论文标题
具有可变顺序的分数方程的CTRW近似值
CTRW approximations for fractional equations with variable order
论文作者
论文摘要
已知标准扩散过程是作为适当随机步行的限制获得的。但是，这些预定的随机步行可能会大不相同。扩散系数可以使跳跃的大小或跳跃强度负责。 “粗糙”扩散极限不会感觉到差异。情况会改变，如果我们通过CTRW使用非指数等待时间建模跳跃型近似值。如果我们使扩散系数负责跳跃的大小，并从$α$稳定的法律的吸引域中取出等待时间，那么标准缩放将导致小跳跃的极限，并且大强度限制到最标准的分数扩散方程。但是，如果我们选择具有固定跳跃尺寸的CTRW近似值，并使用扩散系数区分不同点处的强度，那么我们在极限方程中获得具有可变位置依赖性分数衍生物的方程。在本文中，我们严格地构建了这些近似值，并证明了它们与多维扩散和更一般的Feller过程的相应分数方程的收敛性。
The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an $α$-stable law with a constant intensity $\al$, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this paper we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes.
