Self-intersection local times of Volterra Gaussian bridges with large end-to-end distance

Abstract

Self-intersection local times serve as the Hamiltonian in the Gibbs measure of continuous polymer models, for which the end-to-end distance is a key quantity. We study multiple self-intersection local times of Volterra Gaussian bridges, \emph{i.e.}, Volterra Gaussian processes conditioned on their final location a, as |a|→∞. For d-dimensional (2,ζ)-locally nondeterministic processes with ζ∈(0,2/d), we prove that these local times exist in every L^p. Under two further conditions on the kernel, the p-th moment of the m-multiple self-intersection local time of the bridge is of exact order |a|^−γ for every integer pgeqslant1, with γ=p(m−1)(2−ζd)/(2−ζ). In the plane, for kernels k(t−s) with k(0)≠0 and under strong local nondeterminism, we restrict the double self-intersection local time to {t₂−t₁geqslant|a|^−α}, α>0, and establish a phase transition: its expectation decays like exp(−c|a|^2−α), converges to an explicit positive limit, or grows logarithmically, for α<2, α=2 and α>2 respectively.

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BibTeX RIS

BibTeX

@misc{izyumtseva2026selfintersection,
  title = {{Self-intersection local times of Volterra Gaussian bridges with large end-to-end distance}},
  author = {Olga Izyumtseva and Wasiur R. KhudaBukhsh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35769},
  eprint = {2610.35769},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35769},
}

RIS

TY  - UNPB
TI  - Self-intersection local times of Volterra Gaussian bridges with large end-to-end distance
AU  - Olga Izyumtseva
AU  - Wasiur R. KhudaBukhsh
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35769
UR  - https://arxiv.org/abs/2610.35769
AB  - Self-intersection local times serve as the Hamiltonian in the Gibbs measure of continuous polymer models, for which the end-to-end distance is a key quantity. We study multiple self-intersection local times of Volterra Gaussian bridges, \emph{i.e.}, Volterra Gaussian processes conditioned on their final location a, as |a|→∞. For d-dimensional (2,ζ)-locally nondeterministic processes with ζ∈(0,2/d), we prove that these local times exist in every L^p. Under two further conditions on the kernel, the p-th moment of the m-multiple self-intersection local time of the bridge is of exact order |a|^−γ for every integer pgeqslant1, with γ=p(m−1)(2−ζd)/(2−ζ). In the plane, for kernels k(t−s) with k(0)≠0 and under strong local nondeterminism, we restrict the double self-intersection local time to {t_2−t_1geqslant|a|^−α}, α>0, and establish a phase transition: its expectation decays like exp(−c|a|^2−α), converges to an explicit positive limit, or grows logarithmically, for α<2, α=2 and α>2 respectively.
ER  -

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