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](http://dx.doi.org/10.1007/BF01609446)) and to obtain maximum *a posteriori* state paths (Zeitouni and Dembo, [1987](http://dx.doi.org/10.1080/17442508708833444); Aihara and Bagchi, 1999[a](http://dx.doi.org/1109/ACC.1999.782868),[b](http://dx.doi.org/10.1109/9.793785)). The exponential of the Onsager--Machlup functional is also described by Takahashi ([1981](http://dx.doi.org/10.1007/BFb0088735)) 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.



Fictitious density of paths of diffusion processes outside the Cameron--Martin space
Here's a proof of the statement for $f=0$, so that $X=W$ is a Wiener process. (The proof with general $f$ is a bit more involved, and I give this further below). I'll base the proof on the following simple result. Here, I am using $B\_\epsilon=\left\{\omega\in\mathcal{W}^n\colon\sup\_{t\in[0,1]}\lVert\omega(t)\rVert\le\epsilon\right\}$ for the $\epsilon$-ball in Wiener space.



> 
> **Lemma 1**: For any continuous $\gamma\colon[0,1]\to\mathbb{R}^n$ and $\epsilon > 0$
>  $$
> \mathbb{P}\left(W\in B\_\epsilon+\gamma\right)\le\mathbb{P}\left(W\in B\_\epsilon\right).
> $$
> 
> 
> 


*Proof*: As the Wiener measure is Gaussian, it is [log-concave](http://en.wikipedia.org/wiki/Logarithmically_concave_measure). As $B\_\epsilon$ is convex then its indicator function is log-concave and, as the convolution of log-concave functions is log-concave, this implies that $\gamma\mapsto\mathbb{P}(W\in B\_\epsilon+\gamma)$ is log-concave. As it is also symmetric this gives
$$
\mathbb{P}\left(W\in B\_\epsilon\right)\ge\sqrt{\mathbb{P}\left(W\in B\_\epsilon+\gamma\right)\mathbb{P}\left(W\in B\_\epsilon-\gamma\right)}=\mathbb{P}\left(W\in B\_\epsilon+\gamma\right).
$$
*QED*


Choosing any smooth $\gamma\colon[0,1]\to\mathbb{R}^n$ with $\gamma(0)=0$ then, using a Girsanov transform in the usual way, 
$$
\begin{align}
&\mathbb{P}\left(W\in B\_\epsilon+\phi\right)
=
\mathbb{P}\left(W-\gamma\in B\_\epsilon+\phi-\gamma\right)\\
&\qquad=\mathbb{E}\left[\exp\left(-\int\_0^1\dot\gamma dW-\frac12\int\_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B\_\epsilon+\phi-\gamma\right]\mathbb{P}(W\in B\_\epsilon+\phi-\gamma)\\
&\qquad\le\mathbb{E}\left[\exp\left(\int\_0^1\ddot\gamma W dt-\dot\gamma(1)W(1)-\frac12\int\_0^1\dot\gamma^2dt\right)\Bigg\vert W\in B\_\epsilon+\phi-\gamma\right]\mathbb{P}(W\in B\_\epsilon)
\end{align}
$$
The last inequality here is using the lemma, together with integration by parts in the exponent. So, letting $\epsilon$ go to zero and using uniform convergence in the expectation,
$$
\begin{align}
\limsup\_{\epsilon\to0}\frac{\mathbb{P}(W\in B\_\epsilon+\phi)}{\mathbb{P}(W\in B\_\epsilon)}
&\le\exp\left(\int\_0^1\ddot\gamma (\phi-\gamma) dt-\dot\gamma(1)(\phi(1)-\gamma(1))-\frac12\int\_0^1\dot\gamma^2dt\right)\\
&=\exp\left(\frac12\int\_0^1\dot\gamma^2dt-\int\_0^1\dot\gamma\dot\phi dt\right).
\end{align}
$$
In the last line, the derivative $\dot\phi$ is understood in the sense of distributions. Now, if $\phi$ is not in Cameron--Martin space, then if we let $\gamma$ be smooth approximations to $\phi$ (e.g., convolve with a smooth bump function), the right hand side tends to zero giving the result.




---


Here's a proof of the general case, based on the following lemma which generalizes Lemma 1 above (a proof is given further below).



> 
> **Lemma 2**: Suppose that $X$ satisfies an SDE of the form $dX=g(X\_t,t)dt+dW$, $X\_0=0$, where
>  $g\colon\mathbb{R}^n\times[0,1]\to\mathbb{R}^n$. is (jointly)
>  continuous and $g(x,t)$ has (jointly) continuous first and second
>  order derivatives with respect to $x$. Then, for any continuous path
>  $\phi\colon[0,1]\to\mathbb{R}^d$, $$
> \limsup\_{\epsilon\to0}\frac{\mathbb{P}(X\in
> B\_\epsilon+\phi)}{\mathbb{P}(W\in
> B\_\epsilon)}\le K$$
>  where $K$ is some increasing function of $\lVert Dg(\phi,\cdot)\rVert$ and $\lVert D^2g(\phi,\cdot)\rVert$.
> 
> 
> 


Let's use this to prove the result in the question. For any smooth $\gamma\colon[0,1]\to\mathbb{R}^n$, define $Y$ by $Y\_0=0$ and $dY=f(Y+\gamma)dt+dW$. We see that $X-\gamma$ satisfies the same SDE as $Y$, but with $W-\gamma$ in place of $W$. Applying a Girsanov transformation as above,
$$
\frac{\mathbb{P}(X\in B\_\epsilon+\phi)}{\mathbb{P}(Y\in B\_\epsilon+\phi-\gamma)}=\frac{\mathbb{P}(X-\gamma\in B\_\epsilon+\phi-\gamma)}{\mathbb{P}(Y\in B\_\epsilon+\phi-\gamma)}=\mathbb{E}\left[\exp\left(-\int\_0^1\left(\dot\gamma dW\_t+\frac12\dot\gamma^2dt\right)\right)\Bigg\vert Y\in B\_\epsilon+\phi-\gamma\right]=\mathbb{E}\left[\exp\left(-\int\_0^1\left(\dot\gamma dY\_t-\dot\gamma f(Y\_t+\gamma)dt+\frac12\dot\gamma^2dt\right)\right)\Bigg\vert Y\in B\_\epsilon+\phi-\gamma\right].
$$
Taking limits as $\epsilon$ goes to 0 and applying Lemma 2,
$$
\limsup\_{\epsilon\to0}
\frac{\mathbb{P}(X\in B\_\epsilon+\phi)}{\mathbb{P}(W\in B\_\epsilon)}\le K\exp\left(\int\_0^1\left(\frac12\dot\gamma^2-\dot\gamma\dot\phi-\dot\gamma f(\phi)\right)dt\right).
$$
As above, we use integration by parts to take the limit, and $\dot\phi$ is the derivative in the sense of distributions. The term $K$ is the right hand side of the inequality in Lemma 2, evaluated with $g(x,t)=f(x+\gamma\_t,t)$ evaluated along the path $\phi-\gamma$. This is the same as the right hand side of the inequality evaluated with $g(x,t)=f(x)$ along the path $\phi$, so is independent of $\gamma$. Letting $\gamma$ be smooth approximations to $\phi$, the right hand side of the above inequality can be made arbitrarily close to 0, giving the required result.




---


**Proof of Lemma 2:**


Note that the SDE can be expressed as $W\_t=X\_t-\int\_0^tg(X\_s,s)ds$. Let us define the path $\psi\_t=\phi\_t-\int\_0^tg(\phi\_s,s)ds$. Expanding $g(X\_s,s)$ as a power series to second order in $X-\phi$ gives
$$
W\_t-\psi\_t=X\_t-\phi\_t-\int\_0^t\left(A\_s(X\_s-\phi\_s)+u\_s\right)ds,
$$
where $A\_s=Dg(\phi\_s,s)$ and $u\_s$ is of second order in $X\_s-\phi\_s$. Choosing a fixed $L > \lVert D^2g(\phi\_t,t)\rVert/2$ then $\lVert u\_s\rVert\le L\lVert X\_s-\phi\_s\rVert^2\le L\epsilon^2$ for $X\in B\_\epsilon+\phi$ and small enough $\epsilon$. Now, define $S$ to be set of all paths $\omega$ of the form
$$
\omega\_t=y\_t-\int\_0^t(A\_sy\_s+v\_s)ds
$$
for some $y\in B\_\epsilon$ and $v\in B\_{L\epsilon^2}$. From the above, we have $W\in S+\psi$ whenever $X\in B\_\epsilon+\phi$. As $S$ is convex and symmetric, using log-concavity as in the proof of Lemma 1, $\mathbb{P}(W\in S+\psi)\le\mathbb{P}(W\in S)$.


Now, define a process $Y$ by
$$
dY\_t=dW\_t+AY\_t,\ \ Y\_0=0.
$$
Suppose that $\omega\in S$, $y\in B\_\epsilon$ and $v\in B\_{L\epsilon^2}$ are as above and $W\_t=\omega\_t$ then, we have
$$
d\lVert Y\_t-y\_t\rVert/dt\le\lVert A(Y\_t-y\_t)-v\_t\rVert\le a\lVert Y\_t-y\_t\rVert+L\epsilon^2
$$
where $a=\sup\_t\lVert A\_t\rVert$. From this we can bound $Y\_t-y\_t$ by $L\epsilon^2a^{-1}(e^{at}-1)$, which I will write as $M\epsilon^2$.
So, $Y\in B\_{\epsilon+M\epsilon^2}$, and we have $\mathbb{P}(W\in S)\le\mathbb{P}(Y\in B\_{\epsilon+M\epsilon^2})$.


Next, define the process
$$
dZ = dW - aZdt,\ \ Z\_0=0.
$$
This has drift $-a\lVert Z\rVert$ in the radial direction, whereas $Y$ has radial drift bounded below by $-a\lVert Y\rVert$. By comparing these processes we have $\mathbb{P}(Y\in B\_{\epsilon+M\epsilon^2})\le\mathbb{P}(Z\in B\_{\epsilon+M\epsilon^2})$.


Now, applying the Onsager--Machlup functional to the simple case with $f(x)=-ax$ and $\phi=0$ gives
$$
\lim\_{\epsilon\to0}\frac{\mathbb{P}(Z\in B\_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B\_{\epsilon+M\epsilon^2})}=\exp(a).
$$
We can also use the standard result that $\mathbb{P}(W\in B\_\epsilon)\sim\exp(-c/\epsilon)$ in the limit $\epsilon\to0$ (where $c$ depends only on the dimension $n$) to get
$$
\lim\_{\epsilon\to0}\frac{\mathbb{P}(W\in B\_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B\_{\epsilon})}=\exp(cM).
$$


Putting the above results together gives the following sequence of inequalities, which should be understood in the asymptotic limit $\epsilon\to0$.
$$
\begin{align}
\frac{\mathbb{P}(X\in B\_\epsilon+\phi)}{\mathbb{P}(W\in B\_\epsilon)}
&\le
\frac{\mathbb{P}(W\in S+\psi)}{\mathbb{P}(W\in B\_\epsilon)}
\le
\frac{\mathbb{P}(W\in S)}{\mathbb{P}(W\in B\_\epsilon)}\\
&\le
\frac{\mathbb{P}(Y\in B\_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B\_\epsilon)}
\le
\frac{\mathbb{P}(Z\in B\_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B\_\epsilon)}\\
&\le
\exp(a)\frac{\mathbb{P}(W\in B\_{\epsilon+M\epsilon^2})}{\mathbb{P}(W\in B\_\epsilon)}
\le\exp(a+cM).
\end{align}
$$
**QED**


