Advertisement

Harnack Inequalities and Bounds for Densities of Stochastic Processes

  • Gennaro Cibelli
  • Sergio PolidoroEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 208)

Abstract

We consider possibly degenerate parabolic operators in the form
$$ \mathscr {L}= \sum _{k=1}^{m}X_{k}^{2}+X_{0}-\partial _{t}, $$
that are naturally associated to a suitable family of stochastic differential equations, and satisfying the Hörmander condition. Note that, under this assumption, the operators in the form \(\mathscr {L}\) have a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods to prove analogous upper bounds. We eventually give a list of meaningful examples of operators to which the method applies.

Keywords

Density of a stochastic process Kolmogorov equations Hypoelliptic PDEs Harnack inequality Harnack chain Asymptotic estimates 

Notes

Acknowledgements

We thank the anonymous referee for his/her careful reading of our manuscript and for several suggestions that have improved the exposition of our work.

References

  1. 1.
    Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–896 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aronson, D.G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bally, V.: Lower bounds for the density of locally elliptic Itô processes. Ann. Probab. 34, 2406–2440 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bally, V., Kohatsu-Higa, A.: Lower bounds for densities of Asian type stochastic differential equations. J. Funct. Anal. 258, 3134–3164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bass, R.: Diffusions and elliptic operators. Springer, New York (1998)zbMATHGoogle Scholar
  6. 6.
    Ben, G.: Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale. Probab. Theory Related Fields 90, 175–202 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borodin, A.N., Salminien, P.: Handbook of Brownian motionfacts and formulae, Probability and its Applications, 2nd edn. Birkhauser, Basel (2002)CrossRefGoogle Scholar
  8. 8.
    Boscain, U., Polidoro, S.: Gaussian estimates for hypoelliptic operators via optimal control. Rend. Lincei Math. Appl. 18, 333–342 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cibelli, G., Polidoro, S., Rossi, F.: Sharp estimates for Geman-Yor Processes and Application to Arithmetic Average Asian Option, SubmittedGoogle Scholar
  10. 10.
    Cinti, C., Menozzi, S., Polidoro, S.: Two-sides bounds for degenerate processes with densities supported in subsets of \({\mathbb{R}}^n\), Potential Analysis, pp. 1577–1630 (2014)Google Scholar
  11. 11.
    Cinti, C., Nyström, K., Polidoro, S.: A note on Harnack inequalities and propagation sets for a class of hypoelliptic operators. Potential Anal. 33, 341–354 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cinti, C., Polidoro, S.: Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators. J. Math. Anal. Appl. 338, 946–969 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Davies, E.B.: Explicit constants for Gaussian upper bounds on heat kernels. Am. J. Math. 109, 319–333 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Delarue, F., Menozzi, S.: Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259, 1577–1630 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Di Francesco, M., Polidoro, S.: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Differ. Equ. 11, 1261–1320 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Escauriaza, L.: Bounds for the fundamental solution of elliptic and parabolic equations in nondivergence form. Comm. Partial Differ. Equ. 25, 821–845 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fabes, E.B., Stroock, D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hadamard, J.: Extension à l’ équation de la chaleur d’un théorème de A. Harnack. Rend. Circ. Mat. Palermo 2(3), 337–346 (1954)CrossRefzbMATHGoogle Scholar
  19. 19.
    Il’in, A.M.: On a class of ultraparabolic equations. Dokl. Akad. Nauk SSSR 159, 1214–1217 (1964)MathSciNetGoogle Scholar
  20. 20.
    Jerison, D.S., Sánchez-Calle, A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35, 835–854 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kac, M.: On distributions of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kogoj, A., Polidoro, S.: Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators (to appear on Potential Analysis) (2016)Google Scholar
  23. 23.
    Kohatsu, A.: Higa, Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126, 421–457 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Konakov, V.: Parametrix method for diffusion and Markov chains, Russian preprint available on https://www.hse.ru/en/org/persons/22565341
  25. 25.
    Kolmogoroff, A.: Zufällige Bewegungen (zur Theorie der Brownschen Bewegung). Ann. of Math. 35(2), 116–117 (1934)Google Scholar
  26. 26.
    Krylov, N.V., Safonov, M.V.: A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44, 161–175 (1980)MathSciNetGoogle Scholar
  27. 27.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 391–442 (1987)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: Proceedings of the International Symposium on Stochastic Differential Equations, pp. 195–263. Wiley, New York-Chichester-Brisbane (1978)Google Scholar
  29. 29.
    McKeane Jr., H.P., Singer, I.M.: Curvature and the eigenvalues of t-he Laplacian. J. Differ. Geom. 1, 43–69 (1967)CrossRefGoogle Scholar
  30. 30.
    Metafune, G., Pallara, D., Rhandi, A.: Global properties of invariant measures. J. Funct. Anal. 223, 396–424 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Monti, L., Pascucci, A.: Obstacle problem for arithmetic Asian options. C.R. Math. Acad. Sci. Paris 347, 1443–1446 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Moser, J.: correction to: a Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 20, 231–236 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nualart, D.: The Malliavin Calculus and Related Topics. Probability and its Applications. Springer, New York (2000)Google Scholar
  36. 36.
    Pini, B.: Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Sem. Mat. Univ. Padova 23, 422–434 (1954)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Polidoro, S.: On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Matematiche (Catania) 49, 53–105 (1994)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Polidoro, S.: A global lower bound for the fundamental solution of Kolmogorov–Fokker–Planck equations. Arch. Ration. Mech. Anal. 137, 321–340 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pontryagin, L.S., Mishchenko, E., Boltyanskii, V., Gamkrelidze, R.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)Google Scholar
  40. 40.
    Rothschild, L., Stein, E.M.: E, Hypoelliptic differential operators and nilpotent groups. Acta Math 137, 247–320 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Smirnov, N.: Sur la distribution de 2 (criterium de M. von Mises). C. R. Acad. Sci. Paris 202, 449–452 (1936)Google Scholar
  42. 42.
    Sonin, I.M.: A class o degerate diffusion processes. Teor. Verojatnost. i Primenen 12, 540–547 (1967)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Tolmatz, L.: Asymptotics of the distribution of the integral of the absolute value of the Brownian bridge for large arguments. Ann. Probab. 28, 132–139 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  45. 45.
    Weber, M.: The fundamental solution of a degenerate partial differential equation of parabolic type. Trans. Am. Math. Soc. 71, 24–37 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Università di Modena e Reggio EmiliaModenaItaly

Personalised recommendations