# Analyticity of the semigroup in a degenerate case

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## Abstract

We consider the following class of second order differential operators d d
where

$$
\mathcal{L}_O (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij + } \sum\limits_{i = 1}^d {b_i (x)D_i , x \in \mathbb{R}^d ,} }
$$

*a(x)*is a positive semi-definite symmetric matrix which has quadratic growth and*b(x)*is a vector field of class*C*^{2}which has linear growth. We assume that the mapping*a*: ℝ^{d}→ ℒ(ℝ^{d}) is also of class*C*^{2}with bounded second derivatives, so that it can be written as*a(x)*= ½σ*x*σ^{*}*x*,*x*∈ℝ^{d}, for some matrix valued function σ : ℝ^{d}→ ℒ(ℝ^{d}) which is Lipschitz-continuous (for a proof of this fact see [66] and also [114]. Here we assume further regularity for o; namely we assume that*a*can be factorized by some a which is twice differentiable with bounded derivatives.## Keywords

Degenerate Case Analytic Semigroup Stochastic Partial Differential Equation Quadratic Growth Order Differential Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2001