For certain Markov processes, K. Ito has defined the Poisson point process of excursions away from a fixed point. The law of this process is determined by a certain measure, called its Characteristic measure. He gives a list of conditions this measure must obey. I add to these conditions, obtaining necessary and sufficient conditions for a measure to arise in this way. The main technique is to use a 'last exit decomposition' related to those of Getoor and Sharpe. The more general problem of excursions away from a fixed set is treated using the Exit system of B. Maisonneuve.
This gives a useful technique for constructing new Markov processes from old ones. For example, we obtain a rigorous construction of the Skew Brownian motion of Ito and McKean, and another proof of results of Pittenger and Knight on excision of excursions.
A related question is that of determining whether an entrance point
for a Markov process remains an entrance point for an h-transform of
that process. Let E be an open subset of Euclidean space, with a Green
function, and let X be harmonic measure on the Martin boundary Δ of E.
I show that, except for a λMλ -null-set of (x,y)εΔ², x is an entrance point for Brownian motion conditioned to leave E at y. R.S. Martin gave examples in dimension 3 or higher, for which there exist minimal accessible Martin boundary points x≠y for which this condition fails. I give a similar example in dimension 2. The argument uses recent results of M. Cranston and T. McConnell, together with Schwarz-Christoffel transformations. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/24354 |
| Date | January 1983 |
| Creators | Salisbury, Thomas S. |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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