In
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
on the
unit disc (that is,
where
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
) there exists a unique measure
on the boundary
such that the equality
:
where
is the ''Poisson kernel'',
holds for all
. One way to interpret this is that the functions
for
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
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
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
(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
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
. 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
). This interpretation of
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
.
Definition
The Poisson boundary of a random walk on a discrete group
Let
be a discrete group and
a probability measure on
, which will be used to define a random walk
on
(a discrete-time Markov process whose transition probabilities are
); the measure
is called the ''step distribution'' for the random walk. Let
be another measure on
, which will be the initial state for the random walk. The space
of trajectories for
is endowed with a measure
whose marginales are
(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
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 ...
on
, which identifies
to
if there exists
such that
for all
(the two trajectories have the same "tail"). The ''Poisson boundary'' of
is then the measured space
obtained as the quotient of
by the equivalence relation
.
If
is the initial distribution of a random walk with step distribution
then the measure
on
obtained as the pushforward of
. It is a stationary measure for
, meaning that
:
It is possible to give an implicit definition of the Poisson boundary as the maximal
-set with a
-stationary measure
, satisfying the additional condition that
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
be a
-harmonic function on
, meaning that
. Then the random variable
is a discrete-time martingale and so it converges almost surely. Denote by
the function on
obtained by taking the limit of the values of
along a trajectory (this is defined almost everywhere on
and shift-invariant). Let
and let
be the measure obtained by the constriction above with
(the Dirac mass at
). If
is either positive or bounded then
is as well and we have the ''Poisson formula'':
:
This establishes a bijection between
-harmonic bounded functions and essentially bounded measurable functions on
. In particular the Poisson boundary of
is trivial, that is reduced to a point, if and only if the only bounded
-harmonic functions on
are constant.
General definition
The general setting is that of a ''Markov operator'' on a measured space, a notion which generalises the Markov operator
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
be a random walk on a discrete group. Let
be the probability to get from
to
in
steps, i.e.
. The Green kernel is by definition:
:
If the walk is transient then this series is convergent for all
. Fix a point
and define the Martin kernel by:
.
The embedding
has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point
is usually represented by the notation
.
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
on
such that a Poisson-like formula holds:
:
The measures
are supported on the ''minimal'' Martin boundary, whose elements can also be characterised by being minimal. A positive harmonic function
is said to be ''minimal'' if for any harmonic function
with
there exists