Excursions and Local Time

  • Robert M. Blumenthal
Part of the Probability and Its Applications book series (PA)


For many processes each point in the state space is regular for itself, and so for each point x we have the notions of local time at x and of excursions away from x. Brownian motion in R and most one dimensional diffusion processes are important examples. There has been a great deal of attention focused on the question of how in such cases the local time {L(t, x); t ≥ 0} at x varies with x and on the use of excursion theory, not only to study this question but also to make other important constructions. In this chapter we will give some applications of excursion theory to such matters. We will be considering local time and excursions at several points simultaneously, but the basic notion is still excursions away from a single point, so that no new notions appear. This should be contrasted with the situation in the next chapter. There we will be considering processes essentially in higher dimensional state spaces, where the relevant notion is that of excursions of the path away from a rather general set and where much less is known about specific formulas and constructions.


Brownian Motion Local Time Bessel Process Domain Point Poisson Random Measure 
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Copyright information

© Birkhäuser Boston 1992

Authors and Affiliations

  • Robert M. Blumenthal
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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