Probabilistic Methods for Ultracontractivity

Bernard Roynette
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Bernard Roynette: Université de Nancy I, Département de Mathématiques

A chapter in Harmonic Analysis and Discrete Potential Theory, 1992, pp 111-129 from Springer

Abstract: Abstract We consider the semigroup $$T_t^\mu $$ generated by the operator $$\frac{1}{2}(\Delta f - \nabla u \cdot \nabla f)$$ on the Lebesgue space L 2(ℝ d ;μ) with respect to the measure μ:=e −u. We show by probabilistic methods that $$T_t^\mu $$ is ultracontractive. That is, it maps (for t>0) L 1(μ) to L ∞, whenever the function u satisfies a suitable growth condition at infinity, which essentially amounts (for instance in dimension d=1) to integrability of 1/μ′ at infinity. This is a survey of results obtained by O. Kavian, G. Kerkyacharian and the author in a paper submitted for publication in Journal of Functional Analysis.

Keywords: Brownian Motion; Bounded Operator; Semi Group; Positive Derivative; Independent Brownian Motion (search for similar items in EconPapers)
Date: 1992
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DOI: 10.1007/978-1-4899-2323-3_11

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