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. 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, Ann. Inst. Henri Poincaré, Probab. Stat. 39, No. 1, 69-93 (2003). ZBL1016.60064.