Summary
Let P(x, dy) be the transition probability kernel of a random walk on R n.Let k: R n→[0,1] be a given function. Suppose that the random walk is “killed” according to k: if it is at position x, then it jumps to a special state (the cemetery) with probability k(x); otherwise, it executes a usual random walk step (with law given by P). We say that the killed process is “long-lived” if:
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(a)
P is transient, and there is a non-zero probability of never jumping to the cemetery; or if
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(b)
P is recurrent, and the process started “from infinity” has a chance to get close to the origin before being killed.
This paper develops potential-theoretic characterizations of those functions k which permit long-lived behavior.
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Madras, N. Random walks with killing. Probab. Th. Rel. Fields 80, 581–600 (1989). https://doi.org/10.1007/BF00318907
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DOI: https://doi.org/10.1007/BF00318907
Keywords
- Stochastic Process
- Random Walk
- Probability Theory
- Statistical Theory
- Walk Step