The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\varepsilon$-close to $f$, in the sense that

$$\mathbb P(\sup_{t \in [0, T]} |W_t - f(t)| \leq \varepsilon) > 0.$$

I am interested in what happens to an SDE driven by Brownian motion if we condition on $W$ being ever closer to an arbitrary $f \in C[0, T]$.

Consider the SDE

$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$

with $\mu, \sigma$ as smooth as desired.

Define for every event $E$ of nonzero probability, the conditional measure $\mathbb P_{|E}$ by

$$\mathbb P_{|E}(A) := \frac{\mathbb P(A \cap E)}{\mathbb P(E)}.$$

We write $\mu_\varepsilon$ for the law of $X$ conditional on the event $E_\varepsilon :=\{ \sup_{t \in [0, T]} |W_t - f(t)| \leq \varepsilon\}$, where $f \in C[0, T]$ is deterministic and arbitrary, and set

$$\text{supp}_f (X) := \bigcap_{\varepsilon > 0} \text{supp} (\mu_\varepsilon).$$

Quesiton: Can we describe $\text{supp}_f (X)$? In particular, is it always a singleton for smooth enough $\sigma$?