226 research outputs found
Algebras of invariant differential operators on a class of multiplicity free spaces
Let G be a connected reductive algebraic group and let G'=[G,G] be its
derived subgroup. Let (G,V) be a multiplicity free representation with a one
dimensional quotient (see definition below). We prove that the algebra
D(V)^{G'} of G'-invariant differential operators with polynomial coefficients
on V, is a quotient of a so-called Smith algebra over its center. Over C this
class of algebras was introduced by S.P. Smith as a class of algebras similar
to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of
the Weil representation, where the associative algebra generated by Q(x) and
Q(?) (Q being a non degenerate quadratic form on V) is a quotient of U(sl(2))
Other structure results are obtained when (G,V) is a regular prehomogeneous
vector space of commutative parabolic type
Expansion of the propagation of chaos for Bird and Nanbu systems
The Bird and Nanbu systems are particle systems used to approximate the
solution of the mollied Boltzmann equation. In particular, they have the
propagation of chaos property. Following [GM94, GM97, GM99], we use coupling
techniques and results on branching processes to write an expansion of the
error in the propagation of chaos in terms of the number of particles, for
slightly more general systems than the ones cited above. This result leads to
the proof of the a.s convergence and the centrallimit theorem for these
systems. In particular, we have a central-limit theorem for the empirical
measure of the system under less assumptions then in [M{\'e}l98]. As in [GM94,
GM97, GM99], these results apply to the trajectories of particles on an
interval [0; T]
Decomposition of reductive regular prehomogeneous vector spaces
Let (G,V) be a regular prehomogeneous vector space (abbreviated to PV), where
G is a connected reductive algebraic group over C. If is a decomposition of V into irreducible
representations, then, in general, the PV's are no longer regular.
In this paper we introduce the notion of quasi-irreducible PV (abbreviated to
Q-irreducible), and show first that for completely Q-reducible PV's, the
Q-isotopic components are intrinsically defined, as in ordinary representation
theory. We also show that, in an appropriate sense, any regular PV is a direct
sum of quasi-irreducible PV's. Finally we classify the quasi-irreducible PV's
of parabolic type
Invariant differential operators on a class of multiplicity free spaces
If is a multiplity free space with a one dimensional quotient we give
generators and relations for the non-commutative algebra of
invariant differential operators under the semi-simple part of the
reductive group . More precisely we show that is the quotient of
a Smith algebra by a completely described two-sided ideal.Comment: 31 page
A Numerical Scheme for Invariant Distributions of Constrained Diffusions
Reflected diffusions in polyhedral domains are commonly used as approximate
models for stochastic processing networks in heavy traffic. Stationary
distributions of such models give useful information on the steady state
performance of the corresponding stochastic networks and thus it is important
to develop reliable and efficient algorithms for numerical computation of such
distributions. In this work we propose and analyze a Monte-Carlo scheme based
on an Euler type discretization of the reflected stochastic differential
equation using a single sequence of time discretization steps which decrease to
zero as time approaches infinity. Appropriately weighted empirical measures
constructed from the simulated discretized reflected diffusion are proposed as
approximations for the invariant probability measure of the true diffusion
model. Almost sure consistency results are established that in particular show
that weighted averages of polynomially growing continuous functionals evaluated
on the discretized simulated system converge a.s. to the corresponding
integrals with respect to the invariant measure. Proofs rely on constructing
suitable Lyapunov functions for tightness and uniform integrability and
characterizing almost sure limit points through an extension of Echeverria's
criteria for reflected diffusions. Regularity properties of the underlying
Skorohod problems play a key role in the proofs. Rates of convergence for
suitable families of test functions are also obtained. A key advantage of
Monte-Carlo methods is the ease of implementation, particularly for high
dimensional problems. A numerical example of a eight dimensional Skorohod
problem is presented to illustrate the applicability of the approach
Global solvability of a networked integrate-and-fire model of McKean-Vlasov type
We here investigate the well-posedness of a networked integrate-and-fire
model describing an infinite population of neurons which interact with one
another through their common statistical distribution. The interaction is of
the self-excitatory type as, at any time, the potential of a neuron increases
when some of the others fire: precisely, the kick it receives is proportional
to the instantaneous proportion of firing neurons at the same time. From a
mathematical point of view, the coefficient of proportionality, denoted by
, is of great importance as the resulting system is known to blow-up
for large values of . In the current paper, we focus on the
complementary regime and prove that existence and uniqueness hold for all time
when is small enough.Comment: Published at http://dx.doi.org/10.1214/14-AAP1044 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Path storage in the particle filter
This article considers the problem of storing the paths generated by a
particle filter and more generally by a sequential Monte Carlo algorithm. It
provides a theoretical result bounding the expected memory cost by where is the time horizon, is the number of particles and
is a constant, as well as an efficient algorithm to realise this. The
theoretical result and the algorithm are illustrated with numerical
experiments.Comment: 9 pages, 5 figures. To appear in Statistics and Computin
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