Matsuyo Tomisaki

Convergence of local type Dirichlet forms to a non-local type one

Abstract Convergence of Dirichlet forms of diffusion processes is investigated without assuming that the underlying measures are fixed or compatible with a fixed one. Here we treat the case where the basic processes are skew product of finite dimensional diffusions and one-dimensional ones. We note the corresponding diffusions to the Dirichlet forms can be represented as time changed processes of the basic processes, where the time change is given by the additive functional associated with the underlying measure. Then the convergence of the Markov semigroups of the obtained processes and the Feller property of the limit process are proved by providing some convergence properties on additive functionals. The concrete expression on a core for the limit Dirichlet form is also obtained, which may be of non-local type due to the degeneracy of the underlying measure. Finally, under some regularity assumption, the partial differential equation associated with the limit process is given, which is elliptic on infinitely many disjoint strips with the non-local boundary condition including the boundary values on the neighboring strips.

Martingale approach to limit theorems for jump processes

We consider the weak convergence of laws of cadiag processes determined by a sequence of operators with singularly perturbed terms. We study the problem in the martingale approach, which was formulated to establish weak limit theorems for continuous processes by Papanicolaou, Stroock and Varadhan. However, in this paper, limit processes are not necessarily continuous but cadiag. In particular, we consider a homogenization problem of cadiag processes in the framework of martingale problem.

Lévy Measure Density Corresponding to Inverse Local Time

We are concerned with Levy measure density corresponding to the inverse local time at the regular end point for harmonic transform of a one dimensional di usion process. We show that the Levy measure density is represented as a Laplace transform of the spectral measure corresponding to the original di usion process, where the absorbing boundary condition is posed at the end point if it is regular.

Conditional Processes Induced by Birth and Death Processes

For birth and death processes with finite state space consisting of N + 1 points (N ≥ 2), we consider stochastic processes induced by conditioning on hitting the right boundary point before hitting the left boundary point. We call the induced stochastic processes the conditional processes. We show that the conditional processes are again birth and death processes when the right boundary point is absorbing. On the other hand, it is shown that for N ≥ 3 the conditional processes do not have Markov property and they are not birth and death processes when the right boundary point is reflecting. The conditional processes have been introduced in population genetics. Our results are applied to stochastic models in population genetics and their conditional processes.

Existence of a Strong Solution for an Integro-Differential Equation and Superposition of Diffusion Processes

The purpose of this article is to show the existence and the uniqueness of a strong solution \(u \in {\widehat C^{2,\alpha }}\) to an integro-differential equation \(\left\{ {\mu - \left( {A + B} \right)} \right\}u = f\) for each μ > 0 and \(f \in {\widehat C^\alpha }\), where A is a second order elliptic differential operator, B a Levy type integral operator, \({\widehat C^{k,\alpha }}\) the space of all k-times continuously differentiable functions with α-Holder continuous k-th derivatives and with all j-th derivatives for j ≤ k vanishing at infinity, and \({\widehat C^\alpha } = {\widehat C^{0,\alpha }}\). This ensures the existence of a Feller process associated with the generator A + B in the usual way.

Asymptotic behaviors of moments for one-dimensional generalized diffusion processes

It is well known that the moments of the Brownian motion [B(t), P x ] are given by

Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators

{E_0}{\left|{B\left( t \right)} \right|^\gamma } = {\pi ^{ - 1/2}}{2^{\gamma /2}}\Gamma \left( {\frac{{\gamma + 1}}{2}} \right){t^{\gamma /2}},\gamma > - 1,

Asymptotic conditional distributions related to one-dimensional generalized diffusion processes

(1.1) especially, for γ = 2,