Poisson 噪声驱动的反应扩散系统的 Dean-Kawasaki 方程
A Dean-Kawasaki equation for reaction diffusion systems driven by Poisson noise
AI总结：
本文推导含随机 Poisson 场的 Dean-Kawasaki 型随机偏微分方程，描述 N 粒子一元反应扩散系统的涨落，并基于弱相互作用建立其与 McKean-Vlasov 扩散及强相互作用情形的联系。
摘要：
我们推导了一个随机偏微分方程，用于描述由 N 个粒子组成、经历 Markov 型一元反应的反应扩散系统的涨落行为。该工作通过引入随机 Poisson 场，推广了 Dean [J. Phys. A: Math. and Gen., 29 (24), L613, (1996)] 的研究。我们的方法基于弱相互作用，具有双重优势：所得方程在 N 趋于无穷大的极限下渐近收敛于一种 McKean-Vlasov 扩散的变体，同时仍可通过平凡的尺度变换与类 Dean 强相互作用情形相联系。文中给出了多个示例，并讨论了向更复杂反应机制扩展的可能性。
英文摘要：
We derive a stochastic partial differential equation that describes the fluctuating behaviour of reaction-diffusion systems of N particles, undergoing Markovian, unary reactions. This generalises the work of Dean [J. Phys. A: Math. and Gen., 29 (24), L613, (1996)] through the inclusion of random Poisson fields. Our approach is based on weak interactions, which has the dual benefit that the resulting equations asymptotically converge (in the N to infinity limit) on a variation of a McKean- Vlasov diffusion, whilst still being related to the case of Dean-like strong interactions via a trivial rescaling. Various examples are presented, alongside a discussion of possible extensions to more complicated reaction schemes.
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