In mathematics, the Poisson boundary is a probability space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary, it is in general a purely measure-theoretical object and not a boundary in the topological sense. However, in the case where the random walk is on a topological space, the Poisson boundary can be related to the Martin boundary, which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to harmonic functions on the space via generalisations of the Poisson formula. A related construction is the Furstenberg boundary of a semisimple Lie group.
The case of the hyperbolic plane The Poisson formula states that given a positive harmonic function f {\displaystyle f} on the unit disc D = { z ∈ C : | z | < 1 } {\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}} (that is, Δ f = 0 {\displaystyle \Delta f=0} where Δ {\displaystyle \Delta } is the Laplace–Beltrami operator associated to the Poincaré metric on D {\displaystyle \mathbb {D} } ) there exists a unique measure μ {\displaystyle \mu } on the boundary ∂ D = { z ∈ C : | z | = 1 } {\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} :|z|=1\}} such that the equality
f ( z ) = ∫ ∂ D K ( z , ξ ) d μ ( ξ ) {\displaystyle f(z)=\int _{\partial \mathbb {D} }K(z,\xi )\,d\mu (\xi )} where K ( z , ξ ) = 1 − | z | 2 | ξ − z | 2 {\displaystyle K(z,\xi )={\frac {1-|z|^{2}}{|\xi -z|^{2}}}} is the Poisson kernel, holds for all z ∈ D {\displaystyle z\in \mathbb {D} } . One way to interpret this is that the functions K ( ⋅ , ξ ) {\displaystyle K(\cdot ,\xi )} for ξ ∈ ∂ D {\displaystyle \xi \in \partial \mathbb {D} } are up to scaling all the extreme points in the cone of nonnegative harmonic functions. This analytical interpretation of the set ∂ D {\displaystyle \partial \mathbb {D} } leads to the more general notion of minimal Martin boundary (which in this case is the full Martin boundary). This fact can also be interpreted in a probabilistic manner. If W t {\displaystyle W_{t}} is the Brownian motion on D {\displaystyle \mathbb {D} } with the Poincaré Riemannian metric, then the process f ( W t ) {\displaystyle f(W_{t})} is a continuous-time martingale, and as such converges almost everywhere to a function on the Wiener space of possible (infinite) trajectories for W t {\displaystyle W_{t}} . Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to ∂ D {\displaystyle \partial \mathbb {D} } endowed with the class of Lebesgue measure (note that this identification can be made directly since a path in Wiener space converges almost surely to a point on ∂ D {\displaystyle \partial \mathbb {D} } ). This interpretation of ∂ D {\displaystyle \partial \mathbb {D} } as the space of trajectories for a Markov process is a special case of the construction of the Poisson boundary. Finally, the constructions above can be discretised, i.e. restricted to the random walks on the orbits of a Fuchsian group acting on D {\displaystyle \mathbb {D} } . This gives an identification of the extremal positive harmonic functions on the group, and to the space of trajectories of the random walk on the group (both with respect to a given probability measure), with the topological/measured space D {\displaystyle \mathbb {D} } .
Definition
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