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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.

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  • $\begingroup$ The problem with your setup is that the diffusion part is not compatible with the natural Riemanian structure, and so the existing results do not apply. The simplest case where the question is already challenging is Brownian motion on the Heisenberg group - i.e., $X_1(t),X_2(t)$ are Brownian motion and $X_3(t)$ is the Levy Area. This issue arises generically when the diffusion is not uniformly elliptic. $\endgroup$ – ofer zeitouni Nov 25 '15 at 7:05
  • $\begingroup$ Thanks for the reply @zeitouni. So no theory that could provide the most probable path for the matrix-valued diffusion $X_t$ driven by a one-dimensional Brownian motion $W_t$ is known? Not even if $n = 2$ and $X_t$ is a $2 \times 2$ matrix? What about the situation where the diffusion is vector-valued and driven by a one-dimensional Brownian motion? See added question. $\endgroup$ – tot Nov 26 '15 at 10:14

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