In the above mentionned paper, we fixed a constant K &gt

Return Probabilities for the Reflected Random Walk onN0

ERRATUM

by Rim Essifi & Marc Peign´e.

In the above mentionned paper, we fixed a constant K > 1 and consider the set K(K) of functionsK :Z→R⁺satisfying the following conditions

∀x∈N0 K(x)≥1, RK(x)≤1 and K(x)∼K^x. (1) Unfortunately, the two first conditions readily implyK = 1 since the operatorRis markovian , so that the 3 above conditions cannot be satisfied simultaneously.

In fact, we will simply consider the functionK :s7→K^x. The only one reason of the condition RK(x) ≤1 appeared in the proof of Fact 4.4.1, where the peripherical spectrum of the operators R_s for|s|= 1 and s6= 1 is controlled. With this new choice of function K, one gets the

Fact 4.4.1-For |s|= 1 and s6= 1 one gets kR_sk_K<1; in particular, the spectral radius ofR_s on (C^N0,k · k_K) is <1.

Proof. We could adapt the proof proposed in the paper and show that kR²ⁿ_s k_K ≤ Cρⁿ_s for some ρs<1 whens6= 1. We propose here another simpler argument.

Recall thatR_s acts from (C^N,| · |_K) into (C^N,| · |_∞) and that the identity map is compact from (C^N,| · |_∞) into (C^N,| · |_K). Consequently, the operatorR_K is compact on (C^N,| · |_K) with spectral radius≤1 since it has bounded powers.

Let us fix s∈C\ {1}with modulus 1 and assume thatR_s has spectral radius 1 on (C^N,| · |_K);

since it is compact, there exists a sequencea= (a_x)x∈Z6= 0 and θ∈Rsuch that R_sa=e^iθa, i.e.

∀x∈Z

X

y∈Z

R_s(x, y)a_y =e^iθa_x. (2)

It follows that|a_y|=|a₀| 6= 0 for any y ∈Zsince X

y∈Z

|R_s(x, y)| ≤X

y∈Z

R(x, y) = 1; without loss of generality, we may assume|a_y|= 1, i.e. ay =e^iαy for someαy ∈R. The equality (2) may be thus rewritten

∀x∈Z

X

y∈Z

R_s(x, y)e^iαy =e^iθe^iαx.

By convexity, using again the inequality X

y∈Z

|R_s(x, y)| ≤ X

y∈Z

R(x, y) = 1, one readily gets e^iαy = e^iθe^iαx for any x, y ∈ Z; consequently e^iθ = 1, the sequence a is constant and R_s(x, y) =R(x, y) for any x, y∈Z, which implies in particulers= 1. Contradiction.

Acknowledgments- We thank here S. Gouezel who pointed out to us the contradiction of the 3 conditions (1) and the fact that theR_s are compact on (C,| · |_K).

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