Erratum to: Return Probabilities for the Reflected Random Walk on \(\mathbb {N}_0\)

Journal of Theoretical Probability, Mar 2015

Rim Essifi, Marc Peigné

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Erratum to: Return Probabilities for the Reflected Random Walk on \(\mathbb {N}_0\)

Erratum to: Return Probabilities for the Reflected Random Walk on N0 Rim Essifi Marc Peign In the original publication of this paper, we fix a constant K > 1 and consider the set K(K) of functions K : Z R+ satisfying the following conditions: Unfortunately, the two first conditions readily imply K = 1 since the operator R is markovian, so that the three above conditions cannot be satisfied simultaneously. In fact, we will simply consider the function K : s Kx . The only one reason for the condition RK (x ) 1 appeared in the proof of Fact 4.4.1, where the peripherical spectrum of the operators Rs for |s| = 1 and s = 1 is controlled. With this new choice of function K , one gets rFaadcitus4.o4f.1RFsoorn|s(|C=0N, 1 anKd )sis=<11o.ne gets Rs K < 1; in particular, the spectral Proof We could adapt the proof proposed in the paper and show that Rs2n for some s < 1 when s = 1. We propose here another simpler argument. - x N0 K (x ) 1, RK (x ) 1 and K (x ) Kx . comRpeaccatllfrthoamt R(CsNa,c|ts|fro)min(toC(NC,N| |K ) into (CN , | |) and that the identity map is , | |K ). Consequently, the operator RK is compact on (CN , | |K ) with spectral radius 1 since it has bounded powers. Let us fix s C\{1} with modulus 1 and assume that Rs has spectral radius 1 on (CN , | |K ); since it is compact, there exists a sequence a = (ax )xZ = 0 and R such that Rs a = ei a, i.e. for all x Z : yZ for all x Z : yZ Acknowledgments We thank here S. Gouezel, who pointed out to us the contradiction of the three conditions (1) and the fact that the Rs are compact on (C, | |K ).


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Rim Essifi, Marc Peigné. Erratum to: Return Probabilities for the Reflected Random Walk on \(\mathbb {N}_0\), Journal of Theoretical Probability, 2015, 1250-1251, DOI: 10.1007/s10959-014-0568-6