Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(V(u),\ u\in {\cal{T}})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>u</mi> <mo>∈</mo> <mrow> <mrow> <mi mathvariant="script">T</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> be a (supercritical) branching random walk and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\eta_{u},\ u\in {\cal{T}})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <msub> <mi>η</mi> <mrow> <mi>u</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>∈</mo> <mrow> <mrow> <mi mathvariant="script">T</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> be marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)], for each ray ξ of the tree, we associate the discounted tree sum D(ξ) which is the sum of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\rm{e}^{-V(u)}\eta_{u}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mi>η</mi> <mrow> <mi>u</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> taken along the ray. The paper deals with the finiteness of supξD(ξ). To this end, we study the extreme behaviour of the local time processes of the paths (V(u), u ∈ ξ). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)]. We also present several open questions.

Boundedness of Discounted Tree Sums

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发表日期：

2026年6月15日

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Elie Aïdékon,
Yueyun Hu,
Zhan Shi

摘要

Let $(V(u),\ u\in {\cal{T}})$ $(V (u), u \in T)$ be a (supercritical) branching random walk and $(\eta_{u},\ u\in {\cal{T}})$ $(η_{u}, u \in T)$ be marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)], for each ray ξ of the tree, we associate the discounted tree sum D(ξ) which is the sum of the $\rm{e}^{-V(u)}\eta_{u}$ $e^{- V (u)} η_{u}$ taken along the ray. The paper deals with the finiteness of sup_ξD(ξ). To this end, we study the extreme behaviour of the local time processes of the paths (V(u), u ∈ ξ). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)]. We also present several open questions.