Study of recurrence property of jump type Markov processes

设{A(t)}为无漂移的非负次元，r为非负直线上的左连续函数，该非负直线具有右正极限，满足r(O) = 0, r(x) >，对于x > 0。如果一个随机过程（X(t)）由一个随机微分方程dx (t) = -r(X(t))dt + dA(t)决定，我们说它是一个存储过程。我们的结果如下：(a)在由非负直线上的有界连续函数组成的Basnach空间上，存储过程对应的半群是强连续的，如果(I) 1/r(x)在无穷远处的积分是发散的，(2)Levy测度的总质量是有限的或函数r是非递减的。在Levy测度的总质量是有限的情况下，我们确定了生成子的定义域，并给出了函数r是非递减的情况下生成子的核心。(b)我们证明了存储过程是正循环或零循环或瞬态的，并获得了该过程是瞬态的充分条件和该过程是周期性的充分条件。应当指出，我们既不假定列维测度的总质量是有限的，也不假定r是不减小的。在有些情况下，这两个条件不能适用。然而，我们证明了在特殊情况下，该过程是一个Ornstein-Uhlenbeck型过程，上述暂态充分条件的一部分对于暂态是充分的，也是必要的。

Let {A(t)} be a nonnegative subordinator without drift and r be a left continuous function on the nonnegative line to the nonnegative line with positive right limits satisfying r(O) = 0 and r(x) > 0 for x > 0. We say that a stochastic process {X(t)) is a storage process if it is determined by a stochastic differential equationdX(t) = -r(X(t))dt + dA(t)The following are our results : (a) A semigroup corresponding to the storage process is strongly continuous on the Basnach space consisting of bounded continuous functions on the nonnegative line vanishing at infinity if (I) an integral of 1/r(x) near infinity is divergent and (2) the total mass of the Levy measure is finite or the function r is nondecreasing. We determined the domain of the generator in the Case the total mass of the Levy measure is finite and gave a core for the generator in case the function r is nondecreasing. (b) We showed that the storage process is either positive recurrent or null recurrent or transient and obtained a sufficient condition for the process to be transient and a sufficient condition for the process to be recurrent. It should be remarked that 'we assume neither finiteness of the total mass of the Levy measure nor nondecreasingness of r. There are cases for which these two conditions can not be applied. However, we showed that in special case that the process Is a process of Ornstein-Uhlenbeck type, a part of the above mentioned sufficient condition for transience is sufficient for transience and it is also necessary.