32 research outputs found

### Lenses in Skew Brownian Flow

We consider a stochastic flow in which individual particles follow skew
Brownian motions, with each one of these processes driven by the same Brownian
motion. One does not have uniqueness for the solutions of the corresponding
stochastic differential equation simultaneously for all real initial
conditions. Due to this lack of the simultaneous strong uniqueness for the
whole system of stochastic differential equations, the flow contains lenses,
that is, pairs of skew Brownian motions which start at the same point,
bifurcate, and then coalesce in a finite time. The paper contains qualitative
and quantitative (distributional) results on the geometry of the flow and
lenses.Comment: Published at http://dx.doi.org/10.1214/009117904000000711 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### On the continuity of local times of Borel right Markov processes

The problem of finding a necessary and sufficient condition for the
continuity of the local times for a general Markov process is still open.
Barlow and Hawkes have completely treated the case of the L\'{e}vy processes,
and Marcus and Rosen have solved the case of the strongly symmetric Markov
processes. We treat here the continuity of the local times of Borel right
processes. Our approach unifies that of Barlow and Hawkes and of Marcus and
Rosen, by using an associated Gaussian process, that appears as a limit in a
CLT involving the local time process.Comment: Published at http://dx.doi.org/10.1214/009117906000000980 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### A characterization of the infinitely divisible squared Gaussian processes

We show that, up to multiplication by constants, a Gaussian process has an
infinitely divisible square if and only if its covariance is the Green function
of a transient Markov process.Comment: Published at http://dx.doi.org/10.1214/009117905000000684 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Escape from the boundary in Markov population processes

Density dependent Markov population processes in large populations of size
$N$ were shown by Kurtz (1970, 1971) to be well approximated over finite time
intervals by the solution of the differential equations that describe their
average drift, and to exhibit stochastic fluctuations about this deterministic
solution on the scale $\sqrt N$ that can be approximated by a diffusion
process. Here, motivated by an example from evolutionary biology, we are
concerned with describing how such a process leaves an absorbing boundary.
Initially, one or more of the populations is of size much smaller than $N$, and
the length of time taken until all populations have sizes comparable to $N$
then becomes infinite as $N \to \infty$. Under suitable assumptions, we show
that in the early stages of development, up to the time when all populations
have sizes at least $N^{1-\alpha}$, for $1/3 < \alpha < 1$, the process can be
accurately approximated in total variation by a Markov branching process.
Thereafter, the process is well approximated by the deterministic solution
starting from the original initial point, but with a random time delay.
Analogous behaviour is also established for a Markov process approaching an
equilibrium on a boundary, where one or more of the populations become extinct.Comment: 50 page

### A Skorokhod map on measure valued paths with applications to priority queues

The Skorokhod map on the half-line has proved to be a useful tool
for studying processes with non-negativity constraints. In this work we
introduce a measure-valued analog of this map that transforms each
element ? of a certain class of c`adl`ag paths that take values in the space
of signed measures on [0, ?) to a c\'ad\'ag path that takes values in the
space of non-negative measures on $[0, \infty)$ in such a way that for each
$x>0$, the path $t\to \zeta_t [0, x]$ is transformed via a Skorokhod map on the
half-line, and the regulating functions for different $x>0$ are coupled.
We establish regularity properties of this map and show that the map
provides a convenient tool for studying queueing systems in which tasks
are prioritized according to a continuous parameter. One such priority
assignment rule is the well known earliest-deadline-first priority rule.
We study it both for the single and the many server queueing systems.
We show how the map provides a framework within which to form
fluid model equations, prove uniqueness of solutions to these equations
and establish convergence of scaled state processes to the fluid model.
In particular, for these models, we obtain new convergence results in
time-inhomogeneous settings, which appear to fall outside the purview
of existing approaches and is essential when studying the EDF policy
for many servers queues.
Based on Joint work with Rami Atar, Anup Biswas and Kavita Ramanan.Non UBCUnreviewedAuthor affiliation: TechnionFacult