AI 中文总结
本文提出随机局部化能量方法，将随机多孔介质方程的支撑传播分解为多孔介质演化与随机输运两部分，证明了有限传播速度及等待时间现象的充要条件。
AI 中文摘要
我们发展了一种随机局部化能量方法，用于推导具有线性保守噪声的随机多孔介质方程支撑传播的定性结果。与以往工作中通过空间凸起函数加权来局部化能量的方法不同，我们使用随机输运方程的解作为权重函数。这调制了由保守噪声项引起的支撑传播，使得在此设置中原本失效的能量论证再次适用。由此，我们证明了有限传播速度以及等待时间现象存在的充分必要条件（模随机输运）。这些方法和结果表明，在短时间尺度上，支撑传播可以分解为两个独立的部分：一部分由多孔介质方程的演化引起，另一部分由随机输运引起。
英文摘要
We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments---which otherwise fail in this setting---are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.
Comments28 pages, keywords: Porous media equation, conservative noise, qualitative properties, support propagation, waiting time phenomena, stochastic flows, stochastic transport equation
