This paper studies the identification of an \(\mathbb {R}^d\) -valued diffusion X when a running function of it, say \(h(X_t)\) , is observed. A point-wise observation of the process (in other words, observing \(h(X_t)\) in isolation) cannot identify \(X_t\) unless the h is injective. However observing \(h(X_s)\) on a small interval \([t,t+\varepsilon ]\) can be enough to determine \(X_t\) exactly. The paper contain results that expand on this idea; in particular, a property of ‘fine total asymmetry’ of twice continuously differentiable h is introduced that depends on the fine topology of potential theory and that is both necessary and sufficient for X to be adapted to a natural right-continuous filtration generated by the observations. This particular filtration, though augmented with null sets, does not depend on the distribution of \(X_0\) . For real-analytic h the property reduces to simple asymmetry; that is, there is no nontrivial affine isometry \(\kappa \) on \(\mathbb {R}^d\) such that \(h = h \circ \kappa \) . A second result concerns the case where \(X_0\) is given and h is merely Borel; then X is adapted to an augmented filtration generated by the observation process \((h(X_t))_{t\ge 0}\) if h is ‘locally invertible’ on a subset of \(\mathbb {R}^d\) dense in the fine topology on \(\mathbb {R}^d\) .
