Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:

$$dX_t = f(X_t)\,dt + dW_t,$$

where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class $C^2_b$, i.e., it is twice differentiable and the function and all its derivatives up to order 2 are bounded. For simplicity, assume that $X_0=0$.

It is known that for paths $\phi,\varphi$ in the Cameron--Martin space, i.e., absolutely continuous with $\phi(0)=0$ and $\dot\phi\in L^2([0,1])$,
$$
\lim_{\epsilon\downarrow 0}
\frac{
P(\sup_{t\in[0,1]} \lVert\phi(t)-X_t \rVert<\epsilon)
}{
P(\sup_{t\in[0,1]} \lVert\varphi(t)-X_t \rVert<\epsilon)
} = \exp\Big(J(\phi) - J(\varphi)\Big),
$$
where $J$ is the Onsager--Machlup functional
$$
J(\phi):=
-\frac{1}{2} \int_0^1\lVert\dot\phi(t) - f\big(\phi(t)\big)\rVert^2 dt
-\frac{1}{2} \int_0^1\operatorname{div}f\big(\phi(t)\big)\, dt.
$$
This is used by many to define the "most probable" path of diffusion processes (Dürr and Bach, 1978) and to obtain maximum *a posteriori* state paths (Zeitouni and Dembo, 1987; Aihara and Bagchi, 1999a,b). The exponential of the Onsager--Machlup functional is also described by Takahashi (1981) to be "an *ideal* density with respect to a *fictitious* uniform measure", justifying its use as a *fictitious* density for the purpose of comparing the "probability" of individual paths.

**My question**: for a path $\phi$ outside the Cameron--Martin space and $\varphi$ in the Cameron--Martin space, does
$$
\lim_{\epsilon\downarrow 0}
\frac{
P(\sup_{t\in[0,1]} \lVert\phi(t)-X_t \rVert<\epsilon)
}{
P(\sup_{t\in[0,1]} \lVert\varphi(t)-X_t \rVert<\epsilon)
} = 0,
$$
i.e., does the probability of an $\epsilon$-ball centered paths outside the Cameron--Martin space decays much faster than those centered in paths in the Cameron--Martin space? This seems to be a requirement (and an implicit assumption) for using the Onsager--Machlup function for maximum *a posteriori* estimation of paths of diffusions (my thesis topic).

I've attempted to prove this by taking slightly larger tubes around piecewise-linear interpolations of $\phi$ (which belong to the Cameron--Martin space) and converge uniformly to $\phi$. However, I have problems because some of the bounds I obtained depend on $\lVert\dot\phi_i\rVert$ which explodes. I've also searched extensively and failed to find any mention to this issue. The fact that the functional $J$ includes this quadratic term which depends on $\dot\phi$ seems to imply that the answer is yes.

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