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Dilation Vector Field on Wiener Space

  • Hélène AiraultEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)

Abstract

We consider the heat operator Δ H , heat equation, and heat kernel measures (ν t ) t≥0 on Wiener space Ω as explained in Driver (Contemp. Math. 338:101–141, 2003). We define the notion of heat dilation vector field associated to a family of probability measures (μ t ) t≥0 on Ω. Let ω ∈Ω. The vector field V on Ω is expressed for F(ω)=f(ω(t 1),ω(t 2),,ω(t n )) as VF(ω)=(vf)(ω(t 1), ω(t 2), ,ω(t n )) where \(vf =\sum _{ k=1}^{n}x_{k} \frac{\partial } {\partial x_{k}}\). The vector field V is shown to be a heat vector field for the heat kernel measures (ν t ) t≥0. We project down “through a nondegenerate map Z”, Ornstein–Uhlenbeck operators defined on Ω by \(\mathcal{L}_{t}F = t\Delta _{H}F - V F\). We obtain a first-order partial differential equation for the density of the random vector Z. We compare this differential equation to the heat equation and to Stein’s equation for the density.

Keywords

Wiener space Heat kernel measures Dilation vector fields Ornstein–Uhlenbeck operators Malliavin calculus 

Notes

Acknowledgements

A preliminary version of this note was presented in June 2011 at the “Journée mathématique d’Amiens”. I thank Frederi Viens for friendly and useful discussions in July 2011. I also thank the referee for significant suggestions for the writing.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.INSSET, UMR 7132 CNRS, Université de Picardie Jules VerneSaint Quentin (Aisne)France

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