# Comparing diffusion processes in different metrics

I would like to know if it is possible to compare two diffusion processes defined on the same manifold $$\mathcal{M}$$ but with respect to different metrics say $$g_1$$ and $$g_2$$.

Is there a way to apply a Girsanov theorem for the change of measure in order to express the different metric as a change in the drift?

I.e. Lets say we have a Brownian motion defined in $$(\mathbb{R}^n,g_1)$$ $$dX_t = \sigma dW_t$$,

and a Brownian motion defined in $$(\mathbb{R}^n,g_2)$$ $$dY_t = \sigma d\tilde{W}_t$$.

Is there a way to express $$Y_t$$ in terms of the metric $$g_1$$ as a change in the drift of the process $$Y_t$$?

Update:

Following this paper page 86, diffusions on Riemannian manifolds can be defined through their generators. Thus we can write a Fokker-Planck equation for the evolution of the associated probability density

$$\partial_t p_t(x) = \mathcal{L} p_t(x),$$

where we assume that the expression conserves the integral of $$p_t(x)$$ with respect to the volume element of the manifold $$d Vol_g$$, i.e., $$\int p_t(x) dVol_g = 1$$.

For a Brownian motion on $$(\mathcal{M},g)$$, we use the Laplace-Beltrami operator associated with the metric $$g$$:

$$\Delta_g p_t(x) = \frac{1}{\sqrt{\det g_{ij}} } \partial_i \left( \sqrt{\det g_{ij}} g^{ij} \partial_j p_t(x) \right) ,$$

where $$g_{ij}$$ are the components of the metric in the local chart, and $$g^{ij}$$ are the components of the matrix inverse to $$g_{ij}$$.

So the different metrics $$g_1$$ and $$g_2$$ influence the way the probability density propagates in time for each process. I wonder if one could be able to define this difference in propagation in terms of a change in the drift of a diffusion process.

• maybe also try the ideas developed in the context of Brownian motion in Ricci flow. Here "Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow", they study Brownian motion over a whole family of metrics. Nov 18, 2022 at 20:08
• see here too for some computations relating flat and another metric: math.stackexchange.com/questions/4163105/… Nov 18, 2022 at 20:13