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abstract:
We have developed an overshoot approach to prove recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between $+\\infty$ and $-\\infty$.
In this talk we will concentrate on stable-like processes to illustrate the basic ideas of the approach. In particular we show that a stable-like process with generator $-(-\\Delta)^{\\alpha(x)/2}$ such that $\\alpha(x)=\\alpha$ for $x<-R$ and $\\alpha(x)=\\beta$ for $x>R$ for some $R>0$ and $\\alpha,\\beta\\in(0,2)$ is transient if and only if $\\alpha+\\beta<2$, otherwise it is recurrent.
As a special case this yields also a new proof for the recurrence, point recurrence and transience of symmetric $\\alpha$-stable processes.
References:
An Overshoot Approach to Recurrence and Transience of Markov Processes, 2010, arXiv:1007.2055v1.
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