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.
Both boundaries are related to harmonic functions on the space via generalisations of the Poisson formula.
The Poisson formula states that given a positive harmonic function
are up to scaling all the extreme points in the cone of nonnegative harmonic functions.
(i.e. the Brownian motion on the disc with the Poincaré Riemannian metric), then the process
is a continuous-time martingale, and as such converges almost everywhere to a function on the Wiener space of possible (infinite) trajectories for
Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to
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
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
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
(a discrete-time Markov process whose transition probabilities are
denotes convolution of measures; this is the distribution of the random walk after
almost surely weakly converges to a Dirac mass.
Much of the theory can be developed in this abstract and very general setting.
The Green kernel is by definition: If the walk is transient then this series is convergent for all
Define the Green generating series as Denote by
the radius of convergence of this power series and define for
For a Riemannian manifold the Martin boundary is constructed, when it exists, in the same way as above, using the Green function of the Laplace–Beltrami operator
In this case there is again a whole family of Martin compactifications associated to the operators
Examples where this construction can be used to define a compactification are bounded domains in the plane and symmetric spaces of non-compact type.
corresponding to the constant function is called the harmonic measure on the Martin boundary.
The Poisson and Martin boundaries are trivial for symmetric random walks on nilpotent groups.
[5] On the other hand, when the random walk is non-centered, the study of the full Martin boundary, including the minimal functions, is far less conclusive.
For random walks on a semisimple Lie group (with step distribution absolutely continuous with respect to the Haar measure) the Poisson boundary is equal to the Furstenberg boundary.
[7] The full Martin boundary is also well-studied in these cases and can always be described in a geometric manner.
[8] The Poisson boundary of a Zariski-dense subgroup of a semisimple Lie group, for example a lattice, is also equal to the Furstenberg boundary of the group.
[9] For random walks on a hyperbolic group, under the finite entropy assumption on the step distribution which always hold for a simple walk (a more general condition is that the first moment be finite) the Poisson boundary is always equal to the Gromov boundary when equipped with the hitting probability measure.
For example, the Poisson boundary of a free group is the space of ends of its Cayley tree.