Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity of stochastic differential equations with jumps and singular coefficients

Longjie Xie and Xicheng Zhang

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Abstract

We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drifts. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov’s a priori estimates for SDEs.

Résumé

Nous montrons que les EDS dirigées par un bruit de Lévy multiplicatif général avec des coefficients de diffusion et de saut Sobolev, et une dérive intégrable, sont fortement bien posées. De plus, nous étudions la propriété forte de Feller, l’irréductibilité ainsi que l’ergodicité exponentielle des semi-groupes correspondants quand les coefficients sont indépendants du temps et singulièrement dissipatifs. En particulier, les grands sauts sont autorisés dans l’équation. Pour aboutir au résultat principal, nous présentons une approche générale pour traiter les EDS avec sauts et coefficients singuliers, de telle sorte que nous devons seulement nous intéresser aux estimées a priori de Krylov pour les EDS.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 1 (2020), 175-229.

Dates
Received: 11 August 2017
Revised: 4 December 2018
Accepted: 15 January 2019
First available in Project Euclid: 3 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1580720487

Digital Object Identifier
doi:10.1214/19-AIHP959

Mathematical Reviews number (MathSciNet)
MR4058986

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Pathwise uniqueness Krylov’s estimate Zvonkin’s transformation Ergodicity Heat kernel

Citation

Xie, Longjie; Zhang, Xicheng. Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 1, 175--229. doi:10.1214/19-AIHP959. https://projecteuclid.org/euclid.aihp/1580720487


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References

  • [1] H. Abels and M. Kassmann. The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka J. Math. 46 (2009) 661–683.
  • [2] A. Arapostathis, A. Biswas and L. Caffarelli. The Dirichlet problem for stable-like operators and related probabilistic representations. Comm. Partial Differential Equations 41 (9) (2016) 1472–1511.
  • [3] R. F. Bass, K. Burdzy and Z. Chen. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stochastic Process. Appl. 111 (2004) 1–15.
  • [4] J. Bergh and J. Löfström. An Introduction to Interpolation Spaces. Springer-Verlag, Berlin, 1970.
  • [5] V. I. Bogachev, N. V. Krylov and M. Röckner. On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm. Partial Differential Equations 26 (11–12) (2001) 2037–2080.
  • [6] V. I. Bogachev, N. V. Krylov and M. Röckner. Elliptic and parabolic equations for measures. Uspekhi Mat. Nauk 64 (6) (2009) 5–116 [in Russian]. English transl.: Russian Math. Surveys 64 (6) (2009) 973–1078.
  • [7] V. I. Bogachev, G. D. Prato and M. Röckner. Existence of solutions to weak parabolic equations for measures. Proc. Lond. Math. Soc. 88 (2004) 753–774.
  • [8] V. I. Bogachev, M. Röckner and S. V. Shaposhnikov. On parabolic inequalities for generators of diffusions with jumps. Probab. Theory Related Fields 158 (2014) 465–476.
  • [9] V. I. Bogachev and P. A. Yu. Strong solutions of stochastic equations with Lévy noise and a discontinuous drift coefficient. Dokl. Math. 92 (1) (2015) 471–475.
  • [10] V. I. Bogachev and P. A. Yu. Strong solutions to stochastic equations with a Lévy noise and a non-constant diffusion coefficient. Dokl. Math. 94 (1) (2016) 438–440.
  • [11] Z. Chen, E. Hu, L. Xie and X. Zhang. Heat kernels for non-symmetric diffusions operators with jumps. J. Differential Equations 263 (2017) 6576–6634.
  • [12] Z. Chen, R. Song and X. Zhang. Stochastic flows for Lévy processes with Hölder drift. Rev. Mat. Iberoam. 34 (2018) 1755–1788.
  • [13] Z. Chen and X. Zhang. Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Related Fields 165 (2016) 267–312.
  • [14] Z. Chen and X. Zhang. Heat kernels for time-dependent non-symmetric stable-like operators. J. Math. Anal. Appl. 465 (2018) 1–21.
  • [15] Z. Chen and X. Zhang. Uniqueness of stable like processes. Available at arXiv:1604.02681.
  • [16] K. L. Chung and Z. Zhao. From Brownian Motion to Schrödinger’s Equation. Springer-Verlag, Berlin, 1995.
  • [17] S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei. Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type. Birkhäuser, Basel, 2004.
  • [18] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence of SDEs with non-regular drift. Stochastics 83 (3) (2011) 241–257.
  • [19] E. Fedrizzi and F. Flandoli. Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 (2013) 708–736.
  • [20] F. Flandoli, M. Gubinelli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (1) (2010) 1–53.
  • [21] B. Goldys and B. Maslowski. Exponential ergodicity for stochastic reaction-diffusion equations. In Stochastic Partial Differential Equations and Applications, XVII 115–131. Lect. Notes Pure Appl. Math. 245. Chapman Hall/CRC, Boca Raton, FL, 2006.
  • [22] I. Gyöngy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51 (4) (2001) 763–783.
  • [23] S. Haadem and F. Proske. On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE’s with singular coefficients. J. Funct. Anal. 266 (2014) 5321–5359.
  • [24] M. Hairer. An introduction to stochastic PDEs. Available at http://www.hairer.org/notes/SPDEs.pdf.
  • [25] R. Z. Hasminskii. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Rockville, 1980.
  • [26] V. A. Ju. On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 (1979) 354–366.
  • [27] K.-H. Kim. $L_{q}(L_{p})$-Theory of parabolic PDEs with variable coefficients. Bull. Korean Math. Soc. 45 (2008) 169–190.
  • [28] R. Kruse and M. Scheutzow. A discrete stochastic Gronwall lemma. Math. Comput. Simulation 143 (2018) 149–157.
  • [29] N. V. Krylov. Controlled Diffusion Processes. Applications of Mathematics 14. Springer-Verlag, New York, 1980. Translated from the Russian by A. B. Aries.
  • [30] N. V. Krylov. Nonlinear Elliptic and Parabolic Equations of Second Order. Nauka, Moscow, 1985.
  • [31] N. V. Krylov and M. Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2) (2005) 154–196.
  • [32] A. Kulik. Exponential ergodicity of the solutions to SDE’s with a jump noise. Stochastic Process. Appl. 119 (2) (2009) 602–632.
  • [33] V. P. Kurenok. Stochastic equations with time-dependent drift driven by Lévy processes. J. Theoret. Probab. 20 (2007) 859–869.
  • [34] H. Masuda. Ergodicity and exponential $\beta $-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. 117 (2007) 35–56.
  • [35] P. O. Menoukeu, B. T. Meyer, T. Nilssen, F. Proske and T. Zhang. A variational approach to the construction and Malliavin differentiability of strong solutions of SDEs. Math. Ann. 357 (2013) 761–799.
  • [36] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer-Verlag, Berlin, 1993.
  • [37] R. Mikulevicius and H. Pragarauskas. On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces. Lith. Math. J. 32 (2) (1992) 377–396.
  • [38] S. E. A. Mohammed, T. Nilssen and F. Proske. Sobolev differentiable stochastic flows of SDE’s with singular coefficients: Applications to the transport equation. Ann. Probab. 43 (3) (2015) 1535–1576.
  • [39] H. Pragarauskas. On $L^{p}$-estimates of stochastic integrals. In Probab. Theory and Math. Stat 579–588, 1999.
  • [40] E. Priola. Pathwise uniqueness for singular SDEs driven by stable processes. Osaka J. Math. 49 (2012) 421–447.
  • [41] E. Priola. Stochastic flow for SDEs with jumps and irregular drift term. Banach Center Publ. 105 (2015) 193–210.
  • [42] E. Priola. Davie’s type uniqueness for a class of SDEs with jumps. Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018) 694–725.
  • [43] J. Ren, J. Wu and X. Zhang. Exponential ergodicity of multi-valued stochastic differential equations. Bull. Sci. Math. 134 (2010) 391–404.
  • [44] M. Scheutzow. A stochastic Gronwall’s lemma. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2) (2013) 1350019 (4 pages).
  • [45] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Springer, Berlin, 2005.
  • [46] E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton University Press, Princeton, NJ, 1970.
  • [47] H. Tanaka, M. Tsuchiya and S. Watanabe. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14 (1974) 73–92.
  • [48] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam, 1978.
  • [49] F. Y. Wang. Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift. J. Differential Equations 3 (2016) 2792–2829.
  • [50] F. Y. Wang. Integrability conditions for SDEs and semi-linear SPDEs. Ann. Probab. 45 (2017) 3223–3265.
  • [51] F. Y. Wang and X. Zhang. Degenerate SDE with Hölder–Dini drift and non-Lipschitz noise coefficient. SIAM J. Math. Anal. 48 (3) (2016) 2189–2222.
  • [52] L. Wang, L. Xie and X. Zhang. Derivative formulae for SDEs driven by multiplicative $\alpha $-stable-like processes. Stochastic Process. Appl. 125 (3) (2015) 867–885.
  • [53] L. Xie and X. Zhang. Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients. Ann. Probab. 44 (6) (2016) 3661–3687.
  • [54] X. Zhang. Strong solutions of SDEs with singular drift and Sobolev diffusion coefficients. Stochastic Process. Appl. 115 (2005) 1805–1818.
  • [55] X. Zhang. Stochastic homemomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab. 16 (2011) 1096–1116.
  • [56] X. Zhang. Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 915–1231.
  • [57] X. Zhang. Stochastic differential equations with Sobolev coefficients and applications. Ann. Appl. Probab. 26 (5) (2016) 2697–2732.
  • [58] X. Zhang. Multidimensional singular stochastic differential equations. In Stochastic Partial Differential Equations and Related Fields 391–403. Springer Proc. Math. Stat. 229. Springer, Cham, 2018.