Excursions of Markov Processes
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Let {Xti t ~ O} be a Markov process in Rl, and break up the path X t into (random) component pieces consisting of the zero set ({ tlX = O}) and t the "excursions away from 0," that is pieces of path X. : T ::5 s ::5 t, with Xr- = X = 0, but X. 1= 0 for T s t. When one measures the time in t the zero set appropriately (in terms of the local time) the excursions acquire a measure theoretic structure practically identical to that of processes with stationary independent increments, except the values of the process are paths rather than real numbers. And there is a measure on path space that helps...
Let {Xti t ~ O} be a Markov process in Rl, and break up the path X t into (random) component pieces consisting of the zero set ({ tlX = O}) and t the "excursions away from 0," that is pieces of path X. : T ::5 s ::5 t, with Xr- = X = 0, but X. 1= 0 for T s t. When one measures the time in t the zero set appropriately (in terms of the local time) the excursions acquire a measure theoretic structure practically identical to that of processes with stationary independent increments, except the values of the process are paths rather than real numbers. And there is a measure on path space that helps describe the measure theoretic properties of the excursions in the same way that the Levy measure describes the jumps of a process with independent increments. The entire circle of ideas is called excursion theory. There are many attractive things about the subject: it is an area where one can use to advantage general probabilistic potential theory to make quite specific calculations, it provides a natural setting for apply ing esoteric things like David Williams' path decomposition, it provides a method for constructing processes whose description in terms of an in finitesimal generator or some such analytic object would be complicated. And the ideas seem to be closely related to a good deal of current research in probability.
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