波动输运现象的随机基础：广义Poisson-Kac过程-第III部分动力学理论和输运的扩展和应用
Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport
摘要：
第三部分将广义Poisson-Kac（GPK）过程理论推广到非线性随机模型和连续状态。非线性有两种处理方式：（i）作为随机扰动的参数（随机速度的强度，跃迁率）对状态变量的依赖，类似于非线性朗之万方程的情况，以及（ii）作为随机微动力学运动方程对过程本身的统计描述（非线性福克-普朗克-卡茨模型）的依赖。几个数值和物理例子说明了理论。GPK理论集合了非线性和连续状态，提供了非线性玻尔兹曼方程的随机推导，为Kac的动力学计划提供了一个肯定的答案。从随机微观动力学的GPK范式的框架内的运输理论的过渡也得到了解决。
英文摘要：
This third part extends the theory of Generalized Poisson-Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker-Planck-Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac's program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.
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