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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poisson boundary is a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
associated to a
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
. It is an object designed to encode the
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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 f on the unit disc \mathbb D = \ (that is, \Delta f = 0 where \Delta is the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
associated to the Poincaré metric on \mathbb D) there exists a unique measure \mu on the boundary \partial \mathbb D = \ such that the equality : f(z) = \int_ K(z, \xi) \, d\mu(\xi) where K(z, \xi) = \frac is the ''Poisson kernel'', holds for all z \in \mathbb D. One way to interpret this is that the functions K(\cdot, \xi) for \xi \in \partial \mathbb D are up to scaling all the
extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...
s in the cone of nonnegative harmonic functions. This analytical interpretation of the set \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 is the
Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
associated to \Delta (i.e. the Brownian motion on the disc with the Poincaré Riemannian metric), then the process f(W_t) is a continuous-time martingale, and as such converges almost everywhere to a function on the
Wiener space In mathematics, classical Wiener space is the collection of all Continuous_function#Continuous_functions_between_metric_spaces, continuous functions on a given domain of a function, domain (usually a subinterval of the real line), taking values i ...
of possible (infinite) trajectories for W_t. Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to \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 \partial \mathbb D). This interpretation of \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 \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 \mathbb D.


Definition


The Poisson boundary of a random walk on a discrete group

Let G be a discrete group and \mu a probability measure on G, which will be used to define a random walk X_t on G (a discrete-time Markov process whose transition probabilities are p(x, y) = \mu(xy^)); the measure \mu is called the ''step distribution'' for the random walk. Let m be another measure on G, which will be the initial state for the random walk. The space G^ of trajectories for X_t is endowed with a measure \mathbb P_m whose marginales are m*\mu^ (where * denotes
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of measures; this is the distribution of the random walk after n steps). There is also an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
\sim on G^, which identifies (x_t) to (y_t) if there exists n, m \in \mathbb N such that x_ = y_ for all t \ge 0 (the two trajectories have the same "tail"). The ''Poisson boundary'' of (G, \mu) is then the measured space \Gamma obtained as the quotient of (G^, \mathbb P_m) by the equivalence relation \sim. If \theta is the initial distribution of a random walk with step distribution \mu then the measure \nu_\theta on \Gamma obtained as the pushforward of \mathbb P_\theta. It is a stationary measure for (G, \mu), meaning that :\int_G \nu(g^A) \mu(g) = \nu_\theta(A). It is possible to give an implicit definition of the Poisson boundary as the maximal G-set with a (G, \mu)-stationary measure \nu, satisfying the additional condition that (X_t)_*\nu almost surely weakly converges to a
Dirac mass In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
.


The Poisson formula

Let f be a \mu-harmonic function on G, meaning that \sum_ f(hg)\mu(h) = f(g). Then the random variable f(X_t) is a discrete-time martingale and so it converges almost surely. Denote by \hat f the function on \Gamma obtained by taking the limit of the values of f along a trajectory (this is defined almost everywhere on G^ and shift-invariant). Let x \in G and let \nu_x be the measure obtained by the constriction above with \theta = \delta_x (the Dirac mass at x). If f is either positive or bounded then \hat f is as well and we have the ''Poisson formula'': : f(x) = \int_\Gamma \hat f(\gamma) \, d\nu_x(\gamma). This establishes a bijection between \mu-harmonic bounded functions and essentially bounded measurable functions on \Gamma. In particular the Poisson boundary of (G, \mu) is trivial, that is reduced to a point, if and only if the only bounded \mu-harmonic functions on G are constant.


General definition

The general setting is that of a ''Markov operator'' on a measured space, a notion which generalises the Markov operator f \mapsto \mu*f 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 G, \mu be a random walk on a discrete group. Let p_n(x, y) be the probability to get from x to y in n steps, i.e. \mu^(x^y). The Green kernel is by definition: : \mathcal G(x, y) = \sum_ p_n(x, y). If the walk is transient then this series is convergent for all x, y. Fix a point o \in G and define the Martin kernel by: \mathcal K_o(x, y) = \frac. The embedding y \mapsto \mathcal K_o( \cdot, y) has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point \gamma \in \Gamma is usually represented by the notation \mathcal K(\cdot, \gamma). 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 \nu_ on \Gamma such that a Poisson-like formula holds: : f(x) = \int \mathcal K_o(x, \gamma) \, d\nu_(\gamma). The measures \nu_ are supported on the ''minimal'' Martin boundary, whose elements can also be characterised by being minimal. A positive harmonic function u is said to be ''minimal'' if for any harmonic function v with 0 \le v \le u there exists c \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> such that v = cu. There is actually a whole family of Martin compactifications. Define the Green generating series as :\mathcal G_r(x, y) = \sum_ p_n(x, y)r^n. Denote by R the radius of convergence of this
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
and define for 1\leq r \leq R the r-Martin kernel by \mathcal K_(x, y) = \frac. The closure of the embedding y \mapsto \mathcal K_( \cdot, y) is called the r-Martin compactification.


Martin boundary of a Riemannian manifold

For a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
the Martin boundary is constructed, when it exists, in the same way as above, using the Green function of the Laplace–Beltrami operator \Delta. In this case there is again a whole family of Martin compactifications associated to the operators \Delta + \lambda for 0\le \lambda \le \lambda_0 where \lambda_0 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 \nu_ 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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
(with step distribution absolutely continuous with respect to the Haar measure) the Poisson boundary is equal to the
Furstenberg boundary In potential theory, a discipline within applied mathematics, the Furstenberg boundary is a notion of boundary associated with a group. It is named for Harry Furstenberg, who introduced it in a series of papers beginning in 1963 (in the case of se ...
. 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 In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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 In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
, 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 End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) **End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games **End (gridiron football) ...
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.


Notes


References

* * * * *{{cite news , mr=1815698 , last=Kaimanovich , first=Vadim A. , title=The Poisson formula for groups with hyperbolic properties. , journal=Ann. of Math. , series= 2 , volume=152 , year=2000 , pages=659–692 Harmonic analysis Stochastic processes Compactification (mathematics)