Instantaneous shrinking of supports for stochastic PDEs
Abstract
We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation ∂t u=a(t,x) ∂x2 u + b(t,x) ∂x u + c(t,x) u +σ(u) ξ(t,x), (t,x)∈(0,∞)×ℝ, where ξ is space-time white noise, the coefficients a, b, c may be random, and the noise coefficient σ vanishes at the origin and is sublinear there. The model case is σ(u)=uγ with γ∈(0,1). We show that, under a uniqueness-in-law assumption, if the initial datum has a sufficiently light spatial tail, then every nonnegative solution has compact support at every positive time, even though the initial support is not compact. The initial datum may also be a measure, such as a Dirac mass. When γ∈(0,1/2], finite initial mass suffices; this covers the super-Brownian case γ=1/2. When γ∈(1/2,1), we identify a polynomial moment condition on the initial state whose order diverges as γ↑1, quantifying the trade-off between the strength of the noise near zero and the decay of the initial data required for instantaneous shrinking. As a step of independent interest, we establish weak existence of solutions started from measure-valued initial data for non-Lipschitz σ and random operators. Our results provide a stochastic counterpart of the instantaneous shrinking phenomenon of Evans and Knerr for deterministic parabolic equations with strong absorption, in which the role of the absorption term is played entirely by the noise.
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BibTeX
@misc{han2026instantaneous,
title = {{Instantaneous shrinking of supports for stochastic PDEs}},
author = {Beom-Seok Han and Kunwoo Kim and Jaeyun Yi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01984},
eprint = {2609.01984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01984},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Instantaneous shrinking of supports for stochastic PDEs
AU - Beom-Seok Han
AU - Kunwoo Kim
AU - Jaeyun Yi
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01984
UR - https://arxiv.org/abs/2609.01984
AB - We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation ∂_t u=a(t,x) ∂_x^2 u + b(t,x) ∂_x u + c(t,x) u +σ(u) ξ(t,x), (t,x)∈(0,∞)×ℝ, where ξ is space-time white noise, the coefficients a, b, c may be random, and the noise coefficient σ vanishes at the origin and is sublinear there. The model case is σ(u)=u^γ with γ∈(0,1). We show that, under a uniqueness-in-law assumption, if the initial datum has a sufficiently light spatial tail, then every nonnegative solution has compact support at every positive time, even though the initial support is not compact. The initial datum may also be a measure, such as a Dirac mass. When γ∈(0,1/2], finite initial mass suffices; this covers the super-Brownian case γ=1/2. When γ∈(1/2,1), we identify a polynomial moment condition on the initial state whose order diverges as γ↑1, quantifying the trade-off between the strength of the noise near zero and the decay of the initial data required for instantaneous shrinking. As a step of independent interest, we establish weak existence of solutions started from measure-valued initial data for non-Lipschitz σ and random operators. Our results provide a stochastic counterpart of the instantaneous shrinking phenomenon of Evans and Knerr for deterministic parabolic equations with strong absorption, in which the role of the absorption term is played entirely by the noise.
ER -