1,419 research outputs found
Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes
New algorithms for computing of asymptotic expansions for stationary
distributions of nonlinearly perturbed semi-Markov processes are presented. The
algorithms are based on special techniques of sequential phase space reduction,
which can be applied to processes with asymptotically coupled and uncoupled
finite phase spaces.Comment: 83 page
Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes
A piecewise-deterministic Markov process is a stochastic process whose
behavior is governed by an ordinary differential equation punctuated by random
jumps occurring at random times. We focus on the nonparametric estimation
problem of the jump rate for such a stochastic model observed within a long
time interval under an ergodicity condition. We introduce an uncountable class
(indexed by the deterministic flow) of recursive kernel estimates of the jump
rate and we establish their strong pointwise consistency as well as their
asymptotic normality. We propose to choose among this class the estimator with
the minimal variance, which is unfortunately unknown and thus remains to be
estimated. We also discuss the choice of the bandwidth parameters by
cross-validation methods.Comment: 36 pages, 18 figure
Numerical method for expectations of piecewise-determistic Markov processes
We present a numerical method to compute expectations of functionals of a
piecewise-deterministic Markov process. We discuss time dependent functionals
as well as deterministic time horizon problems. Our approach is based on the
quantization of an underlying discrete-time Markov chain. We obtain bounds for
the rate of convergence of the algorithm. The approximation we propose is
easily computable and is flexible with respect to some of the parameters
defining the problem. Two examples illustrate the paper.Comment: 41 page
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
Entropy: The Markov Ordering Approach
The focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with respect to monotonic
transformations and analyze two invariant "additivity" properties: (i)
existence of a monotonic transformation which makes the functional additive
with respect to the joining of independent systems and (ii) existence of a
monotonic transformation which makes the functional additive with respect to
the partitioning of the space of states. All Lyapunov functionals for Markov
chains which have properties (i) and (ii) are derived. We describe the most
general ordering of the distribution space, with respect to which all
continuous-time Markov processes are monotonic (the {\em Markov order}). The
solution differs significantly from the ordering given by the inequality of
entropy growth. For inference, this approach results in a convex compact set of
conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of
the various entropy additivity properties and separation of variables for
independent subsystems in MaxEnt problem is added in Section 4.2.
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