254,003 research outputs found
MEXIT: Maximal un-coupling times for stochastic processes
Classical coupling constructions arrange for copies of the \emph{same} Markov
process started at two \emph{different} initial states to become equal as soon
as possible. In this paper, we consider an alternative coupling framework in
which one seeks to arrange for two \emph{different} Markov (or other
stochastic) processes to remain equal for as long as possible, when started in
the \emph{same} state. We refer to this "un-coupling" or "maximal agreement"
construction as \emph{MEXIT}, standing for "maximal exit". After highlighting
the importance of un-coupling arguments in a few key statistical and
probabilistic settings, we develop an explicit \MEXIT construction for
stochastic processes in discrete time with countable state-space. This
construction is generalized to random processes on general state-space running
in continuous time, and then exemplified by discussion of \MEXIT for Brownian
motions with two different constant drifts.Comment: 28 page
Convergence analysis of Adaptive Biasing Potential methods for diffusion processes
This article is concerned with the mathematical analysis of a family of
adaptive importance sampling algorithms applied to diffusion processes. These
methods, referred to as Adaptive Biasing Potential methods, are designed to
efficiently sample the invariant distribution of the diffusion process, thanks
to the approximation of the associated free energy function (relative to a
reaction coordinate). The bias which is introduced in the dynamics is computed
adaptively; it depends on the past of the trajectory of the process through
some time-averages.
We give a detailed and general construction of such methods. We prove the
consistency of the approach (almost sure convergence of well-chosen weighted
empirical probability distribution). We justify the efficiency thanks to
several qualitative and quantitative additional arguments. To prove these
results , we revisit and extend tools from stochastic approximation applied to
self-interacting diffusions, in an original context
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