Consider two-dimensional brownian motion, but in a maze, such that there are "walls" which prevent the path from taking certain steps (based on this tweet).
I'm curious about algorithms to efficiently sample from two different distributions (for a particular maze):
- The distribution of the path of the first particle to reach the end, out of a population of N particles.
- The distribution of paths which reach the exit at exactly time $t$. This is like a two dimensional Brownian bridge, but with extra constraints
Of course since it's impossible to represent an actual Brownian motion, an algorithm that converges to the samples from these distributions would be fine.
Is it possible to do better than actually simulating N particles for the first one? For the second, how can one sample from the distribution at all?