ANISOTROPIC HARMONIC ANALYSIS ON TREES 7

Note that if x G G, then d(x, e) (with respect to the Cayley graph) is equal

to the length of x as a reduced word in the generators. In general we have

d(x, y) =

\x~1y\1

where | • | denotes the length of a reduced word.

We introduce here some notation associated with G. We shall be interested

in the function spaces

0(G) = {/: G - C; ||/||p = ( £ |/(x)|") ' oo}. .

x€G

The group G acts on these spaces according to (x • f)(y) =

f(x~ly)

and this

representation is called the {left) regular representation. When

£P(G)

=

£2(G),

this representation is unitary. If 6X G

^X(G),

is given by

then x • / = 6X* f. Sometimes we write 6xy with the same meaning as Sx(y).

Finally

(,l(G)

is a convolution algebra with 6e as the identity. All the above

applies to any discrete group.

A random walk on G is defined by giving an infinite matrix R = (R(y, x))y,xeG

with nonnegative coefficients, satisfying ^ZyR(y^x) = 1. The random walk is

called symmetric if R(x, y) = R(y, x) for all x and y.

The number R(y, x) represents the probability of taking a step from x to y.

The matrix R = (R(y,x)) is called the transition matrix of the random walk.

Observe that the conditions R(y, x) 0 and ]Ty R(y, x) = 1 imply that we can

define the powers of R as infinite matrices

Rn

=

(Rn(y,x)),

where

Rn(y,x)

=

^2tRn~1(y,t)R(t,x). Induction shows that

]"(*,, x) =

J ^ J T * - 1

(»,*)*(*,*) =

I

.

y y t

The number Rn(y, x) represents the probability of going from x to y in n steps.

The conditions R(y, x) 0 and ^ R(y, x) — 1 imply that the matrix R

defines a bounded linear operator on ^{G); (Rf)(y) = ^2X R{y'»x)

jT(x),

and

P/III^EE^^I/^IHI/III-

y£Gx£G

We say that the random walk R is G-invariant if R{ty,tx) = R(y,x) for all

x,y,t G G. A G-invariant random walk may also be invariant for other isometries

of the tree. An important special case is that of a random walk which is invariant

with respect to any isometry. In the latter case R(y, x) will depend only on ^he

distance between x and y. Such a random walk is called isotropic.

From now on we consider only G-invariant random walks. Define a probability

measure \i on G by \±{x) — R(x,e). Since R is G-invariant, R(y,x) =

ji(x~ly),

and Rf(y) = ^2X

fi(x~1y)f(x)

= (/ • fi)(y). Moreover if

/in

denotes the n-th

convolution power of /x, then the coefficients of Rn are jin{x~ly). A G-invariant