Masaaki Tsuchiya

Lévy measure with generalized polar decomposition and the associated sde with jumps

We treat a class of Levy measures which are decomposed in the sense of generalized polar decomposition. According to the decomposition, we construct coefficients of the jump term in the associated stochastic differential equations; we then give a condition on such Levy measures under which the coefficients of the jump term satisfy the Lipschitz condition. For example, the condition is fulfilled by the Levy measure of the Levy generator with Lipschitz continuous exponent . The condition is a nearly best one, which is illustrated with several examples

Stochastic processes and semigroups associated with degenerate Lévy generating operators

We consider three types of Levy generating operators with smooth data (diffusion matrix, drift vector, and Levy measure). These operators are characterized by the types of their Levy measures: the first type is given in general form, but a strong integrability condition is assumed; the second and third types are given as certain generalizations of the Levy measures of stable processes. For such Levy generating operators with degenerate diffusion matrix, we prove a uniqueness theorem for the martingale problem and study a smoothness preserving property of the time-inhomogeneous semigroups associated with the operators.

Uniqueness in shape identification of a time-varying domain and related parabolic equations on non-cylindrical domains

The paper deals with an inverse problem determining the shape of a time-varying Lipschitz domain by boundary measurements of the temperature; such a domain is treated as a non-cylindrical domain in the time-space. Here we focus on the uniqueness of the shape identification. As a general treatment to show the uniqueness, a comparability condition on a pair of domains is introduced; the condition holds automatically in the time-independent case. Based on the condition, we provide several classes of domains in which the uniqueness of the shape identification holds under an appropriate initial shape condition or initial temperature condition. Each of such classes is characterized by a certain geometric condition on its each single element; in particular, it is verified that the class of polyhedral domains and any class of domains with C1 smoothness and with a common initial shape fulfil the uniqueness property. The inverse problem is studied via a parabolic equation with a mixed boundary condition. Then the unique continuation property of weak solutions and the uniqueness of weak solutions to an induced parabolic equation with the homogeneous Dirichlet boundary condition on a non-cylindrical non-Lipschitz domain play key roles.

An estimation problem for the shape of a domain varying with time via parabolic equations

This paper treats the inverse problem determining the shape of a multi-dimensional domain varying with time from measurements of the temperature on an accessible portion of the boundary. On the shape with only Lipschitz continuity on the unknown portion of the boundary, we provide the unique identification theorem and a reconstruction procedure based on a suitably linearized problem, which is formulated by an initial-boundary value problem for a parabolic equation with a mixed boundary condition. For the reconstruction procedure, its convergence and stability are verified, and further its effectiveness is examined in numerical examples.

Markov semigroups associated with one-dimensional Le´vy operators—regularity and convergence

Many diffusion operators appearing in the diffusion approximation to discrete models are degenerate. For example, the diffusion operator ½ x(1-x)(d/dx)2 + [u-(u+v)x](d/dx) (0 < x < 1) arises as a diffusion approximation for the Wright-Fisher model with mutation and migration. To obtain an error estimate for such diffusion approximation, it is useful to show the smoothness of solutions of the diffusion equations. Ethier has obtained important results for the smoothness problem, especially in the one-dimensional case. Motivated by his work, we will study the smoothness problem for certain degenerate integro-differential operators appearing in the theory of Markov processes. These operators include diffusion operators and are called Levy operators (in this note, we treat the case without boundary). Our result can be used to get some limit theorem for stochastic processes with discontinuous paths. We will discuss the generation of the semigroups by one-dimensional Levy operators and the differentiability preserving properties of the semigroups. Convergence problems for the semigroups are also treated.