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On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

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Abstract

The paper contains a representation formula for positive solutions of linear degenerate second-order equations of the form

$$\partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \quad (x,t)\in \mathbb{R}^N \times\, ]- \infty ,T[,$$

where the X j are smooth vector fields satisfying the Hörmander condition. It is assumed that X j are invariant under left translations of a Lie group and the corresponding paths satisfy a local admissibility criterion. The representation formula is established by an analytic approach based on Choquet theory. As a consequence we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.

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Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Alessia E. Kogoj

  2. Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Yehuda Pinchover

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

    Sergio Polidoro

Authors
  1. Alessia E. Kogoj
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  2. Yehuda Pinchover
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  3. Sergio Polidoro
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Correspondence to Alessia E. Kogoj.

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Kogoj, A.E., Pinchover, Y. & Polidoro, S. On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators. J. Evol. Equ. 16, 905–943 (2016). https://doi.org/10.1007/s00028-016-0325-7

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