Formulae for the derivatives of degenerate diffusion semigroups

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• 作者：Enrico Priola
• 关键词：Diffusion semigroups ; degenerate parabolic equations ; Malliavin Calculus
• 刊名：Journal of Evolution Equations
• 出版年：2006
• 出版时间：December, 2006
• 年：2006
• 卷：6
• 期：4
• 页码：577-600
• 全文大小：216 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1424-3202

文摘

We consider the diffusion semigroup P_t associated to a class of degenerate elliptic operators A \texton \mathbbRn\mathcal{A}\; \text{on}\; \mathbb{R}^n . This class includes the hypoelliptic Ornstein-Uhlenbeck operator but does not satisfy in general the well-known H?rmander condition on commutators for sums of squares of vector fields. We establish probabilistic formulae for the spatial derivatives of P_t f up to the third order. We obtain L∞-estimates for the derivatives of P_t f and show the existence of a classical bounded solution for the parabolic Cauchy problem involving A\mathcal{A} and having f ? C_b(\mathbbRn)f\in C_b(\mathbb{R}^n) as initial datum.