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Analytic semigroups and degenerate elliptic operators with unbounded coefficients: A probabilistic approach. (English) Zbl 0957.35053
The paper deals with second order degenerate elliptic operators $A_0u(x)= \textstyle {1\over 2}\sum^d_{i,j=1} a_{ij}(x)D_{ij} u(x)+\sum^d_{i=1} b_i(x)D_i u(x),\quad x\in\mathbb{R}^d,$ where the coefficients $$a_{ij}(x)$$ and $$b_i(x)$$ can be undefined. The matrix $$a(x)=\{a_{ij}(x)\}$$ is positive semi-definite and has quadratic growth, and the vector $$b(x)$$ has linear growth and is of class $$C^2$$. $$A_0$$ is the diffusion operator associated with the stochastic differential problem $d\xi(t)= b\bigl(\xi(t) \bigr)dt+\sqrt {a\bigl(\xi(t) \bigr)} dw(t),\;\xi(0)=x,$ where $$w(t)=(w_1(t), \dots,w_d(t))$$ is a standard dimensional Brownian motion. Based on this fact and using the technique of investigating stochastic differential equations, the author proves that the operator $$A_0$$ generates an analytic semigroup $P_t\varphi(x) =E\varphi \bigl( \xi(t,x)\bigr),\;t\geq 0,\;x\in\mathbb{R}^D$ in the space $$C_b(\mathbb{R}^d)$$ in the sense that if $$\varphi\in C_b(\mathbb{R}^d)$$, then $$P_t\varphi\in D(A_0)$$. The mapping of $$P_t\varphi(x)$$ is differentiable and one has $\sup_{x\in \mathbb{R}^d}\left |{d \over dt}(P_t\varphi) (x)\right|= \bigl\|A_0(P_t \varphi)\bigr \|_0 \leq c(t\wedge 1)^{-1} \|\varphi \|_0.$

##### MSC:
 35J70 Degenerate elliptic equations 60H30 Applications of stochastic analysis (to PDEs, etc.) 47D03 Groups and semigroups of linear operators
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