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Harnack Inequalities and Bounds for Densities of Stochastic Processes

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Modern Problems of Stochastic Analysis and Statistics (MPSAS 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 208))

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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.

Dedicated to Valentin Konakov in occasion of his 70th birthday.

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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.

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Authors and Affiliations

  1. Università di Modena e Reggio Emilia, Via Campi 213/b, 41125, Modena, Italy

    Gennaro Cibelli & Sergio Polidoro

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  1. Gennaro Cibelli
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  2. Sergio Polidoro
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Correspondence to Sergio Polidoro .

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Editors and Affiliations

  1. Department of Statistics and Data Analysis, Higher School of Economics, Moscow, Russia

    Vladimir Panov

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Cibelli, G., Polidoro, S. (2017). Harnack Inequalities and Bounds for Densities of Stochastic Processes. In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_4

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