Study of recurrence property of jump type Markov processes

设{A(T)}是无漂移的非负从属函数，r是非负线上非负线上的左连续函数，右极限满足r(O)=0和r(X)&gt；0。我们称一个随机过程{X(T))是一个存储过程，如果它由一个随机微分方程dX(T)=-r(X(T))dt+da(T)决定。我们的结果如下：(A)在由非负线上有界连续函数组成的Basnach空间上，对应于该存储过程的半群是强连续的，如果(I)1/r(X)的一个近无穷积分发散，(2)Levy测度的总质量是有限的或函数r是非减的。在Levy测度的总质量是有限的情况下，我们确定了生成元的定义域，并在函数r不减的情况下，给出了生成元的核。(B)证明了存储过程要么是正循环的，要么是零循环的，要么是暂态的，要么是暂态的，并且得到了过程是常返的充分条件。需要指出的是，我们既不假定Levy测度的总质量是有限的，也不是r的不减性，在某些情况下，这两个条件都不适用.然而，我们证明了在过程是Ornstein-Uhlenbeck型过程的特殊情况下，上述瞬变充分条件的一部分是瞬变的充分条件，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.