Onsager-Machlup functional when drift is time-dependent

Question

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.

Answer 1

Using the argument of http://users.sussex.ac.uk/~md326/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 $z\in W_0^{1,2}$.