Potentially useful background info

For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \end{align*} where $A: \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ and $B: \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}\times \mathbb{R}^{n}$ are smooth and $W_{t}$ is a $n$-dimensional standard Brownian motion. Equip $\mathbb{R}^{m}$ with the metric $g = (BB^{T})^{-1}$ and the Levi-Cevita connection and consider $X_{t}$ to be a diffusion on the Riemannian manifold $M = (\mathbb{R}^{m},g)$ with generator $\frac{1}{2} \Delta_{M} + f$, where $\Delta_{M}$ is the Laplace–Beltrami operator and $f:\mathbb{R}^{m} \rightarrow M$ is given by a complicated expression involving $A$, $BB^{T}$ and $(BB^{T})^{-1}$. Then for any smooth curve $u:[0,T] \rightarrow M$ it holds that \begin{align*} P\bigl( \rho( X_{t} , u(t) ) < \epsilon \text{ for all } t \in [0,T] \bigr)\underset{\epsilon \rightarrow 0^{+}}{\sim} e^{-\frac{1}{2} \int_{0}^{T} \mathcal{L}(u(t),u'(t)) dt } \qquad (1) \end{align*} where $\rho$ is the Riemannian distance and $\mathcal{L}$ is a function on the tangent bundle $TM$ given by \begin{align*} \mathcal{L}(u,u') = \lVert f(u) - u' \rVert_{u}^{2} + \text{div } f(u) - \frac{1}{6} R(u) \end{align*} Here $\lVert \cdot \rVert_{u}$ is the Riemannian norm on the tangent space $T_{u}(M)$ and $R(u)$ is the scalar curvature. $\mathcal{L}$ is called the Onsager-Machlup function.

For the simple case where $m = n$ and $B = I$ (the identity), the Riemannian structure induced by the diffusion is just the Euclidean one and theorem reduces to: For Then for any smooth curve $u:[0,T] \rightarrow \mathbb{R}^{m}$ it holds that \begin{align*} P\bigl( \lvert X_{t} - u(t) \rvert < \epsilon \text{ for all } t \in [0,T] \bigr)\underset{\epsilon \rightarrow 0^{+}}{\sim} e^{-\frac{1}{2} \int_{0}^{T} \mathcal{L}(u(t),u'(t)) dt } \end{align*} where $\lvert \cdot \rvert$ is the Euclidean distance and \begin{align*} \mathcal{L}(u,u') = \sum_{i = 1}^{m} \bigl( A_{i}(u) - u_{i}' \bigr)^{2} + \sum_{i=1}^{m} \frac{\partial A_{i}}{\partial x_{i}}(u) \end{align*}

References

Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland, pp. 532-539, 1989.

Fujita, T. and Kotani, S.: The Onsager–Machlup function for diffusion processes, J. Math. Kyoto Univ. 22: 115–130, 1982.

Capitaine, M.: On the Onsager Machlup functional for elliptic diffusion processes. In Seminaire de Probabilites 34, Lecture Notes in Math., Springer, 2000, Vol. 1729.

Question 1

Suppose that $X_{t}$ is a complex, matrix-valued diffusion given by

\begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (2) \end{align*} where $A, B: \mathbb{C}^{n}\times \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}\times \mathbb{C}^{n}$ and $W_{t}$ is a real, one-dimensional standard Brownian motion. What is the equivalent of (1) and what is the Onsager-Machlup function for $X_{t}$?

Question 2

Suppose that $X_{t}$ is a complex, vector-valued diffusion given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (3) \end{align*} where $A, B: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ and $W_{t}$ is a real, one-dimensional standard Brownian motion. What is the equivalent of (1) and what is the Onsager-Machlup function for $X_{t}$?

Any information would be much appreciated.

Reasons for asking

Recently physicists have been trying to describe the most probable time evolution ("path") of quantum systems subject to continuous-in-time (homodyne) measurements. The state of such a system is governed by either (2) (for impure states and imperfect detection) or (3) (for pure states and perfect detection). These physicists use non-rigorous path integral methods to obtain the most likely path. And it has been known for a long time that path integral methods sometimes yield results different from the rigorous Onsager-Machlup theory described above. See Dürr, D. and Bach, A.: The Onsager–Machlup function as Lagrangian for the most probable path of a diffusion process, Commun. Math. Phys. 60: 153–170, 1978.