SOME REMARKS AND EXAMPLES CONCERNING THE TRANSIENCE AND RECURRENCE OF RANDOM DIFFUSIONS

Let k (t) be an ergodic Markov chain on E = {1,2,...,n} and let L(k) = 1/2 SIGMA(i,j=1)d a(ij)(x; k) partial derivative 2/partial derivative x(i) partial derivative x(j) + SIGMA(i=1)d b(i)(x; k) partial derivative/partial derivative x(i), for k = 1,2,...,n. Then for each realization k (t) = k (t, omega) of the Markov chain, L(k(t)) may be thought of as a time inhomogeneous diffusion generator. The process it generates, X (t) = X (t; k (.)), will be called a random diffusion. We give examples such that each L(k) generates a positive recurrent (transient) diffusion but such that the random diffusion is a. s. transient (positive recurrent). We also give some results and examples for transience and recurrence in the special case that (X (t), k (t)) is a reversible process. Finally, we consider the effects of speeding up or slowing down the jump rate of the Markov chain.