椭圆型算子的扩张及■半群

THE EXTENSION OF AN ELLIPTIC DIFFERENTIALOPERATOR AND ■ SEMIGROUPS

如何构造一个马尔科夫过程以G为生成元素(在适当的定义域上)的问题,在[1a]中已经给出回答.[1a]中假定系数a_(ij)(x),b_j(x),c(x)皆有界.可以证明在此条件下,过程的极限分布必为零;当l>2时,对应的过程必为非常返的(定义见[2]).

Denote by C the space of continuous functions defined on the l-dimensional Euclidean space R_l, and by C the subspace of C spanned by the funetions with limit zero at infinity. In this paper, we consider the differential expression (in general, unsymmetric and with unbounded coefficients) We have proved that G induces a closed operator A on a subspace C of C(C C C). A is shown to satisfy the Hille-yosida conditions, so it generates a contraction semigroup T_t on C.If the ergodic limit of Γ_tf(x) does not vanish (theorem 5), it can be proved that T_t possesses an invariant measure θ(dx) and is ergodie in the sense of L_p(θ(dx)). The semigroup T_t is also shown to have a Markov transition function p(t, x, Γ) which furnishes the basis for constructing a Markov process.