Comparison principles for stochastic reaction-diffusion equations on metric measure spaces

Abstract

We study parabolic stochastic partial differential equations on metric measure spaces (𝕏, d,m) of the form ∂_t u(t,x) = ℒ^* u(t,x) + b(t,x,u(t,x)) + σ(t,x,u(t,x)) Ẇ(t,x), t>0, x ∈ 𝕏, where ℒ is the generator of a Markov process which possesses transition densities, and Ẇ is a Gaussian noise that is white in time and possibly with spatial correlation. We assume the coefficients b and σ are Lipschitz and satisfy the linear growth condition. We formulate general and checkable assumptions on (𝕏,ℒ,Ẇ) that ensure existence and uniqueness of probabilistically strong, continuous, tempered mild solutions. We then prove comparison principles (including a strong comparison principle) and strict positivity, relative to initial conditions. Our framework allows non-symmetric heat kernels and includes diffusion-type and stable-type examples, such as metric graphs and fractal spaces with sub-Gaussian heat kernel estimates.

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BibTeX RIS

BibTeX

@misc{fan2026comparison,
  title = {{Comparison principles for stochastic reaction-diffusion equations on metric measure spaces}},
  author = {Louis Wai-Tong Fan and Zhenyao Sun and Johnny and Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07790},
  eprint = {2610.07790},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07790},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Comparison principles for stochastic reaction-diffusion equations on metric measure spaces
AU  - Louis Wai-Tong Fan
AU  - Zhenyao Sun
AU  - Johnny
AU  - Yang
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07790
UR  - https://arxiv.org/abs/2610.07790
AB  - We study parabolic stochastic partial differential equations on metric measure spaces (𝕏, d,m) of the form ∂_t u(t,x) = ℒ^* u(t,x) + b(t,x,u(t,x)) + σ(t,x,u(t,x)) Ẇ(t,x), t>0, x ∈ 𝕏, where ℒ is the generator of a Markov process which possesses transition densities, and Ẇ is a Gaussian noise that is white in time and possibly with spatial correlation. We assume the coefficients b and σ are Lipschitz and satisfy the linear growth condition. We formulate general and checkable assumptions on (𝕏,ℒ,Ẇ) that ensure existence and uniqueness of probabilistically strong, continuous, tempered mild solutions. We then prove comparison principles (including a strong comparison principle) and strict positivity, relative to initial conditions. Our framework allows non-symmetric heat kernels and includes diffusion-type and stable-type examples, such as metric graphs and fractal spaces with sub-Gaussian heat kernel estimates.
ER  -

Abstract

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