Harnack inequalities and bounds for densities of stochastic processes

DOI10.1007/978-3-319-65313-6_4zbMATH Open1382.60088arXiv1610.08792OpenAlexW2545846319MaRDI QIDQ1703892

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Publication date: 8 March 2018

Abstract: We consider possibly degenerate parabolic operators in the form

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 H"ormander condition. Note that, under this assumption, the operators in the form

L

has 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.

Full work available at URL: https://arxiv.org/abs/1610.08792

zbMATH Keywords

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Mathematics Subject Classification ID

Applications of stochastic analysis (to PDEs, etc.) (60H30) Random operators and equations (aspects of stochastic analysis) (60H25) Stochastic partial differential equations (aspects of stochastic analysis) (60H15)