Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations
Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations
复制标题
速度跳跃过程和非局部 Hamilton-Jacobi 方程的大偏差
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Grégoire Nadin
中科院分区：
文献类型：
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作者：
E. Bouin;V. Calvez;E. Grenier;Grégoire Nadin
我们建立了一个速度跳跃过程的大偏差理论，其中新的随机速度从高斯分布中以恒定的速率选取。与此过程相关的Kolmogorov前向方程是一个线性动力学输运方程，其中BGK算子考虑了速度的变化。我们分析了它的渐近极限后，一个适当的重新标度兼容的WKB展开。这就产生了一种新的汉密尔顿雅可比方程，它是关于速度变量的非局部的。我们引入了一个专门的概念的粘度解决方案的极限问题，我们证明了适定性的粘度意义。的基本解是明确计算，产生定量估计的大偏差的基本速度跳跃过程一拉Freidlin-Wentzell。作为这一理论的应用，我们推测了某些非线性动力学反应-迁移方程的精确加速率。
We establish a large deviation theory for a velocity jump process, where new random velocities are picked at a constant rate from a Gaussian distribution. The Kolmogorov forward equation associated with this process is a linear kinetic transport equation in which the BGK operator accounts for the changes in velocity. We analyse its asymptotic limit after a suitable rescaling compatible with the WKB expansion. This yields a new type of Hamilton Jacobi equation which is non local with respect to velocity variable. We introduce a dedicated notion of viscosity solution for the limit problem, and we prove well-posedness in the viscosity sense. The fundamental solution is explicitly computed, yielding quantitative estimates for the large deviations of the underlying velocity-jump process a la Freidlin-Wentzell. As an application of this theory, we conjecture exact rates of acceleration in some nonlinear kinetic reaction-transport equations.