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)


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.


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



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.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Università di Modena e Reggio EmiliaModenaItaly

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