具有回火稳定噪声的随机系统非局部Fokker-Planck方程的动力学行为
Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise
摘要：
通过研究随机动力系统的概率密度演化，刻画了具有回火稳定噪声的随机动力系统的特征.该密度函数满足非局部福克-普朗克方程。首先，我们证明了一个叠加原理，即该非局部Fokker-Planck方程的概率测度值解等价于由逆随机流组成的鞅解.这个结果与Schauder估计一起得到了非局部Fokker-Planck方程强解的存在唯一性.其次，我们设计了一个收敛的有限差分法，通过求解非局部的Fokker-Planck方程来模拟密度函数。最后，我们将上述理论和数值结果应用于非线性滤波系统，通过模拟非局部Zakai方程。
英文摘要：
We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.
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