38] David Nualart, The Malliavin calculus and related topics, second ed., Probability and its Applications (New York), Springer-Verlag, Berlin, Berlin, 2006.
  • [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.
  • [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.
  • [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.
  • [43] Marco Romito, Time regularity of the densities for the Navier–Stokes equations with noise, J. Evol. Equations 16 (2016), no. 3, 503–518.
  • [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.
  • [47] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983.
  • [48] Hans Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992.