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Suppose the graph G has N boundary nodes then the hitting probability matrix is such that the entry h(ij) equals to the probability that the next boundary vertex that a particle starting its random walk at the boundary vertex v_i occupies is the boundary vertex v_j. The columns of the matrix H(G) add up to 1. We will derive an explicit formula for the matrix H(G) in terms of the blocks of Laplace matrix L(G) of the graph G.
The stationary distribution of a particle under Brownian motion is described by harmonic functions. It follows from the averaging property of the Laplace operator. It is conformaly invariant.
![]() | This is not a Wikipedia article: It is an individual user's work-in-progress page, and may be incomplete and/or unreliable. For guidance on developing this draft, see
Wikipedia:So you made a userspace draft. Find sources:
Google (
books ·
news ·
scholar ·
free images ·
WP refs) ·
FENS ·
JSTOR ·
TWL |
Suppose the graph G has N boundary nodes then the hitting probability matrix is such that the entry h(ij) equals to the probability that the next boundary vertex that a particle starting its random walk at the boundary vertex v_i occupies is the boundary vertex v_j. The columns of the matrix H(G) add up to 1. We will derive an explicit formula for the matrix H(G) in terms of the blocks of Laplace matrix L(G) of the graph G.
The stationary distribution of a particle under Brownian motion is described by harmonic functions. It follows from the averaging property of the Laplace operator. It is conformaly invariant.