Electronic Journal of Probability

Regularity of stochastic kinetic equations

Abstract

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 48, 42 pp.

Dates
Accepted: 3 May 2017
First available in Project Euclid: 31 May 2017

https://projecteuclid.org/euclid.ejp/1496196076

Digital Object Identifier
doi:10.1214/17-EJP65

Mathematical Reviews number (MathSciNet)
MR3661662

Zentralblatt MATH identifier
1371.35372

Citation

Fedrizzi, Ennio; Flandoli, Franco; Priola, Enrico; Vovelle, Julien. Regularity of stochastic kinetic equations. Electron. J. Probab. 22 (2017), paper no. 48, 42 pp. doi:10.1214/17-EJP65. https://projecteuclid.org/euclid.ejp/1496196076

References

• [1] Beck L. and Flandoli F.: A regularity theorem for quasilinear parabolic systems under random perturbations. Journal of Evolution Equations 13, (2013), 829–874.
• [2] Beck L., Flandoli F., Gubinelli M. and Maurelli M.: Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness. arXiv:1401.1530.
• [3] Bergh J. and Lofström J.: Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976.
• [4] Bouchut F.: Hypoelliptic regularity in kinetic equations. J. Math Pures Appl. 81, (2002), 1135–1159.
• [5] Bramanti M., Cupini G., Lanconelli E. and Priola E.: Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators. Math. Z. 266, (2010), 789–816.
• [6] Chaudru de Raynal P.E.: Strong existence and uniqueness for degenerate SDE with Hölder drift. Ann. Inst. H. Poincaré Probab. Statist. 53 (1), (2017), 259–286.
• [7] Chaudru de Raynal P.E.: Weak well posedness for hypoelliptic stochastic differential equations with singular drift: a sharp result. arXiv:1606.05458.
• [8] Coghi M. and Flandoli F.: Propagation of chaos for interacting particles subject to environmental noise. Ann. Appl. Probab. 26 (3), (2016), 1407–1442.
• [9] Cwikel M.: On $(L^{p_0}(A_{0}),\, L^{p_{1}}(A_{1}))_{\theta ,\,{q}}$. Proc. Amer. Math. Soc. 44, (1974), 286–292.
• [10] Da Prato G., Flandoli F., Priola E. and Röckner M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 (5), (2013), 3306–3344.
• [11] Delarue F., Flandoli F. and Vincenzi D.: Noise prevents collapse of Vlasov-Poisson point charges. Comm. Pure Appl. Math. 67 (10), (2014), 1700–1736.
• [12] DiPerna R. and LionsP.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, (1989), 511–547.
• [13] Falkovich G., Gawedzki K. and Vergassola M.: Particles and fields in fluid turbulence. Rev. Modern Phys. 73 (4), (2001), 913–975.
• [14] Fedrizzi E. and Flandoli F.: Pathwise uniqueness and continuous dependence for SDEs with non-regular drift. Stochastics 83 (3), (2011), 241–257.
• [15] Fedrizzi E. and Flandoli F.: Hölder Flow and Differentiability for SDEs with Nonregular Drift. Stochastic Analysis and Applications 31 (4), (2013), 708–736.
• [16] Fedrizzi E. and Flandoli F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264, (2013), 1329–1354.
• [17] Fedrizzi E., Flandoli F., Priola E. and Vovelle J.: Regularity of stochastic kinetic equations. Preprint version arXiv:1606.01088.
• [18] Flandoli F.: Random perturbation of PDEs and fluid dynamic models. Saint Flour Summer School Lectures, 2010, Lecture Notes in Math. 2015, Springer, Berlin, 2011.
• [19] Flandoli F., Gubinelli M. and Priola E.: Well–posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (1), (2010), 1–53.
• [20] Golse F. and Saint-Raymond L.: Velocity averaging in $L^1$ for the transport equation. C. R. Acad. Sci. Paris Sér. I Math. 334, (2002), 557–562.
• [21] Han-Kwan D.: $L^{1}$ averaging lemma for transport equations with Lipschitz force fields. Kinet. Relat. Models 3 (4), (2010), 669–683.
• [22] Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes. North Holland-Kodansha, II edition, 1989.
• [23] Khas’minskii R.Z.: On positive solutions of the equation $Au+Vu=0$. Theoret. Probab. Appl. 4, (1959), 309–318.
• [24] Krylov N.V. and Röckner M.: Strong solutions to stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, (2005), 154–196.
• [25] Kunita H.: Stochastic differential equations and stochastic flows of diffeomorphisms. XII École d’été de probabilités de Saint-Flour - 1982. Lecture Notes in Math. 1097, 143–303. Springer, Berlin, 1984.
• [26] Kunita H.: Stochastic flows and stochastic differential equations. Cambridge Stud. Adv. Math., Cambridge University Press, 1990.
• [27] Lions J.L. and Peetre J.: Sur une classe d’espaces d’interpolation. Inst. Hautes Etudes Sci. Publ. Math. 19, (1964), 5–68.
• [28] Liptser R. and Shiryaev A.N.: Statistics of Random Processes I. General Theory. Springer, 2001.
• [29] Lunardi A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, 1995.
• [30] Priola E.: Pathwise uniqueness for singular SDEs driven by stable processes. Osaka Journal of Mathematics 49, (2012), 421–447.
• [31] Priola E.: On weak uniqueness for some degenerate SDEs by global $L^p$ estimates. Potential Anal. 42 (1), (2015), 247–281.
• [32] Stein E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
• [33] Strichartz R.S.: Fubini-type theorems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 22, (1968), 399–408.
• [34] Sznitman A.-S.: Brownian motion, obstacles, and random media. Springer, 1998.
• [35] Triebel H.: Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland, 1978.
• [36] Veretennikov A.J.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 111 (153) (3), (1980), 434–452.
• [37] Wang F.-Y. and Zhang X.: Degenerate SDEs in Hilbert Spaces with Rough Drifts. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (4), (2015).
• [38] Wang F.-Y. and Zhang X.: Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient. SIAM J. Math. Anal. 48 (3), (2016), 2189–2226.
• [39] Xie L. and Zhang X.: Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients. Ann. Probab. 44 (6), (2016), 3661–3687.
• [40] Zhang X.: Stochastic partial differential equations with unbounded and degenerate coefficients. J. Differential Equations 250, (2011), 1924–1966.
• [41] Zhang X.: Stochastic Hamiltonian flows with singular coefficients. arXiv:1606.04360.