Kiyoshi Kawazu

On birth and death processes in symmetric random environment

We prove a limit theorem for a process in a random one-dimensional medium, which has been considered before as a model for hopping conduction in a disordered medium. To the edge between the two integersj and (j+ 1) a rate λj > 0 is attached. Theseλj:j integral are taken as independent, identically distributed random variables, and represent the medium. For given values λj, X(t) is a Markov chain in continuous time which jumps fromj to (j + 1) and from (j + 1) toj at the same rate λj. We show that in many cases there exists normalizing constants y(t) (which tend to oo witht) such that the distribution of X(t)/γ(t), or more generally of the whole processX(st)/γ(t)S⩾0, converges to a limit as t→ ∞. The limit process is continuous and self-similar.

Limit theorems for one-dimensional diffusions and random walks in random environments

SummaryThe limiting behavior of one-dimensional diffusion process in an asymptotically self-similar random environment is investigated through the extension of Broxs method. Similar problems are then discussed for a random walk in a random environment with the aid of optional sampling from a diffusion model; an extension of the result of Sinai is given in the case of asymptotically self-similar random environments.

A one-dimensional birth and death process in random environment

To each integern, a random variableηn is attached at the integer point {n} andλn is attached at the interval (n, n+1) as random environment. We give the global limit theorem of one dimensional birth and death processes in these random media, that is, preparing two independent ratesg(n) andh(n) which are induced by λ’s and η’s, respectively, we prove that the process {X(g(n)h(n)t)/n} converges to a continuous self-similar process.

Branching one-dimensional periodic diffusion processes

Let X be a nonsingular conservative one-dimensional periodic diffusion process, [lambda]0 its principal eigenvalue and X a binary splitting branching diffusion process with nonbranching part X. For each [alpha] > [lambda]0 we have two positive martingales Wit([alpha]), i = 1, 2, of X attached to the two positive [alpha]-harmonic functions of X. The main purpose of this paper is to show that their limit random variables are positive for all [alpha] [epsilon] ([lambda]0, [alpha]i), where [alpha]i are some constants greater than [lambda]0.

A limit theorem for branching one-dimensional periodic diffusion processes

We give almost sure convergence of appropriately normalized particle numbers in bounded domains of locally supercritical branching diffusion processes with one-dimensional periodic diffusions as their non-branching part processes. Some spectral properties of periodic diffusion operators including Hills ones are also studied.