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PreprintCC BY 4.0arXiv

Intermittency for the parabolic Anderson model on spaces with sub-Gaussian heat kernel estimates

  • Fabrice Baudoin
  • Hongyi Chen
  • Li Chen
  • Søren Kjærsgaard
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Abstract

We study the parabolic Anderson model ∂t u=Δu+βu Ẇα on an unbounded Ahlfors dh-regular Dirichlet metric measure space whose heat kernel satisfies two-sided sub-Gaussian estimates with walk dimension dw. The noise is white in time and, in space, either white (α=0) or colored through the Riesz kernel of (−Δ)−2α, where 0≤ 2α<dh/dw<1+2α. No recurrence is assumed. We prove existence, uniqueness and Hölder regularity of the mild Skorohod solution for bounded initial conditions. For nonzero nonnegative bounded initial conditions, the solution is positive and its moment Lyapunov exponents exist for every real p≥1, independently of the spatial point and the initial condition. For p≥2, these exponents have order p (β2(p−1))dw/((1+2α)dw−dh), with upper and lower bounds matching up to constants.

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Cite

BibTeXRIS
BibTeX
@misc{baudoin2026intermittency,
  title = {{Intermittency for the parabolic Anderson model on spaces with sub-Gaussian heat kernel estimates}},
  author = {Fabrice Baudoin and Hongyi Chen and Li Chen and Søren Kjærsgaard},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20421},
  eprint = {2610.20421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20421},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Intermittency for the parabolic Anderson model on spaces with sub-Gaussian heat kernel estimates
AU  - Fabrice Baudoin
AU  - Hongyi Chen
AU  - Li Chen
AU  - Søren Kjærsgaard
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20421
UR  - https://arxiv.org/abs/2610.20421
AB  - We study the parabolic Anderson model ∂_t u=Δu+βu Ẇ_α on an unbounded Ahlfors d_h-regular Dirichlet metric measure space whose heat kernel satisfies two-sided sub-Gaussian estimates with walk dimension d_w. The noise is white in time and, in space, either white (α=0) or colored through the Riesz kernel of (−Δ)^−2α, where 0≤ 2α<d_h/d_w<1+2α. No recurrence is assumed. We prove existence, uniqueness and Hölder regularity of the mild Skorohod solution for bounded initial conditions. For nonzero nonnegative bounded initial conditions, the solution is positive and its moment Lyapunov exponents exist for every real p≥1, independently of the spatial point and the initial condition. For p≥2, these exponents have order p (β^2(p−1))^d_w/((1+2α)d_w−d_h), with upper and lower bounds matching up to constants.
ER  -