Defining a probability distribution on each tangent space of a manifold?

Question

I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to each tangent space, and make sure these probability distributions vary smoothly, then this gives us a very clear way to generalize probability distributions to manifolds. Does such a concept exist in literature?

I want to emphasize I'm being very loose with my definitions. What does it mean to assign a probability distribution to a manifold? Well, I suppose for each tangent space $T_xM$, we assign to it a probability distribution $p_x :T_xM \to [0,\infty)$ and $p_x$ "varies smoothly" for $x \in M$. Please understand, this is a soft question and I am being loose with my definitions since I am trying to get a better idea of what's going on.

Any suggestions of papers or textbooks, I would greatly appreciate.

Answer 1

First let’s generalize the Gaussian distribution to $\mathbb{R}^n$: we can define the Gaussian centered at $\mu\in \mathbb{R}^n $ with standard deviation $\sigma\in \mathbb{R}$ as the unique distribution which is rotationally symmetric about $\mu$ and has $$P[|x-\mu|<r]=\frac{\int_{u=0}^r e^{-u^2/2\sigma^2}dV(u)}{\int_{u=0}^\infty e^{-u^2/2\sigma^2}dV(u)}$$ where $V(u)$ is the hypervolume of the solid ball of radius $u$ in $R^n$. This is the same as the product of the Gaussian distributions with standard deviation $\sigma$ in each dimension.

So for a suitable Riemannian manifold $M$, let $V_\mu(u)$ be the volume of the ball $B(\mu,u)$ of radius $u$ around center $\mu$. Then we can define the Gaussian centered at $\mu\in M$ with standard deviation $\sigma$ to be the unique distribution with $$P[d(x,\mu)<r]=\frac{\int_{u=0}^r e^{-u^2/2\sigma^2}dV_\mu(u)}{\int_{u=0}^\infty e^{-u^2/2\sigma^2}dV_\mu(u)}$$ and where any regions $S,S’$ of equal volume in $B(\mu,r)-B(\mu,r’)$ have probabilities that differ by a ratio of at most $e^{-(r^2-r’^2)/2\sigma^2}$.

This should apply whenever $V_\mu(r)$ is unbounded and has a constant $C$ with $V_\mu(r)<C e^r$. (The second condition may hold for all Riemannian manifolds.) For that matter, it should also apply to any metric spaces which meet the above criteria and where $V_\mu(r)$ is a differentiable function of $R$. If you also want the distributions to vary smoothly with $\mu$, then you should probably require that $M$ has no conjugate points.