Poisson boundary

In mathematics, the Poisson boundary is a measure 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.
The case of the hyperbolic plane.
The Poisson formula states that given a positive harmonic function formula_1 on the unit disc formula_2 (that is, formula_3 where formula_4 is the Laplace–Beltrami operator associated to the Poincaré metric on formula_5) there exists a unique measure formula_6 on the boundary formula_7 such that the equality 
holds for all formula_10. One way to interpret this is that the functions formula_11 for formula_12 are up to scaling all the extreme points in the cone of nonnegative harmonic functions. This analytical interpretation of the set formula_13 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 formula_14 is the Markov process associated to formula_4 (i.e. the Brownian motion on the disc with the Poincaré Riemannian metric), then the process formula_16 is a continuous-time martingale, and as such converges almost everywhere to a function on the Wiener space of possible (infinite) trajectories for formula_14. Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to formula_13 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 formula_13). This interpretation of formula_13 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 formula_5. 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 formula_5.
Definition.
The Poisson boundary of a random walk on a discrete group.
Let formula_23 be a discrete group and formula_6 a probability measure on formula_23, which will be used to define a random walk formula_26 on formula_23 (a discrete-time Markov process whose transition probabilities are formula_28); the measure formula_6 is called the "step distribution" for the random walk. Let formula_30 be another measure on formula_23, which will be the initial state for the random walk. The space formula_32 of trajectories for formula_26 is endowed with the measure formula_34 (where formula_35 denotes convolution of measures). There is also an equivalence relation formula_36 on formula_32, which identifies formula_38 to formula_39 if there exists formula_40 such that formula_41 for all formula_42 (the two trajectories have the same "tail"). The "Poisson boundary" of formula_43 is then the measured space formula_44 obtained as the quotient of formula_45 by the equivalence relation formula_36.
If formula_47 is the initial distribution of a random walk with step distribution formula_6 then the measure formula_49 on formula_44 obtained as the pushforward of formula_51. It is a stationary measure for formula_43, meaning that 
It is possible to give an implicit definition of the Poisson boundary as the maximal formula_23-set with a formula_43-stationary measure formula_56, satisfying the additional condition that formula_57 almost surely weakly converges to a Dirac mass.
The Poisson formula.
Let formula_1 be a formula_6-harmonic function on formula_23, meaning that formula_61. Then the random variable formula_62 is a discrete-time martingale and so it converges almost surely. Denote by formula_63 the function on formula_44 obtained by taking the limit of the values of formula_1 along a trajectory (this is defined almost everywhere on formula_32 and shift-invariant). Let formula_67 and let formula_68 be the measure obtained by the constriction above with formula_69 (the Dirac mass at formula_70). If formula_1 is either positive or bounded then formula_63 is as well and we have the "Poisson formula":
This establishes a bijection between formula_6-harmonic bounded functions and essentially bounded measurable functions on formula_44. In particular the Poisson boundary of formula_43 is trivial, that is reduced to a point, if and only if the only bounded formula_6-harmonic functions on formula_23 are constant.
General definition.
The general setting is that of a "Markov operator" on a measured space, a notion which generalises the Markov operator formula_79 associated to a random walk. Much of the theory can be developed in this abstract and very general setting.
The Martin boundary.
Martin boundary of a discrete group.
Let formula_80 be a random walk on a discrete group. Let formula_81 be the probability to get from formula_70 to formula_83 in formula_84 steps, i.e. formula_85. The Green kernel is by definition:
If the walk is transient then this series is convergent for all formula_87. Fix a point formula_88 and define the Martin kernel by: 
formula_89. 
The embedding formula_90 has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point formula_91 is usually represented by the notation formula_92.
The Martin kernels are positive harmonic functions and every positive harmonic function can be expressed as an integral of functions on the boundary, that is for every positive harmonic function there is a measure formula_93 on formula_44 such that a Poisson-like formula holds:
The measures formula_93 are supported on the "minimal" Martin boundary, whose elements can also be characterised by being minimal. A positive harmonic function formula_97 is said to be "minimal" if for any harmonic function formula_98 with formula_99 there exists formula_100 such that formula_101.
There is actually a whole family of Martin compactifications. Define the Green generating series as
Denote by formula_103 the radius of convergence of this power series
and define for formula_104 the formula_105-Martin kernel by
formula_106.
The closure of the embedding formula_107 is called the formula_105-Martin compactification.
Martin boundary of a Riemannian manifold.
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 formula_4. In this case there is again a whole family of Martin compactifications associated to the operators formula_110 for formula_111 where formula_112 is the bottom of the spectrum. Examples where this construction can be used to define a compactification are bounded domains in the plane and symmetric spaces of non-compact type.
The relationship between Martin and Poisson boundaries.
The measure formula_113 corresponding to the constant function is called the "harmonic measure" on the Martin boundary. With this measure the Martin boundary is isomorphic to the Poisson boundary.
Examples.
Nilpotent groups.
The Poisson and Martin boundaries are trivial for symmetric random walks in nilpotent groups. 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.
Lie groups and discrete subgroups.
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. The Poisson boundary of the Brownian motion on the associated symmetric space is also the Furstenberg boundary. The full Martin boundary is also well-studied in these cases and can always be described in a geometric manner. For example, for groups of rank one (for example the isometry groups of hyperbolic spaces) the full Martin boundary is the same as the minimal Martin boundary (the situation in higher-rank groups is more complicated).
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.
Hyperbolic groups.
For random walks on a hyperbolic group, under rather weak assumptions 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. For example, the Poisson boundary of a free group is the space of ends of its Cayley tree. The identification of the full Martin boundary is more involved; in case the random walk has finite range (the step distribution is supported on a finite set) the Martin boundary coincides with the minimal Martin boundary and both coincide with the Gromov boundary.