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

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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 ℝ n, k∈ℕ ∗ is even and x=(x 1,n ,x n+1)∈ℝ 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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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126, Bologna, Italy

    Chiara Cinti

  2. Laboratoire d’Analyse et Probabilités, Université d’Evry Val d’Essonne, 23 Boulevard de France, 91037, Evry Cedex, France

    Stéphane Menozzi

  3. Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, via Campi 213/b, 41125, Modena, Italy

    Sergio Polidoro

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  1. Chiara Cinti
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  2. Stéphane Menozzi
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  3. Sergio Polidoro
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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 ℝ N . Potential Anal 42, 39–98 (2015). https://doi.org/10.1007/s11118-014-9424-7

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