臺灣大學數學系專題演講博士生論文口試

演講者：劉聚仁
講  題：Partial Differential Equations with Random Initial Data
時  間：2013年6月7日 (星期五) 10:20
地  點：臺灣大學天數館201室
摘  要：
Given a time-dependent partial differential equation, the evolution of the solution is determined by its initial data. Due to errors caused by measurements, we cannot identify the one that will actually materialize in an experiment; in general, we are only able to say that some functions in the set are more likely to be observed than some others, which motivates us to model the initial data by random fields.
Previous studies show that the non-Gaussian scenario may exist in the large scaling limit of the solution of the diffusion equation with random initial data. The random initial value problem for the time-fractional relativistic diffusion system was considered in this thesis. The main different point with respect to the previous studies is that the solution is a vector-valued spatial-temporal random field, so each component of the solution is affected simultaneously by two random fields, which may be dependent. We found that when the strength of long range dependence of the random initial data exceeds some threshold, the rescaled solution will converge to a non-Gaussian field. And there is a competition relationship between the effects on the solution induced by the random initial data.  
In this thesis, we also propose a new small-scaling procedure that not only emphasizes the important role of small-scale parameters but also displays the multi-scale feature of the random solution. In our main results, the initial data are modeled by nonlinear functions of homogeneous Gaussian random fields. We use the spectral representation method and the Hermite expansion to analyze the random solution field and apply the multiple Wiener integrals to demonstrate the existence of non-Gaussian limits.