Makoto Yamazato

Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Processes of Ornstein-Uhlenbeck type on d are analogues of the Ornstein-Uhlenbeck process on d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on 1.

Absolute continuity of operator-self-decomposable distributions on Rd

It is shown that every genuinely d-dimensional operator-self-decomposable distribution is absolutely continuous.

Multidimensional process of Ornstein-Uhlenbeck type with nondiagonalizable matrix in linear drift terms

Let R d be the d -dimensional Euclidean space where each point is expressed by a column vector. Let | x | and ‹ x, y › denote the norm and the inner product in R d . Let Q = ( Q jk ) be a real d × d -matrix of which all eigenvalues have positive real parts. Let X be a process of Ornstein-Uhlenbeck type (OU type process) on R d associated with a Levy process {Z : t ≥ 0} and the matrix Q . Main purpose of this paper is to give a recurrence-transience criterion for the process X when Q is a Jordan cell matrix and to compare it with the case when Q is diagonalizable. Here by a Levy process we mean a stochastically continuous process with stationary independent increments, starting at 0.

Hitting time distributions of single points for

In this paper, we will characterize the class of (conditional) hitting time distributions of single points of one dimensional generalized diffusion processes and give their tail behaviors in terms of speed measures of the generalized diffusion processes.

1

A distribution μ on R + = [0, ∞) is said to be a distribution if there are an increasing (in the strict sense) sequence of positive real numbers such that, for each j = 0, …, m , there is at least one a k satisfying b j a k b +1 and the Laplace transform of μ is represented as

-dimensional generalized diffusion processes

Limit theorems are obtained for suitably normalized hitting times of single points for 1-dimensional generalized diffusion processes as the hitting points tend to boundaries under an assumption which is slightly stronger than that the existence of limits 7 + 1 of the ratio of the mean and the variance of the hitting time. Laplace transforms of limit distributions are modifications of Bessel functions. Results are classified by the one parameter {7}, each of which is the degree of corresponding Bessel function. In case the limit distribution is degenerate to one point, by changing the normalization, we obtain convergence to the normal distribution. Regarding the starting point as a time parameter, we obtain convergence in finite dimensional distributions to self-similar processes with independent increments under slightly stronger assumption. §