MULTI-SCALING LIMITS FOR RELATIVISTIC DIFFUSION EQUATIONS WITH RANDOM INITIAL DATA

作者：Liu Gi Ren^*; Shieh Narn Rueih

来源：Transactions of the American Mathematical Society, 2015, 367(5): 3423-3446.

• Large-scale limits
• small-scale limits
• relativistic diffusion equations
• random initial data
• multiple Ito-Wiener integrals
• subordinated Gaussian fields
• Hermite ranks

Let u(t, x), t > 0, x is an element of R-n, be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatial-fractional parameter alpha is an element of (0, 2) and the mass parameter m > 0, subject to a random initial condition u(0, x) which is characterized as a subordinated Gaussian field. In this article, we study the large-scale and the small-scale limits for the suitable space-time re-scalings of the solution field u(t, x). Both the Gaussian and the non-Gaussian limit theorems are discussed. The small-scale scaling involves not only scaling on u(t, x) but also re-scaling the initial data; this is a new type result for the literature. Moreover, in the two scalings the parameter alpha is an element of (0, 2) and the parameter m > 0 play distinct roles for the scaling and the limiting procedures.

• 出版日期2015-5