UBC Theses and Dissertations
Construction of strong Markov processes through excursions, and a related Martin boundary Salisbury, Thomas S.
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.
Item Citations and Data