# User:Daviddaved/On random processes

#### Random walk[edit]

- We consider the following random walk of a particle on a graph G with discrete time.

- At moment t = 0 the particle occupies a boundary vertex v of G.
- At moment t = n+1 the particle moves to a neighbor of its position at moment t = n.

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.

#### Brownian motion[edit]

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.