Analytic semigroups and degenerate elliptic operators with unbounded coefficients: A probabilistic approach
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 \[ \sup_{x\in \mathbb{R}^d}\left |{d \over dt}(P_t\varphi) (x)\right|= \bigl\|A_0(P_t \varphi)\bigr \|_0 \leq c(t\wedge 1)^{-1} \|\varphi \|_0. \]
scientific article; zbMATH DE number 1517170
scientific article