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

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 ${\mathrm{\Lambda }}_s$ depending on the canonical process only up to time $s$, ${\mathbb{Q}}_T({\mathrm{\Lambda }}_s)\to \mathbb{Q}({\mathrm{\Lambda }}_s)$. Moreover, we prove that, if $\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s{)}_{s\in {\mathbb{R}}_{+}}$ such that $$\mathbb{Q}$(\Lambda_s) =$\mathbb{E}${$\mathbb{P}$}(\mathbh{1}{\Lambda_s}D_s)$, and we give an explicit expression for this martingale.