Construction of an Edwards' probability measure on $\mathcal{C}({\mathbb{R}}_{+},\mathbb{R})$

Abstract

In this article, we prove that the measures ${\mathbb{Q}}_T$ associated to the one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit measure $\mathbb{Q}$ when $T$ goes to infinity, in the following sense: for all $s\ge 0$ and for all events ${\Lambda }_s$ depending on the canonical process only up to time $s$, ${\mathbb{Q}}_T({\Lambda }_s)\to \mathbb{Q}({\Lambda }_s)$. Moreover, we prove that, if $\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s{)}_{s\in {\mathbb{R}}_{+}}$ such that $\mathbb{Q}(Λ_s) =\mathbb{E}_{\mathbb{P}}(\mathbh{1}_{Λ_s}D_s)$, and we give an explicit expression for this martingale.

Published in at http://dx.doi.org/10.1214/10-AOP540 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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