987 research outputs found
A Free Boundary Characterisation of the Root Barrier for Markov Processes
We study the existence, optimality, and construction of non-randomised
stopping times that solve the Skorokhod embedding problem (SEP) for Markov
processes which satisfy a duality assumption. These stopping times are hitting
times of space-time subsets, so-called Root barriers. Our main result is,
besides the existence and optimality, a potential-theoretic characterisation of
this Root barrier as a free boundary. If the generator of the Markov process is
sufficiently regular, this reduces to an obstacle PDE that has the Root barrier
as free boundary and thereby generalises previous results from one-dimensional
diffusions to Markov processes. However, our characterisation always applies
and allows, at least in principle, to compute the Root barrier by dynamic
programming, even when the well-posedness of the informally associated obstacle
PDE is not clear. Finally, we demonstrate the flexibility of our method by
replacing time by an additive functional in Root's construction. Already for
multi-dimensional Brownian motion this leads to new class of constructive
solutions of (SEP).Comment: 31 pages, 14 figure
What is the probability of intersecting the set of Brownian double points?
We give potential theoretic estimates for the probability that a set
contains a double point of planar Brownian motion run for unit time. Unlike the
probability for to intersect the range of a Markov process, this cannot be
estimated by a capacity of the set . Instead, we introduce the notion of a
capacity with respect to two gauge functions simultaneously. We also give a
polar decomposition of into a set that never intersects the set of Brownian
double points and a set for which intersection with the set of Brownian double
points is the same as intersection with the Brownian path.Comment: Published in at http://dx.doi.org/10.1214/009117907000000169 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Explicit formulae in probability and in statistical physics
We consider two aspects of Marc Yor's work that have had an impact in
statistical physics: firstly, his results on the windings of planar Brownian
motion and their implications for the study of polymers; secondly, his theory
of exponential functionals of Levy processes and its connections with
disordered systems. Particular emphasis is placed on techniques leading to
explicit calculations.Comment: 14 pages, 2 figures. To appear in Seminaire de Probabilites, Special
Issue Marc Yo
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