Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by


\begin{align}
\mathcal{D} = \nabla^2 + \sum\_{i=1}^d b\_i(x) \frac{\partial}{\partial x\_i},
\end{align}


where $b\_i(x) \in \mathcal{C}\_b^2(\mathbb{R}^d)$, $i=1,2,\ldots,d$, and initial condition $x \in \mathbb{R}^d$. Ikeda and Watanabe prove the following theorem in [$\ast$, Chapter VI, $\S$9]


**Theorem 1:** Let $\varphi: [0,T] \rightarrow \mathbb{R}^d$ be a $\mathcal{C}^2$ curve such that $\varphi(0) = x$. Furthermore let $\lambda\_1$ be the first eigenvalue of the boundary value problem
\begin{align}
\begin{cases}\big(\frac{1}{2}\nabla^2 + \lambda\big)\psi = 0\\ \psi |\_{\partial D} = 0, \end{cases}
\end{align}
with associated eigenfunction $c$. Here $D = \{x: |x| < 1\}$. Then we have


\begin{align}
P\_x(w: \|w - \varphi\|\_T < \epsilon) \sim c \exp\left(\int\_0^T L(\varphi, \dot\varphi) \mathrm{d}t\right)\exp\left(-\frac{\lambda\_1T}{\epsilon^2}\right), \quad \text{as} \quad \epsilon \downarrow 0, 
\end{align}


where $L(x,\dot x)$ is the Onsager-Machlup (OM) function of $X(t)$, given by


\begin{align} 
L(x,\dot x)= -\frac{1}{2} |\dot x - b(x) | ^2 - \frac{1}{2} (\nabla \cdot b)(x). 
\end{align}


In the above $\| \cdot \|\_T = \sup\_{t \in [0,T]} |w(t)| $ and $w \in \mathcal{C}^0([0,T] \rightarrow \mathbb{R}^d)$.


I am interested in an extension of the above result that describes the OM function for diffusion processes where $b = b(t,x)$ is a not a time-homogeneous drift. Ideally, the result should cover cases where $b$ is *not* a continuous function of $t$ (but it can be bounded). In particular, I am interested in the case where $b(t,x) = b(t)$ is a sample path for a [telegraph process](https://en.wikipedia.org/wiki/Telegraph_process). The most relevant result I have found is in a paper by Bardina et al. [$\ast\ast$] where they extend the result for diffusion process $X(t)$ on a real separable Hilbert space $H$, given by


\begin{align} 
\begin{cases} 
dX(t) = (AX(t) + F(t,X(t))dt + B dW(t), \quad t \in [0,1] \\ 
X(0) = x \in H.
\end{cases} 
\end{align}


For sufficiently nice $A$ and $B$, this admits a unique solution. Let $W^A(t)$ be the solution to the above when $F \equiv 0, x = 0$. Then the authors show that, given suitable assumptions on $A$,$B$, and $F$,


\begin{align} 
\lim\_{\epsilon \downarrow 0} \frac{P(\|X-\varphi\| < \epsilon)}{P(\|W^A\| < \epsilon)} = \exp \left(\hat{L}(\varphi,\dot \varphi)\right)
\end{align}


where


\begin{align}\hat{L}(\varphi,\dot\varphi) = -\frac{1}{2}\int\_0^1 \left \lvert B^{-1} \left(A \varphi (t) + F(t, \varphi(t)) - \dot\varphi (t) \right)\right\lvert\_H \mathrm{d}t - Tr(\mathcal{S}\_{PR^\ast}) 
\end{align}


with $\mathcal{S}\_{PR^\ast}$ a certain bounded linear operator determined by $\nabla\_x F$ and $\varphi$. This seems to be very close to what I need, however, two questions remain for me:


1. In the paper, the authors are not clear on the meaning of $\nabla\_x$ when $F$ is not differentiable. I presume it then refers to the distributional derivative but it is not clear.
2. The authors assume in their derivation that $F$ is Lipschitz, but the sample paths of the telegraph process are discontinuous. Are there any similar results that allow for discontinuities in the drift?


**References**


$\ast$*Ikeda, Nobuyuki; Watanabe, Shinzo*, *Stochastic differential equations and diffusion processes*, North-Holland Mathematical Library.


$\ast\ast$*Bardina, Xavier; Rovira, Carles; Tindel, Samy*, [**Onsager-Machlup functional for stochastic evolution equations**](http://dx.doi.org/10.1016/S0246-0203(02)00009-2), Ann. Inst. Henri Poincaré, Probab. Stat. 39, No. 1, 69-93 (2003). [ZBL1016.60064](https://zbmath.org/?q=an:1016.60064).



Onsager-Machlup functional when drift is time-dependent
Using the argument of [http://users.sussex.ac.uk/~md326/MAP.pdf](http://users.sussex.ac.uk/%7Emd326/MAP.pdf) or <https://arxiv.org/abs/2209.04523>


We have that if $\mu\_0$ is a centered Gaussian measure then its Onsager-Machlup function is $\operatorname{OM}\_{\mu\_0}(z)=\frac{1}{2}\|z\|\_{\mu\_0}^2$. If $\mu$ is equivalent to $\mu\_0$ with density $\frac{d\mu}{d\mu\_0}\propto e^{-\Phi}$, then


\begin{align\*}
\frac{\mu(B\_\varepsilon(z\_1))}{\mu(B\_\varepsilon(z\_2))}&=\frac{\int\_{B\_\varepsilon(z\_1)}\mu(du)}{\int\_{B\_\varepsilon(z\_2)}\mu(du)}\\
&=\frac{\int\_{B\_\varepsilon(z\_1)}e^{-\Phi(u)}\mu\_0(du)}{\int\_{B\_\varepsilon(z\_2)}e^{-\Phi(u)}\mu\_0(du)}\\
&=\frac{\int\_{B\_\varepsilon(z\_1)}e^{-\Phi(u)+\Phi(z\_1)-\Phi(z\_1)}\mu\_0(du)}{\int\_{B\_\varepsilon(z\_2)}e^{-\Phi(u)+\Phi(z\_2)-\Phi(z\_2)}\mu\_0(du)}\\
&=\frac{e^{-\Phi(z\_1)}}{e^{-\Phi(z\_2)}}\frac{\int\_{B\_\varepsilon(z\_1)}e^{-\Phi(u)+\Phi(z\_1)}\mu\_0(du)}{\int\_{B\_\varepsilon(z\_2)}e^{-\Phi(u)+\Phi(z\_2)}\mu\_0(du)}
\end{align\*}


If $\Phi$ is say, locally Lipschitz, in $z$ then you can bound $|\Phi(z\_i)-\Phi(u)|\leq L |z\_i-u|\lt L\varepsilon$. Therefore the Onsager-Machlup function is


\begin{equation}
\operatorname{OM}\_\mu(z)=
\begin{cases}
\Phi(z)+\frac{1}{2}\|z\|\_{\mu\_0}^2 &\text{ if }z\in \mathcal H\_{\mu\_0}\\
\infty &\text{ else}.
\end{cases}
\end{equation}


By Girsanov, the law of $$dX(t)=b(t,X(t))dt+c dB(t),$$


$\mu^c$, has a density with respect to the law of $$cB(t),$$


$\mu\_0^c$, given by


$$\frac{d\mu^c}{d\mu\_0^c}=\exp\left(\frac{1}{c^2}\left(\int\_0^T b(t,B(t))dB(t)-\frac{1}{2}\int\_0^T b^2(t,B(t))dt\right)\right).$$


We must convert the Ito integral to Stratonovich which is defined pathwise. Therefore we have that the density is $\mu\_0^c$-a.s. equal to


$$\frac{d\mu^c}{d\mu\_0^c}=\exp\left(\frac{1}{c^2}\left(\int\_0^T b(t,B(t))\circ dB(t)-\frac{c^2}{2}\int\_0^T b\_x(t,B(t)) dt-\frac{1}{2}\int\_0^T b^2(t,B(t))dt\right)\right),$$


where $\circ dB(t)$ represents then Stratonovich integral. So long as the this exponent is locally Lipschitz in $B$ then we have that


\begin{equation}
\operatorname{OM}\_\mu(z)=
\begin{cases}
\frac{1}{2c^2}\int\_0^T((z'(t)-b(t,z(t)))^2+c^2b\_x(t,z(t)))dt &\text{ if }z\in \mathcal H\_{\mu\_0}\\
\infty &\text{ else}.
\end{cases}
\end{equation}


Also, in the case where $b$ is independent of $X$, then the process $X(t)=\int\_0^t b(s) ds+B(t)$ is just a mean shifted Brownian motion which is a Gaussian process. In this case, the Onsager-Machlup function is just $\frac{1}{2}\|z-\int\_0^\cdot b(s)ds\|\_{W\_0^{1,2}}^2$ for $z\in W\_0^{1,2}$.


