# Onsager-Machlup functional when drift is time-dependent

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.

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

$$$$\operatorname{OM}_\mu(z)= \begin{cases} \Phi(z)+\frac{1}{2}\|z\|_{\mu_0}^2 &\text{ if }z\in \mathcal H_{\mu_0}\\ \infty &\text{ else}. \end{cases}$$$$

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

$$$$\operatorname{OM}_\mu(z)= \begin{cases} \frac{1}{2c^2}\int_0^T((z'(t)-b(t,z(t)))^2+c^2b_x(t,z(t)))dt &\text{ if }z\in \mathcal H_{\mu_0}\\ \infty &\text{ else}. \end{cases}$$$$

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 $$z\in W_0^{1,2}$$.

• A couple of questions: is $\mathcal{H}_{\mu_0}$ the Cameron-Martin space of $\mu_0$? Also, is $W^{1,2}_0$ the Sobolev space with elements zero at the boundary? If it is, then does this result answer the question? Since the sample paths of the telegraph process are not elements of $W^{1,2}_0$. Oct 4, 2022 at 11:36
• Yes, it is the Cameron-Martin space. And yes it is indeed the Sobolev space. This does answer the question as it is the Onsager-Machlup function. However, as you have noticed, the Onsager-Machlup function is only finite on a set of measure zero (the Cameron-Martin space). It is infinite outside the Cameron-Martin space. That is the unfortunate thing about Onsager-Machlup theory... Oct 4, 2022 at 12:33
• I see. So in the case of the basic result proved by Ikeda and Watanabe, for example, the OM function is finite for $\mathcal{C}^2$ curves because these curves happen to be in the Cameron-Martin space of the Weiner measure? Oct 6, 2022 at 13:58
• Also, I assume that when writing $|| \cdot ||_{W^{1,2}_0}$ at the end you are referring to the semi-norm? Oct 6, 2022 at 15:13
• @Enforce Yes, $C^2$ curves are in the CM space (however the CM space is a bit larger). And in this case $W_0^{1,2}$ is an actual norm because the space we care about are the continuous functions with $f(0)=0$. Oct 6, 2022 at 22:10