Convergence of scaled asymptotically-free self-interacting random walks to Brownian motion perturbed at extrema

2024/02/19 by Liu, Xiaoyu, Wang, Zhe
#60J85 (Secondary) #60K35 (Primary) 60F17 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2402.11828

Abstract

We consider a family of one-dimensional self interacting walks whose dynamics characterized by a monotone weight function w on ℕ∪ \0\. The weight function takes the form w(n) = (1 + 2^p Bn^-p + O(n^-1-κ))^-1, for some B ∈ ℝ , κ>0 and p∈ (0,1]. Our main model parameter is p, and for p∈ (0,1/2] we show the convergence of the SIRW to Brownian motion perturbed at extrema under the diffusive scaling. This completes the functional limit theorem in [8] for the asymptotically free case and extends the result to the full parameter range (0,1]. Our method depends on the generalized Ray-Knight theorems ([T96], [KMP23]) for the rescaled local times of this walk. The directed edge local times, described by the branching-like processes, are used to analyze the total drift experienced by the walker.

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