Hilbert Space Valued Generalized Random Processes – Part I (original) (raw)

Structure theorems for generalized random processes

Acta Mathematica Hungarica, 2007

Generalized random processes are classified by various types of continuity. Representation theorems of a generalized random process on K{Mp} on a set with arbitrary large probability, as well as representations of a correlation operator of a generalized random process on K{Mp} and L r (R), r > 1, are given. Especially, Gaussian generalized random processes are proven to be representable as a sum of derivatives of classical Gaussian processes with appropriate growth rate at infinity. Examples show the essence of all the proposed assumptions. In order to emphasize the differences in the concept of generalized random processes defined by various conditions of continuity, the stochastic differential equation y (ω, t) = f (ω, t) is considered, where y is a generalized random process having a point value at t = 0 in sense of Lojasiewicz.

Probabilistic properties of generalized stochastic processes in algebras of generalized functions

Monatshefte für Mathematik, 2017

Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established, which uniquely characterize the process, and relying on this result we prove the measurability of the corresponding random variables with values in the Colombeau algebra of compactly supported generalized constants endowed with the topology generated by sharp open balls. The generalized characteristic function and the generalized correlation function of Colombeau stochastic processes are introduced and their properties are investigated. It is shown that the characteristic function of classical stochastic processes can be embedded into the space of generalized characteristic functions. The generalized expectation and the generalized correlation function can be retrieved from the generalized characteristic function. The structural representation of the correlation function Communicated by A. Constantin.

Generalized stochastic processes in algebras of generalized functions

Journal of Mathematical Analysis and Applications, 2009

Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given.

Gauss-Markov processes on Hilbert spaces

Transactions of the American Mathematical Society, 2015

K. Itô characterised in 1984 zero-mean stationary Gauss–Markov processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô in the case of Hilbert spaces: Gauss–Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core, thus providing an infinite-dimensional extension of the classical result of Courrège in the case of Gauss–Markov semigroups.

Stochastic Analysis and Applications, 19(4), 481–498 (2001) Differential Representation and Markov Property of Generalized Random

2013

Using the geometric properties of Sobolev spaces of integer order and a duality condition, the covariance operators of a generalized random field and its dual can be factorized. Via this covariance factorization, a representation of the generalized random field is obtained as a stochastic equation driven by generalized white noise. This stochastic equation becomes a differential equation under the orthogonality of the dual random field. The solution to this equation satisfies the weak-sense Markov property of integer order. Furthermore, such a solution admits a mean-square series expansion in terms of the eigenfunctions associated with the pure point spectrum of the corresponding covariance operator. From this representation, the relationship between the covariance function and