Traces of random variables on Wiener space and the Onsager-Machlup functional
type: Item
instance of: scholarly article
title: Traces of random variables on Wiener space and the Onsager-Machlup functional
zbMATH Open document ID: 0760.60062
DOI: 10.1016/0022-1236(92)90116-Z
published in: Journal of Functional Analysis
publication date: 1992-09-27T00:00:00Z
review text: Let \(F\) be a random variable defined over the classical Wiener space. The knowledge about \(F\) is restricted to sets of full measure; in particular, it does not give sense to speak about the value of \(F\) at a fixed point \(\omega_ 0\) of the Cameron-Martin space. The authors study sufficient conditions for the existence of the approximate limit of \(F\) at \(\omega_ 0\) (in the sense of Denjoy) and consider some weaker notion for defining \(F(\omega_ 0)\). The results are first proved for a multiple integral \(I_ n(f_ n)\), \(n\) even. Using a refined version of Young's inequality due to Brascamp and Lieb for the main estimate, the authors show that the approximate limit exists whenver \(f_ n\in L^ p([0,1]^ n)\), \(p>1+n\), and the trace of \(f_ n\) exists. Then, by controlling the summation of the limits of the multiple integrals in the Wiener chaos decomposition, the results are extended to sums \[ F=\sum_{n=0}^ \infty I_ n n(f_ n), \quad \sum_{n=0}^ \infty \sqrt{n!} K^ n\| f_ n\|_ \infty<\infty,\quad \text{for all real}\quad K>0. \] The last section of the paper applies these results to the computation of the Onsager-Machlup functional of a special nonanticipative stochastic differential equation.
reviewed by: Rainer Buckdahn
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 65349
zbMATH Keywords: approximation limit
zbMATH Keywords: multiple trace
zbMATH Keywords: Cameron-Martin space
zbMATH Keywords: multiple integrals
zbMATH Keywords: Wiener chaos decomposition
zbMATH Keywords: Onsager-Machlup functional
author: René Carmona
author: David Nualart
MaRDI profile type: Publication
cites work: Best constants in Young's inequality, its converse, and its generalization to more than three functions
cites work: Q5667298
cites work: Q5565773
cites work: Q5683506
cites work: The Onsager-Machlup function for diffusion processes
cites work: Q3874902
cites work: Q3361661
cites work: Fluctuations and Irreversible Process. II. Systems with Kinetic Energy
cites work: A note on conditional exponential moments and Onsager-Machlup functionals
cites work: Q5671456
cites work: Q3324711
cites work: On the Onsager-Machlup functional of diffusion processes around non \(C^ 2\) curves
full work available at URL: 0022-1236(92)90116-z
OpenAlex ID: W2017511647
description: scientific article; zbMATH DE number 65349