1
$\begingroup$

The question "Brownian bridge interpreted as Brownian motion on the circle" reminds me of a question along similar lines. Say that you have a weighted undirected graph $G$, and you have constrained Brownian motion on the graph where edges have lengths equal to the reciprocal of their weights. So for example a cycle with weight 1 (and therefore length 1) would be like the Brownian bridge. By 'constrained' I mean that when edges meet at a vertex, the endpoints of each edge should have the same value.

So my question is how to compute the matrix whose $(i,j)$ entry is the variance of the difference between values at vertices $i$ and $j$, given a matrix representation of the graph, for example the graph Laplacian or weighted adjacency matrix.

For example, if the graph is a tree, then the $(i,j)$ entry should be proportional to the sum of edge lengths along the unique path between vertices $i$ and $j$. But when there are cycles in the graph, the variance should be lower because of the additional brownian-bridge-like constraints.

$\endgroup$
2
$\begingroup$

You're basically talking about the Gaussian Free Field on a graph. There are many recent works on that and also on similar random embedding of graphs, e.g. hypercube.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.