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

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.

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Correspondence to Stéphane Menozzi.

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Cinti, C., Menozzi, S. & Polidoro, S. Two-Sided Bounds for Degenerate Processes with Densities Supported in Subsets of N . Potential Anal 42, 39–98 (2015). https://doi.org/10.1007/s11118-014-9424-7

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Keywords

  • Harnack inequality
  • Malliavin Calculus
  • Hörmander condition
  • Two-sided bounds

Mathematics Subject Classifications (2010)

  • Primary 35H10
  • 60J60
  • secondary 31C05
  • 60H07