Parametrix techniques and martingale problems for some degenerate Kolmogorov equations

Stephane Menozzi (Laboratoire de Probabilités et Modèles Aléatoires. Université Paris Diderot)

Abstract


We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [BP09] to prove uniqueness of the martingale problem in the framework of non- degenerate elliptic operators and the Mc Kean and Singer [MS67] parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi [DM10]).

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 234-250

Publication Date: May 2, 2011

DOI: 10.1214/ECP.v16-1619

References

  1. Bass, R.F.; Perkins, E.A. A new technique for proving uniqueness for martingale problems. From Probability to Geometry (I): Volume in Honor of the 60th Birthday of Jean-Michel Bismut (2009), 47--53. MR2642351
  2. Barucci, E.; Polidoro, S.; Vespri, V. Some results on partial differential equations and Asian options. Math. Models Methods Appl. Sci. 11 (2001), no. 3, 475--497. MR1830951 (2002f:35233)
  3. Delarue, François; Menozzi, Stéphane. Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259 (2010), no. 6, 1577--1630. MR2659772 (Review)
  4. Eckmann, J.-P.; Pillet, C.-A.; Rey-Bellet, L. Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Comm. Math. Phys. 201 (1999), no. 3, 657--697. MR1685893 (2000d:82025)
  5. Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp. MR0181836 (31 #6062)
  6. Hérau, Frédéric; Nier, Francis. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 (2004), no. 2, 151--218. MR2034753 (2005f:82085)
  7. Hörmander, Lars. Hypoelliptic second order differential equations. Acta Math. 119 1967 147--171. MR0222474 (36 #5526)
  8. Ishii, H.; Lions, P.-L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990), no. 1, 26--78. MR1031377 (90m:35015)
  9. Konakov, Valentin; Mammen, Enno. Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Related Fields 117 (2000), no. 4, 551--587. MR1777133 (2001j:60141)
  10. Konakov, Valentin; Mammen, Enno. Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl. 8 (2002), no. 3, 271--285. MR1931967 (2004e:60098)
  11. Konakov, V.; Menozzi, S.. Weak error for stable driven stochastic differential equations: Expansion of the densities. To appear in Journal of Theoretical Probability, (2010).
  12. Konakov, V.; Menozzi, S.; Molchanov, S. Explicit parametrix and local limit theorems for some degenerate diffusion processes. Annales de l'Institut Henri Poincarée, Série B. 46--4 (2010), 908--923. doi:10.1214/09-AIHP207
  13. McKean, H. P., Jr.; Singer, I. M. Curvature and the eigenvalues of the Laplacian. J. Differential Geometry 1 1967 no. 1 43--69. MR0217739 (36 #828)
  14. Norris, James. Simplified Malliavin calculus. Séminaire de Probabilités, XX, 1984/85, 101--130, Lecture Notes in Math., 1204, Springer, Berlin, 1986. MR0942019 (89f:60058)
  15. Rey-Bellet, Luc; Thomas, Lawrence E. Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys. 215 (2000), no. 1, 1--24. MR1799873 (2001k:82061)
  16. Soize, C. The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions. Series on Advances in Mathematics for Applied Sciences, 17. World Scientific Publishing Co., Inc., River Edge, NJ, 1994. xvi+321 pp. ISBN: 981-02-1755-2 MR1287386 (95j:60125)
  17. Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR0532498 (81f:60108)
  18. Talay, D. Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Inhomogeneous random systems (Cergy-Pontoise, 2001). Markov Process. Related Fields 8 (2002), no. 2, 163--198. MR1924934 (2003e:60129)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.