0

The Adler paper "On the existence of paths between points in high level excursion sets of Gaussian random fields" discusses the asymptotic limit of path probabilities in Gaussian random field excursion sets.

But they make the assumption that a path does indeed exist: "..(the implicit assumption being that T contains some path between a and b)..."

Is there a way to calculate the probability that a path exists between a and b where a and b are both in a given excursion set for a Gaussian random field?

Link to paper: https://projecteuclid.org/download/pdfview_1/euclid.aop/1395838123

Heathcliffe
  • 1
  • 1

1 Answers1

0

The assumption is that T is a (fixed) compact subset of R^d. We might have that T is two disjoint closed disks. If then a lies in the first disk and b lies in the other disk, we do not have that T contains some path between a and b. The assumption is that this is not the case - i.e. in the case with two disjoint disks, we have that a and b must be in the same disks.

Abm
  • 159
  • 3
  • So it is the probability that a and b lie within some open subset of T? I'm struggling to see how this would be calculated/computed for a real Gaussian random field though. – Heathcliffe Mar 09 '20 at 12:23
  • The a, b and subset T are fixed, and have nothing to do with the Gaussian random field assumption. As they are fixed, we cannot put a probability on the event that T contains some path between a and b - either it is true or not true. In the paper, it is implicitly assumed that T indeed contains a path between a and b. – Abm Mar 09 '20 at 13:30
  • Okay. For excursion level u>-inf T always contains a path between a and b. Are you saying that we cannot find the probability that there is a path for arbitrary u? – Heathcliffe Mar 09 '20 at 14:14