Stochastic Hamiltonian flows with singular coefficients

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Abstract

In this paper, we study the following stochastic Hamiltonian system in ℝ2d (a second order stochastic differential equation):
$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$
where b(x; v) : ℝ2d → ℝ d and σ(x; v): ℝ2d → ℝ d ⊗ ℝ d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and bH p 2/3,0 and ∇σL p for some p > 2(2d+1), where H p α,β is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that \((x,v) \mapsto {Z_t}(x,v): = ({X_t},{\dot X_t})(x,v)\) forms a stochastic homeomorphism flow, and \(\left( {x,v} \right) \mapsto {Z_t}\left( {x,v} \right)\) is weakly differentiable with \(ess.{\sup _{x,v}}E(su{p_{t \in [o,T]}}|\nabla {Z_t}(x,v){|^q}) < \infty \) for all q ⩾ 1 and T ⩾ 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coeffcients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).

Keywords

stochastic Hamiltonian system weak differentiability Krylov’s estimate Zvonkin’s transformation kinetic Fokker-Planck operator 

MSC(2010)

60H10 

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References

  1. 1.
    Bass R, Chen Z-Q. Brownian motion with singular drift. Ann Probab, 2003, 31: 791–817MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bergh J, Löfström J. Interpolation Spaces: An Introduction. Berlin: Springer-Verlag, 1976CrossRefMATHGoogle Scholar
  3. 3.
    Bogachev V I, Krylov N V, Röckner M. On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm Partial Differential Equations, 2001, 26: 2037–2080MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bogachev V I, Krylov N V, Röckner M. Elliptic and parabolic equations for measures. Russian Math Surveys, 2009, 64: 973–1078MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bouchut F. Hypoeliptic regularity in kinetic equations. J Math Pures Appl (9), 2002, 81: 1135–1159MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bramanti M, Cupini G, Lanconelli E, et al. Global L p-estimate for degenerate Ornstein-Uhlenbeck operators. Math Z, 2010, 266: 789–816MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chaudru de Raynal P E. Strong existence and uniqueness for stochastic differential equation with Hölder drift and degenerate noise. Ann Inst H Poincaré, 2017, 53: 259–286CrossRefGoogle Scholar
  8. 8.
    Chen Z Q, Zhang X. L p-maximal hypoelliptic regularity of nonlocal kinetic Fokker-Planck operators. ArXiv: 1608.05502, 2016Google Scholar
  9. 9.
    Cherny A S. On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. Theory Probab Appl, 2006, 46: 483–497Google Scholar
  10. 10.
    Crippa G, De Lellis C. Estimates and regularity results for the DiPerna-Lions ow. J Reine Angew Math, 2008, 616: 15–46MathSciNetMATHGoogle Scholar
  11. 11.
    Fedrizzi E, Flandoli F. Noise prevents singularities in linear transport equations. J Funct Anal, 2013, 264: 1329–1354MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fedrizzi E, Flandoli F. Hölder flow and differentiability for SDEs with non regular drift. Stoch Anal Appl, 2013, 31: 708–736MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fedrizzi E, Flandoli F, Priola E, et al. Regularity of stochastic kinetic equations. ArXiv:1606.01088, 2016MATHGoogle Scholar
  14. 14.
    Figalli A. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coeffcients. J Funct Anal, 2008, 254: 109–153MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jin P. Brownian motion with singular time-dependent drift. J Theoret Probab, 2017, 30: 1499–1538MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Karatza I, Shreve S E. Brownian Motion and Stochastic Calculus. New York: Springer-Verlag, 1988CrossRefGoogle Scholar
  17. 17.
    Krylov N V. Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, vol. 96. Providence: Amer Math Soc, 2008Google Scholar
  18. 18.
    Krylov N V, Röckner M. Strong solutions of stochastic equations with singular time dependent drift. Probab Theory Related Fields, 2005, 131: 154–196MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kunita H. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge: Cambridge University Press, 1990Google Scholar
  20. 20.
    Menoukeu-Pamen O, Meyer-Brandis T, Nilssen T, et al. A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math Ann, 2013, 357: 761–799MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Menozzi S. Martingale problems for some degenerate Kolmogorov equations. Stochastic Process Appl, 2017, 128: 756–802MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mohammed S E A, Nilssen T, Proske F. Sobolev differentiable stochastic flows for SDEs with singular coeffcients: Applications to the transport equation. Ann Probab, 2015, 43: 1535–1576MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Priola E. On weak uniqueness for some degenerate SDEs by global Lp-estimate. Potential Anal, 2015, 42: 247–281MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Röckner M, Zhang X. Weak uniqueness of Fokker-Planck equations with degenerate and bounded coeffcients. C R Math Acad Sci Paris, 2010, 348: 435–438MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Soize C. The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. Series on Advances in Mathematics for Applied Sciences, vol. 17. Singapore: World Scientic, 1994Google Scholar
  26. 26.
    Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970MATHGoogle Scholar
  27. 27.
    Stroock D, Varadhan S R S. Multidimensional Diffusion Processes. Berlin: Springer-Verlag, 1997CrossRefMATHGoogle Scholar
  28. 28.
    Talay D. Stochastic Hamiltonian systems: Exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Process Related Fields, 2002, 8: 1–36MathSciNetMATHGoogle Scholar
  29. 29.
    Trevisan D. Well-posedness of multidimensional diffusion processes with weakly differentiable coeffcients. Electron J Probab, 2016, 21, doi: 10.1214/16-EJP4453Google Scholar
  30. 30.
    Wang F, Zhang X. Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coeffcient. SIAM J Math Anal, 2016, 48: 2189–2222MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Xie L, Zhang X. Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coeffcients. Ann Probab, 2016, 44: 3661–3687MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zhang X. Strong solutions of SDEs with singular drift and Sobolev diffusion coeffcients. Stochastic Process Appl, 2005, 115: 1805–1818MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhang X. Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coeffcients. Electron J Probab, 2011, 16: 1096–1116MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Zhang X. Stochastic partial differential equations with unbounded and degenerate coeffcients. J Differential Equations, 2011, 250: 1924–1966MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zhang X. Stochastic differential equations with Sobolev diffusion and singular drift. Ann Appl Probab, 2016, 26: 2697–2732MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Zvonkin A K. A transformation of the phase space of a diffusion process that removes the drift. Mat Sb, 1974, 93: 129–149MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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