# Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For example, $f$ could represent a color or temperature over the surface.

How can I numerically compute a (generalized) "Gaussian blur" of $f$ on the surface?

($E$ could be such that the mesh is topologically similar to a rectangular grid or if convenient edges could be added so that faces are subdivided further into triangular, and therefore planar, faces.)

My thoughts, for what they're worth:

The probability distribution of a particle's position after a random walk in the mesh is a Gaussian blur of the initial probability distribution, I think. This probability distribution can be reached by starting with $f$ and repeatedly updating for random 1-hops in the mesh, Markov-chain-style, but I don't know how to assign probabilities to transitions along each outgoing edge from a vertex.

In symmetrical cases such as a planar square grid, it is obvious that each outgoing edge carries equal weight/transition-probability, but how to determine these weights/transition-probabilities in the case of a general/irregular/curved mesh? I feel the expected position after a hop from a given vertex should lie somewhere along the span of the mean curvature normal (which can be calculated like this) vector from that vertex, and that this is the generalized version of the Martingale property of planar random walks. This constraint would fully determine the relative weights/transition-probabilities of each edge if the vertex in question has valence 3, but in the far more common case where vertices have valence >3, the weights/transition-probabilities are under-determined without some additional constraint/optimization. Could it be that the weights should be decided in this case by maximizing the post-hop variance?