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.