Strongly self-interacting processes on the circle

ABSTRACT

The purpose of this paper is to investigate the long-time behaviour for a self-interacting diffusion and a self-interacting velocity jump process. While the diffusion case has already been studied for some particular potential function, the second one, which belongs to the family of piecewise deterministic processes, is new. Depending on the underlying potential function's shape, we prove either the almost sure convergence or the recurrence for a natural extended process given by a change a variable.     

Time reversal for infinite-dimensional diffusions

Summary In this paper we analize the reversibility of the diffusion property for the solution of certain infinite-dimensional systems of stochastic differential equations. Necessary and sufficient conditions ensuring this reversibility are given. The proofs use the techniques of the stochastic calculus of variations.This work was partly done when the first author was visiting the Centre de Recerca Matemàtica at Barcelona     

Invertibility in infinite-dimensional spaces

An interesting result of Doyle and Hocking states that a topological -manifold is invertible if and only if it is a homeomorphic image of the -sphere . We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.

     

Connection between deriving bridges and radial parts from multidimensional Ornstein-Uhlenbeck processes

Summary First we give a construction of bridges derived from a general Markov process using only its transition densities. We give sufficient conditions for their existence and uniqueness (in law). Then we prove that the law of the radial part of the bridge with endpoints zero derived from a special multidimensional Ornstein--Uhlenbeck process equals the law of the bridge with endpoints zero derived from the radial part of the same Ornstein--Uhlenbeck process. We also construct bridges derived from general multidimensional Ornstein--Uhlenbeck processes.     

Time aggregated Markov decision processes via standard dynamic programming

This note addresses the time aggregation approach to ergodic finite state Markov decision processes with uncontrollable states. We propose the use of the time aggregation approach as an intermediate step toward constructing a transformed MDP whose state space is comprised solely of the controllable states. The proposed approach simplifies the iterative search for the optimal solution by eliminating the need to define an equivalent parametric function, and results in a problem that can be solved by simpler, standard MDP algorithms.     

An explicit representation of local times of anticipative processes

In this work we give an explicit representation of the local time of an anticipative Ito process using the stochastic calculus of variations and the theory of the nuclear space-valued stochastic processes     

Area-interaction point processes

We introduce a new Markov point process that exhibits a range of clustered, random, and ordered patterns according to the value of a scalar parameter. In contrast to pairwise interaction processes, this model has interaction terms of all orders. The likelihood is closely related to the empty space functionF, paralleling the relation between the Strauss process and Ripley'sK-function. We show that, in complete analogy with pairwise interaction processes, the pseudolikelihood equations for this model are a special case of the Takacs-Fiksel method, and our model is the limit of a sequence of auto-logistic lattice processes.     

Filtering and change point estimation for hidden Markov-modulated Poisson processes

A continuous-time Markov chain which is partially observed in Poisson noise is considered, where a structural change in the dynamics of the hidden process occurs at a random change point. Filtering and change point estimation of the model is discussed. Closed-form recursive estimates of the conditional distribution of the hidden process and the random change point are obtained, given the Poisson process observations     

Large and moderate deviations for infinite-dimensional autoregressive processes

We consider large and moderate deviations for the empirical mean and covariance of hilbertian autoregressive processes. As an application we obtain moderate deviations principles for the eigenvalues and associated projectors of the empirical covariance.     

Homotopic invariance of infinite-dimensional homotopic groups of Hilbert space subsets

The paper discusses some equivalent ways of construction of infinite-dimensional homotopic groups of subsets and pairs of subsets in real Hilbert spaces. In the admissible class of K ₀-continuous mappings, the homotopic invariance of the mentioned groups and their isomorphism are demonstrated in the case where the basic points belong to the same component of K ₀-linear connectivity.     

Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures.One application is a simple exact sampling algorithm for qvolume-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand–Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.     

Aggregation of Markov processes: Axiomatization

We give two simple axioms that characterize a simple functional form for aggregation of column stochastic matrices (i.e., Markov processes). Several additional observations are made about such aggregation, including the special case in which the aggregated process is Markovian relative to the original one.     

Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps

We consider Markov processes built from pasting together pieces of strong Markov processes which are killed at a position dependent rate and connected via a transition kernel. We give necessary and sufficient conditions for local absolute continuity of probability laws for such processes on a suitable path space and derive an explicit formula for the corresponding likelihood ratio process. The main tool is the consideration of the process between successive jumps – what we call ‘elementary experiments’ – and criteria for absolute continuity of laws of the process there. We apply our results to systems of branching diffusions with interactions and immigrations. This revised version was published online in June 2006 with corrections to the Cover Date.     

两个相关更新过程的一种新描述

研究了两个相关点过程之间相关性的描述。我们指出当两个点过程是泊淞过程时,相关系数能充分描述它们之间的相互关系。当两个点过程是更新过程时,相关系数随着时间窗口的变化而变化,是时间窗口的递增函数。于是我们提出用相关系数曲线来描述两个更新过程之间的相互关系。这些结论对优化神经网络的设计将是很有用的。     

Deformations of the infinite-dimensional Lie algebra L 3

For the nilpotent infinite-dimensional Lie algebra L ₃, we compute the second cohomology group H ²(L ₃, L ₃) with coefficients in the adjoint module. Nontrivial cocycles are found in closed form, and Massey powers are computed for them.     

Harnack and HWI inequalities on infinite-dimensional spaces

In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.     

Optimal control of one dimensional non-conservative quasi-diffusion processes

An extension of the work of P. Mandl concerning the optimal control of time-homogeneous diffusion processes in one dimension is given. Instead of a classical second order differential operator as infinitesimal generator, Feller's generalized differential operator D_mD⁺_p with a possibly nondecreasing weight function m is used. In this manner an optimal control of a wider class of one dimensional Marcov processes-including diffusions as well as birth and death processes-is realized.     

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.