发表机构
Sungshin Women’s University; Pohang University of Science and Technology (POSTECH); Korea Institute for Advanced Study (KIAS)（成均馆大学; 浦项科技大学; 韩国高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对一类含时空白噪声的随机偏微分方程，在特定初始条件下证明其非负解的支集会瞬时收缩，为确定性抛物方程的对应现象提供了随机版本。
AI 中文摘要
我们研究随机偏微分方程 \\( \partial_t u=a(t,x)\\,\partial_x^2 u + b(t,x)\\,\partial_x u + c(t,x)\\,u +\sigma(u)\\,\xi(t,x) \\)（其中 \\( (t,x)\in(0,\infty)\times\mathbb R \\)，\\( \xi \\) 是时空白噪声，系数 \\( a,b,c \\) 可为随机，噪声系数 \\( \sigma \\) 在原点处消失且为次线性，典型情形为 \\( \sigma(u)=u^\gamma \\)，\\( \gamma\in(0,1) \\)）非负解的支集瞬时收缩问题。在唯一性律假设下，若初始数据具有足够轻的空间尾，即使初始支集非紧，每个非负解在任意正时刻都具有紧支集；初始数据也可为测度，如 Dirac 质量。当 \\( \gamma\in(0,1/2] \\) 时，有限初始质量即满足，包含超布朗运动情形 \\( \gamma=1/2 \\)；当 \\( \gamma\in(1/2,1) \\) 时，我们确定初始状态的多项式矩条件，其阶随 \\( \gamma \\) 趋近 1 而发散，量化了零附近噪声强度与初始数据衰减对瞬时收缩的权衡关系。作为独立研究步骤，我们针对非 Lipschitz 型 \\( \sigma \\) 和随机算子，建立了从测度值初始数据出发的解的弱存在性。我们的结果为确定性抛物方程（具强吸收项，吸收项作用完全由噪声承担）的 Evans 和 Knerr 所研究的瞬时收缩现象提供了随机对应。
英文摘要
We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +σ(u)\,ξ(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] 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 $γ\in(0,1)$. 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 $γ\in(0,1/2]$, finite initial mass suffices; this covers the super-Brownian case $γ=1/2$. When $γ\in(1/2,1)$, we identify a polynomial moment condition on the initial state whose order diverges as $γ\uparrow1$, 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.
Comments48 pages
