1,161 research outputs found
Quantitative ergodicity for some switched dynamical systems
We provide quantitative bounds for the long time behavior of a class of
Piecewise Deterministic Markov Processes with state space Rd \times E where E
is a finite set. The continuous component evolves according to a smooth vector
field that switches at the jump times of the discrete coordinate. The jump
rates may depend on the whole position of the process. Under regularity
assumptions on the jump rates and stability conditions for the vector fields we
provide explicit exponential upper bounds for the convergence to equilibrium in
terms of Wasserstein distances. As an example, we obtain convergence results
for a stochastic version of the Morris-Lecar model of neurobiology
Ergodicity of the zigzag process
The zigzag process is a Piecewise Deterministic Markov Process which can be
used in a MCMC framework to sample from a given target distribution. We prove
the convergence of this process to its target under very weak assumptions, and
establish a central limit theorem for empirical averages under stronger
assumptions on the decay of the target measure. We use the classical
"Meyn-Tweedie" approach. The main difficulty turns out to be the proof that the
process can indeed reach all the points in the space, even if we consider the
minimal switching rates
Stability of Piecewise Deterministic Markovian Metapopulation Processes on Networks
The purpose of this paper is to study a Markovian metapopulation model on a
directed graph with edge-supported transfers and deterministic intra-nodal
population dynamics. We first state tractable stability conditions for two
typical frameworks motivated by applications: constant jump rates with
multiplicative transfer amplitudes, and coercive jump rates with unitary
transfers. More general criteria for boundedness, petiteness and ergodicity are
then given
On the long time behavior of the TCP window size process
The TCP window size process appears in the modeling of the famous
Transmission Control Protocol used for data transmission over the Internet.
This continuous time Markov process takes its values in , is
ergodic and irreversible. It belongs to the Additive Increase Multiplicative
Decrease class of processes. The sample paths are piecewise linear
deterministic and the whole randomness of the dynamics comes from the jump
mechanism. Several aspects of this process have already been investigated in
the literature. In the present paper, we mainly get quantitative estimates for
the convergence to equilibrium, in terms of the Wasserstein coupling
distance, for the process and also for its embedded chain.Comment: Correction
Exponential Ergodicity of the Bouncy Particle Sampler
Non-reversible Markov chain Monte Carlo schemes based on piecewise
deterministic Markov processes have been recently introduced in applied
probability, automatic control, physics and statistics. Although these
algorithms demonstrate experimentally good performance and are accordingly
increasingly used in a wide range of applications, geometric ergodicity results
for such schemes have only been established so far under very restrictive
assumptions. We give here verifiable conditions on the target distribution
under which the Bouncy Particle Sampler algorithm introduced in \cite{P_dW_12}
is geometrically ergodic. This holds whenever the target satisfies a curvature
condition and has tails decaying at least as fast as an exponential and at most
as fast as a Gaussian distribution. This allows us to provide a central limit
theorem for the associated ergodic averages. When the target has tails thinner
than a Gaussian distribution, we propose an original modification of this
scheme that is geometrically ergodic. For thick-tailed target distributions,
such as -distributions, we extend the idea pioneered in \cite{J_G_12} in a
random walk Metropolis context. We apply a change of variable to obtain a
transformed target satisfying the tail conditions for geometric ergodicity. By
sampling the transformed target using the Bouncy Particle Sampler and mapping
back the Markov process to the original parameterization, we obtain a
geometrically ergodic algorithm.Comment: 30 page
A piecewise deterministic scaling limit of Lifted Metropolis-Hastings in the Curie-Weiss model
In Turitsyn, Chertkov, Vucelja (2011) a non-reversible Markov Chain Monte
Carlo (MCMC) method on an augmented state space was introduced, here referred
to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization
process in the Curie-Weiss model is derived for LMH, as well as for
Metropolis-Hastings (MH). The required jump rate in the high (supercritical)
temperature regime equals for LMH, which should be compared to
for MH. At the critical temperature the required jump rate equals for
LMH and for MH, in agreement with experimental results of Turitsyn,
Chertkov, Vucelja (2011). The scaling limit of LMH turns out to be a
non-reversible piecewise deterministic exponentially ergodic `zig-zag' Markov
process
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity
We consider a time inhomogeneous jump Markov process with state
dependent jump intensity, taking values in Its infinitesimal generator
is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial
f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial
x_i } (x) \int_{E_1} c_1^i ( t, z, x) \gamma_1 ( t, z, x ) \mu_1 (dz ) \\ +
\sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] \gamma_l ( t, z, x)
\mu_l (dz ) , \end{multline*} where are sigma-finite measurable spaces describing three different jump
regimes of the process (fast, intermediate, slow).
We give conditions proving that the long time behavior of can be related
to the one of a time homogeneous limit process Moreover, we
introduce a coupling method for the limit process which is entirely based on
certain of its big jumps and which relies on the regeneration method. We state
explicit conditions in terms of the coefficients of the process allowing to
control the speed of convergence to equilibrium both for and for $\bar X.
- …