ABSTRACT
For our purposes, a diffusion, X(t), is a real-valued strong Markov process with continuous sample paths. (See [2, Chapter 2] for background information and terminology regarding diffusions.) We assume that the process is defined for all t > 0 and that its state space, I = [TO' TIl, is closed and bounded. A central role in the description and analysis of a finite Markov chain, Xn , is played by its transition matrix, P, with components
Pxy = P(Xn + 1 = ylXn = x). Associated with this matrix is a transformation of functions (regarded as column vectors) defined by matrix multiplication,
or, equivalently, Pf(x) = E(f(Xn+I)IXn = x). Both the transition matrix and the transition operator can be generalized to diffusions, but the transition operator, defined by
Ttf(x) = E(f(Xt+s)IXs = x), is far more tractable and thus occupies a more prominent place in the theory of diffusions.
