The girsanov theorem and weak solutions of stochastic differential equations in the dual of a nuclear space

A version of the Girsanov theorem for the inhomogeneous Wiener process in the dual of a nuclear space is proved.It is then applied to the investigation of the existence and uniqueness of weak solutions of stochastic differential equations in this space.     

Langevin equations for L′-Valued Gaussian processes and fluctuation limits of infinite particle systems

Summary L Gaussian processes of a certain class are shown to satisfy generalized Langevin equations. Examples are fluctuation limits of several infinite particle systems, in particular infinite particle branching Brownian motions with immigration under various scalings and the voter model with hydrodynamic scaling.Partially supported by CONACyT grants PCCBBNA 002042 and 140102 G203-006 (México) and a grant of the NSERC (Canada)     

Time-Localization of Random Distributions on Wiener Space II: Convergence,Fractional Brownian Density Processes

For a random element X of a nuclear space of distributions on Wiener space C([0,1],R ^d ), the localization problem consists in projecting X at each time t[0,1] in order to define an S(R ^d )-valued process X={X(t),t[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S(R ^d )) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R ^d ). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S(R ^d )-valued Gaussian processes, the fractional Brownian density processes.     

The Skorohod integral in conuclear spaces

We define an anticipative stochastic integral with respect to a nonhomogeneous Wiener process in a dual of a nuclear space and investigate its basic properties. The theory is developed without the use of chaos expansions.     

The anticipative Stratonovich integral in conuclear spaces

We define an anticipative stochastic integral of Stratonovich type with respect to a nonhomogeneous Wiener process in the dual of a nuclear space and investigate its basic properties.This research was partially supported by Komitet Bada Naukowych, Grant 2 1094 91 01.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.