Tokuzo Shiga

Stochastic partial differential equations for some measure-valued diffusions

SummaryWe consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R1, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.

Stationary states and their stability of the stepping stone model involving mutation and selection

On considere un modele de diffusion qui est un processus de diffusion de dimension infinie. On etudie la regularite des distributions marginales de dimension finie des etats stationnaires

Stepping Stone Models in Population Genetics and Population Dynamics

We summarize the results of two kinds of stepping stone models arising in population genetics and population dynamics. Although these two describe different phenomena they are closely related through a duality relation. We further attempt to generalize this framework as much as possible.

A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type

SummaryA Markov process of Ornstein-Uhlenbeck type was introduced in [5] as a Markov process onRd generated by a Lévy process generator plus a drift term

A FLEMING-VIOT PROCESS WITH UNBOUNDED SELECTION, II

- \sum\limits_{j = 1}^d {\sum\limits_{k = 1}^d {Q_{jk} } x_k \frac{\partial }{{\partial x_j }}}

Interacting diffusion systems over Z d

with the matrixQ=(Qjk) having eigenvalues with positive real parts. A criterion for positive recurrence of processes of this type was given by Sato-Yamazato [5]. This paper gives a criterion for null recurrence and transience by a integral condition involving the Lévy measure in the case of one dimension. Multi-dimensional cases are also discussed.

Infinite dimensional stochastic differential equations and their applications

SummaryWe study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={xi(t),i ∈ ℤd} in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems,