Harnack inequalities for hypoelliptic evolution operators: geometric issues and applications

Sergio Polidoro


We consider linear second order Partial Differential Equations in the form of "sum of squares of Hörmander vector fields plus a drift term" on a given domain. We prove that an Harnack inequality holds for every compact subset of the interior of the attainable set defined in terms of the vector fields that define the Partial Differential Equation considered. We then ally Harnack's inequalities to prove asymptotic lower bounds of the joint density of a wide class of stochastic processes. Analogous upper bound are proved by Mallilavin's calculus.


hypoelliptic equations, Harnack inequality; potential theory; Malliavin calculus

Full Text:

PDF (Italiano)


D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), pp. 890–896.

H. Bauer, Harmonische R¨aume und ihre Potentialtheorie, Ausarbeitung einer im Sommersemester 1965 an der Universit¨at Hamburg gehaltenen Vorlesung. Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin, 1966.

G. Ben Arous, R. L´eandre, D´ecroissance exponentielle du noyau de la chaleur sur la diagonale.

II, Probab. Theory Related Fields, 90 (1991), pp. 377–402.

A. Bonfiglioli, E. Lanconelli, On left invariant H¨ormander operators in RN. Applications to Kolmogorov-Fokker-Planck equations, Proceeding of the Fifth International Conference on Differential and Functional Differential Equations, Moscow, August 17-24, 2008 (to appear).

A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007.

A. N. Borodin, P. Salminen, Handbook of Brownian motion—facts and formulae, Probability and its Applications, Birkh¨auser Verlag, Basel, second ed., 2002.

U. Boscain, S. Polidoro, Gaussian estimates for hypoelliptic operators via optimal control, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), pp. 333–342.

C. Cinti, S. Menozzi, S. Polidoro, Two-sided bounds for degenerate processes with densities supported in subsets of Rn, (preprint), (2012).

C. Cinti, K. Nystr¨om, S. Polidoro, A note on Harnack inequalities and propagation sets for a class of hypoelliptic operators, preprint, (2010).

C. Constantinescu, A. Cornea, Potential theory on harmonic spaces, Springer-Verlag, New York, 1972. With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158.

M. Di Francesco, S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in non-divergence form, Advances in Differential Equations, 11 (2006), pp. 1261–1320.

N. Garofalo, E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), pp. 775–792.

M. Kac, On distributions of certain Wiener functionals, Trans. Amer. Math. Soc., 65 (1949), pp. 1–13.

L. P. Kupcov, Harnack’s inequality for generalized solutions of second order degenerate elliptic equations, Differencial′nye Uravnenija, 4 (1968), pp. 110–122.

, The fundamental solutions of a certain class of elliptic-parabolic second order equations, Differencial′nye Uravnenija, 8 (1972), pp. 1649–1660, 1716.

, The mean value property and the maximum principle for second order parabolic equations, Dokl. Akad. Nauk SSSR, 242 (1978), pp. 529–532.

, On parabolic means, Dokl. Akad. Nauk SSSR, 252 (1980), pp. 296–301.

E. Lanconelli, S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), pp. 29–63. Partial differential equations, II (Turin, 1993).

E. B. Lee, L. Markus, Foundations of optimal control theory, Robert E. Krieger Publishing Co. Inc., Melbourne, FL, second ed., 1986.

S. Polidoro, A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations, Arch. Rational Mech. Anal., 137 (1997), pp. 321–340.

N. Smirnov, Sur la distribution de omega^2 (criterium de M. von Mises), C. R. Acad. Sci. Paris, 202 (1936), pp. 449–452.

L. Tolmatz, Asymptotics of the distribution of the integral of the absolute value of the Brownian bridge for large arguments, Ann. Probab., 28 (2000), pp. 132–139.

N. T. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and geometry on groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992.

DOI: 10.6092/issn.2240-2829/3415


  • There are currently no refbacks.

Copyright (c) 2012 Sergio Polidoro

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

Creative Commons License 

ISSN 2240-2829
Registration at Bologna Law Court no. 8138 on 24th November, 2010

The journal is hosted and mantained by ABIS-AlmaDL [privacy]