Source: Ann. Probab. Volume 31, Number 1
(2003), 486-527.
We develop a technique for "partially
collapsing'' one Markov process to produce another. The state space of the new Markov process is
obtained by a pinching operation that identifies points of the original state space
via an equivalence relationship. To ensure that the new process is Markovian we need to
introduce a randomized twist according to an appropriate probability kernel. Informally,
this twist randomizes over the uncollapsed region of the state space when the process
leaves the collapsed region. The "Markovianity'' of the new process is
ensured by suitable intertwining relationships between the semigroup of
the original process and the pinching and twisting operations.
We construct the new Markov process, identify its resolvent and transition function and, under some natural assumptions, exhibit a core for its generator. We also investigate its excursion decomposition.
We apply our theory to a number of examples, including Walsh's spider
and a process similar to one introduced by Sowers in
studying stochastic averaging.
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UNIVERSITY OF CALIFORNIA, BERKELEY 367 EVANS HALL
BERKELEY, CALIFORNIA 94720-3860 E-MAIL: evans@stat.berkeley.edu WEB: http://www.stat.berkeley.edu/users/evans DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS
URBANA, ILLINOIS 60201 E-MAIL: r-sowers@uiuc.edu WEB: http://www.math.uiuc.edu/ r-sowers/
