By Cox R.T., Hubbard J.C.

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H) deﬁned by ϕβp(N ),0 = 1, (1 − p−β )pβ if 0 < i ≤ N, −1 if i = 0, ⎧ −β βm ⎪ if m − 1 < i ≤ N , ⎨(1 − p )p β ϕp(N ),m (i, j) = −pβ(m−1) if i = m − 1, ⎪ ⎩ 0 if 0 ≤ i < m − 1, ϕβp(N ),1 (i, j) = (m ≥ 2). and ϕβZp ,0 = φZp , ϕβZp ,m = pβm φpm Zp − pβ(m−1) φpm−1 Zp (m ≥ 1). We call ϕβZp ,m the p-Laguerre basis, it is the analogue of the Laguerre polynomial. Note that if β = 1, the γ-measure can be written as τZ1p = φZp (x)|x|1p d∗ x/ ζp (1) = dx, where dx is the Haar measure of the additive group Qp normalized to be a probability measure by dx(Zp ) = 1.

Note that the constant function 1 is clearly harmonic. Up to a constant multiplication, this is equivalent to the equation f (x) = P (x, x )f (x ) x We denote by Harm(X) the collection of all harmonic functions. Notice that Harm(X) is convex. Namely, f0 , f1 ∈ Harm(X) =⇒ λ0 f0 + λ1 f1 ∈ Harm(X). λ0 , λ1 ≥ 0, λ0 + λ1 = 1 The set Harm(X) is also compact for the topology of pointwise convergence. If we can take λ0 , λ1 > 0, then such a function is called non-extremal and we let Harm(X)non-ext be the set of all non-extremal harmonic function; Harm(X)non-ext := λ0 f0 + λ1 f1 f0 , f1 ∈ Harm(X), λ0 , λ1 > 0, λ0 + λ1 = 1 .

Let X = n Xn , X0 = {x0 } be the state space, P : n Xn × Xn+1 → [0, 1] the transition probability. Then we have Probability measure Green kernel Martin kernel τn (x) = (P ∗ )n δx0 (x) G(x, y) = P m−n (x, y) G(x, y) . K(x, y) = G(x0 , y) (x ∈ Xn ), (x ∈ Xn , y ∈ Xm ), The Martin kernel gives a metric. The sequence {yn } is a Cauchy sequence if {K(x, yn )} is a Cauchy sequence of R for all x and {yn } ∼ {yn } if {K(x, yn )} ∼ {K(x, yn )}. Then we obtain the compactiﬁcation X = {Cauchy sequence of X}/∼ = X ∂X.