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Published in:
2013 | OriginalPaper | Chapter
4. Dilation Vector Field on Wiener Space
Author : Hélène Airault
Published in: Malliavin Calculus and Stochastic Analysis
Publisher: Springer US
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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.