Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $$\mathcal{R}^d$$

$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$

with $$\sigma(x) \in \mathbb{R}^{d \times m}$$ and $$m$$ dimensional Brownian motion can be considered that induces a Riemannian structure on $$\mathcal{R}^d$$. In particular we can consider $$X_t$$ as a diffusion on the Riemannian manifold $$\mathcal{R}^d, g)$$, where the metric is defined as $$g = \left( \sigma \sigma^{\top} \right)^{-1}.$$

My question is why doesn't the drift of the process influence the induced Riemannian structure?

If we consider the process $$dY_t = dW_t,$$ with $$Y_t \in \mathcal{R}^d$$ then given two states $$Y_0$$ and $$Y_T$$ at times $$t=0,T$$, the most probable path of $$Y_t$$ between these two states can be obtained from the Onsager-Machlup (OM) function. For diffusion processes on manifolds the most probable path can be obtained as the minimiser of the following Lagrangian $$L(\gamma, \dot{\gamma}) = \frac{1}{2} \| \dot{\gamma}(t) \|_g^2 - \frac{1}{12} S(\gamma(t)),$$ where $$S(\cdot)$$ indicates the scalar curvature at each point on the manifold. The first term minimises the energy of the curve, and is the same term used to identify geodesics on manifolds. Thus for flat manifolds, the most probable path of diffusion process between two states can be obtained from the geodesic curve that connects these two points.

Going back to our example, the induced manifold by the considered SDE would be the Euclidean space in $$\mathcal{R}^d$$, i.e., $$(\mathcal{R}^d, \text{Id})$$. Thus the most probable path between two states of the process will be the geodesic that connects them, that is here, a straight line.

If I now add some drift on the process $$f(\cdot)$$, that let's say in the absence of noise represents limit cycle dynamics. I keep the same unit diffusion coefficient as previously. Since I didn't change the diffusion the induced space is the same Euclidean space, and thus the geodesics between two states should be again straight lines.

However, we know that the most probable path will be somewhere close to the associated deterministic path, i.e., on the limit cycle, that is obviously not a straight line.

Thus I would expect that the drift also contributes to the change of the induced metric of the diffusion. But apparently it is not.

What is wrong in my perspective?

1: https://eudml.org/doc/114044 Capitaine, Mireille. "On the Onsager-Machlup functional for elliptic diffusion processes." Séminaire de Probabilités XXXIV (2000): 313-328.

• The notion of 'most probable path' is very problematic, see the 1999 paper by Andersson and Driver. Also, your formula for the OM functional is only correct in the reversible (i.e. driftless) case. The general formula is on page 2 of the paper you're linking to. Commented Nov 20, 2022 at 16:16
• @MartinHairer What exactly do you mean by the notion of "most probable path" being problematic? Commented Nov 20, 2022 at 16:27
• The 'most probable' path depends on what small neighbourhoods you choose to consider around them, there isn't really a canonical choice. Anyway, the main point is the second one. Commented Nov 20, 2022 at 20:14

1 Answer

What is missing above perspective is that by adding drift the most probable path for a diffusion is