Persistent occupation and critical marked sums on branching random walks

Abstract

We study two path functionals of a branching random walk at an exponential normalization. For centered integer displacements bounded above by one, we construct on survival one ray whose local time at every large spatial level is bounded below by a positive multiple of the square of that level. For displacements in $\{-1,0,1\}$, overlapping forward profiles give a single ray with lower coefficient $4t/(σ^2j_{1,1}^2)$, where $σ^2$ is the tilted variance and $j_{1,1}$ the first positive zero of $J_1$. The profile energy quotient is optimized exactly. For discounted marked path sums with real displacements, we prove critical-moment sufficiency without a drift assumption, and necessity under a finite ascending-ladder mean and a weighted sibling moment. At positive tilted drift, a stopping-line argument replaces the sibling condition by an $L\log L$ hypothesis. A sparse mark construction identifies an exact universality obstruction in the constant-displacement subclass. At negative tilted drift, weighted powers give explicit sufficient logarithmic-tail conditions for every exponent $t>0$ and examples excluding a universal critical-moment converse. Under our sufficient conditions, path sums converge uniformly and attain their maxima. The exact optimal persistent local-time constant and a general negative-drift converse remain open.