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Potential Analysis

, Volume 42, Issue 1, pp 39–98 | Cite as

Two-Sided Bounds for Degenerate Processes with Densities Supported in Subsets of N

  • Chiara Cinti
  • Stéphane MenozziEmail author
  • Sergio Polidoro
Article

Abstract

We obtain bounds for the density of \(\left (x_{1,n}+W_{t}, x_{n+1}+{{\int }_{0}^{t}} |x_{1,n}+\right .\\\left .W_{s}|^{k}ds\right )\) where (W t ) t≥0 is a standard Brownian motion of n , k is even and x=(x 1,n ,x n+1)∈ n+1. This process satisfies a weak Hörmander condition but the support of its density is not the whole space. Also, the Density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hörmander’s theorem). The proofs of lower and upper bounds are based on Harnack inequalities and Malliavin calculus respectively. The case of the joint law of Brownian motion and the integral of odd powers of its coordinates is also considered.

Keywords

Harnack inequality Malliavin Calculus Hörmander condition Two-sided bounds 

Mathematics Subject Classifications (2010)

Primary 35H10 60J60 secondary 31C05 60H07 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Chiara Cinti
    • 1
  • Stéphane Menozzi
    • 2
    Email author
  • Sergio Polidoro
    • 3
  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.Laboratoire d’Analyse et ProbabilitésUniversité d’Evry Val d’EssonneEvry CedexFrance
  3. 3.Dipartimento di Scienze Fisiche, Informatiche e MatematicheUniversità di Modena e Reggio EmiliaModenaItaly

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