Probability and Its Applications

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. 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=""></ 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.>