71 research outputs found
Competition interfaces and second class particles
The one-dimensional nearest-neighbor totally asymmetric simple exclusion
process can be constructed in the same space as a last-passage percolation
model in Z^2. We show that the trajectory of a second class particle in the
exclusion process can be linearly mapped into the competition interface between
two growing clusters in the last-passage percolation model. Using technology
built up for geodesics in percolation, we show that the competition interface
converges almost surely to an asymptotic random direction. As a consequence we
get a new proof for the strong law of large numbers for the second class
particle in the rarefaction fan and describe the distribution of the asymptotic
angle of the competition interface.Comment: Published at http://dx.doi.org/10.1214/009117905000000080 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the location of the maximum of a continuous stochastic process
In this short note we will provide a sufficient and necessary condition to
have uniqueness of the location of the maximum of a stochastic process over an
interval. The result will also express the mean value of the location in terms
of the derivative of the expectation of the maximum of a linear perturbation of
the underlying process. As an application, we will consider a Brownian motion
with variable drift. The ideas behind the method of proof will also be useful
to study the location of the maximum, over the real line, of a two-sided
Brownian motion minus a parabola and of a stationary process minus a parabola.Comment: More general results, including a proof of Groeneboom-Janson formula
for the variance of the location of maximum of a Brownian motion minus a
parabola, and a proof of the uniqueness of the location of the maximum of an
Airy process minus a parabola. Final version, accepted for publication in the
Journal of Applied Probabilit
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