79 research outputs found
Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts
The long time behavior of an absorbed Markov process is well described by the
limiting distribution of the process conditioned to not be killed when it is
observed. Our aim is to give an approximation's method of this limit, when the
process is a 1-dimensional It\^o diffusion whose drift is allowed to explode at
the boundary. In a first step, we show how to restrict the study to the case of
a diffusion with values in a bounded interval and whose drift is bounded. In a
second step, we show an approximation method of the limiting conditional
distribution of such diffusions, based on a Fleming-Viot type interacting
particle system. We end the paper with two numerical applications : to the
logistic Feller diffusion and to the Wright-Fisher diffusion with values in
conditioned to be killed at 0
Uniform tightness for time-inhomogeneous particle systems and for conditional distributions of time-inhomogeneous diffusion processes
In this article, we consider time-inhomogeneous diffusive particle systems,
whose particles jump from the boundary of a bounded open subset of ,
. We give a sufficient criterion for the family of empirical
distributions of such systems to be uniformly tight, independently of the jump
location of the particles. As an application, we show that the conditional
distribution of a family of time-inhomogeneous and environment-dependent
diffusions conditioned not to hit the boundary of a bounded open subset of
is uniformly tight.Comment: 20 page
Minimal quasi-stationary distribution approximation for a birth and death process
In a first part, we prove a Lyapunov-type criterion for the -positive
recurrence of absorbed birth and death processes and provide new results on the
domain of attraction of the minimal quasi-stationary distribution. In a second
part, we study the ergodicity and the convergence of a Fleming-Viot type
particle system whose particles evolve independently as a birth and death
process and jump on each others when they hit . Our main result is that the
sequence of empirical stationary distributions of the particle system converges
to the minimal quasi-stationary distribution of the birth and death process.Comment: The new version provides an original Lyapunov-type criterion for the
-positive recurrence of a birth and death process. An original result
on the domain of attraction of the minimal quasi-stationary distribution of a
birth and death processes is also included. (26 pages
Interacting particle systems and Yaglom limit approximation of diffusions with unbounded drift
We study the existence and the exponential ergodicity of a general
interacting particle system, whose components are driven by independent
diffusion processes with values in an open subset of \mathds{R}^d, .
The interaction occurs when a particle hits the boundary: it jumps to a
position chosen with respect to a probability measure depending on the position
of the whole system. Then we study the behavior of such a system when the
number of particles goes to infinity. This leads us to an approximation method
for the Yaglom limit of multi-dimensional diffusion processes with unbounded
drift defined on an unbounded open set. While most of known results on such
limits are obtained by spectral theory arguments and are concerned with
existence and uniqueness problems, our approximation method allows us to get
numerical values of quasi-stationary distributions, which find applications to
many disciplines. We end the paper with numerical illustrations of our
approximation method for stochastic processes related to biological population
models
General approximation method for the distribution of Markov processes conditioned not to be killed
We consider a strong Markov process with killing and prove an approximation
method for the distribution of the process conditioned not to be killed when it
is observed. The method is based on a Fleming-Viot type particle system with
rebirths, whose particles evolve as independent copies of the original strong
Markov process and jump onto each others instead of being killed. Our only
assumption is that the number of rebirths of the Fleming-Viot type system
doesn't explode in finite time almost surely and that the survival probability
of the original process remains positive in finite time. The approximation
method generalizes previous results and comes with a speed of convergence. A
criterion for the non-explosion of the number of rebirths is also provided for
general systems of time and environment dependent diffusion particles. This
includes, but is not limited to, the case of the Fleming-Viot type system of
the approximation method. The proof of the non-explosion criterion uses an
original non-attainability of result for pair of non-negative
semi-martingales with positive jumps.Comment: 33 pages, additional correction
Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates
This article studies the quasi-stationary behaviour of multidimensional birth
and death processes, modeling the interaction between several species, absorbed
when one of the coordinates hits 0. We study models where the absorption rate
is not uniformly bounded, contrary to most of the previous works. To handle
this natural situation, we develop original Lyapunov function arguments that
might apply in other situations with unbounded killing rates. We obtain the
exponential convergence in total variation of the conditional distributions to
a unique stationary distribution, uniformly with respect to the initial
distribution. Our results cover general birth and death models with stronger
intra-specific than inter-specific competition, and cases with neutral
competition with explicit conditions on the dimension of the process.Comment: 18 page
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