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研究生:劉聚仁
研究生(外文):Gi-Ren Liu
論文名稱:偏微分方程組具隨機初始值
論文名稱(外文):Partial Differential Equations with Random Initial Data
指導教授:謝南瑞
口試委員:許元春黃啟瑞張志中
口試日期:2013-06-07
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2013
畢業學年度:101
語文別:英文
論文頁數:151
中文關鍵詞:偏微分方程隨機場
外文關鍵詞:partial differential equationsrandom field
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In this thesis, we study the limiting distributions of linear systems of partial differential
equations with subordinated Gaussian random initial data. When the initial
data is non-random, the solutions of the linear systems are given by the convolution
of the Green kernels and the initial data. Therefore, the evolution of the solutions
is totally determined by their initial data. However, the information regarding the
initial data is obtained through some process of measurement, resulting in measurement
error. In our work, we use the second-order homogeneous random field to model
these measurement error and apply the spectral representation method to study the
covariance matrix functions of the random solution vector fields. In view of that the
solution fields can be thought of as the weighted sum of correlated random variables,
we will also consider the limiting distributions of the random solution fields from
di↵erent viewpoints, including macroscopic scales and microscopic scales. When the
random initial data is weakly dependent, our results can be thought of as a generalized
central limit theorem. There are two contributions for the new results. The first
one is that the initial data is modeled by two cross-correlated subordinated Gaussian
random fields. We use the method of Feynman diagrams to analyze the asymptotic
behavior of the covariance matrix function of the random solution field induced by
the random initial data. Second, the limit of the random solution vector field under
the macroscopic/microscopic coordinate systems is represented by a L2-convergent
series of mutually independent Gaussian random fields. We also study the limiting
distributions of the solution vector field when its random initial data is long-range
dependent. Compared to the previous case, the limiting law of the rescaled solution
vector field is non-Gaussian, which is represented by multiple Wiener integrals. In
contrast to the existing mathematical literature we found that there is a competition
relationship between the effect coming from two components of the random initial
data. That is, one of the two components of the random initial data will be determined dominantly the structure of the limiting distribution of the random part of the
solution vector field.

Contents
Acknowledgements i
Abstract ii
Contents iii
1 Literature Review 1
2 Introduction and Preliminary 9
2.1 Spectral representation of homogeneous random vector fields . . . . . 9
2.1.1 Random solutions . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Proof of Proposition 2.4 . . . . . . . . . . . . . . . . . . . . . 20
2.3 Appendix: proof of (2.14) . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Non-Gaussian Scenarios for the Fractional Kinetic System with Long-
Range Dependent Initial Data 25
3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Large scaling limits for the random solutions . . . . . . . . . . . . . . 29
3.3 Small scaling limits for the random solutions . . . . . . . . . . . . . . 35
3.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.3.1 Proof of Proposition 3.8 (1) . . . . . . . . . . . . . . 53
3.4.4 Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . 56
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5.1 Slutsky’s Argument and the Cramer-Wold Device . . . . . . . 57
3.5.2 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Relativistic Di↵usion Equations with Weakly Dependent Random
Initial Data 61
4.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.1 Relativistic Schr¨odinger operators and its Green functions . . 64
4.2 Large scaling limits under Condition WW . . . . . . . . . . . . . . . 66
4.3 Large scaling limits under Condition WL . . . . . . . . . . . . . . . . 68
4.4 Large scaling limits of the relativistic di↵usion system subject to interrelated
random initial data . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.1 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.2 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.3 Proof of Theorem 4.6 . . . . . . . . . . . . . . . . . . . . . . . 86
4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6.1 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . 95
4.6.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . 96
5 Time-Fractional Di↵usion-Wave Equations with Random Initial Data 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Mittage-Le✏er functions . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 General random initial data . . . . . . . . . . . . . . . . . . . 102
5.3 Large scaling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3.1 Non-Gaussian limits for the fractional wave system (! 2 (1, 2)) 103
5.3.2 Gaussian limiting theorems for the fractional wave systems (! 2
(1, 2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.3 Hyperbolic scaling limit for linear wave equations (! = 2) . . . 110
5.4 Extension: Time-fractional relativistic di↵usion system (! 2 (0, 1)) . . 114
5.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5.1 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . 118
5.5.2 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . 119
5.5.3 Proof of Lemma 5.3 . . . . . . . . . . . . . . . . . . . . . . . . 122
5.5.4 Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . 124
5.5.5 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . 128
5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6.1 Proof of (5.11) . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.6.2 Proof of (5.14) . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6 Conclusion and Future Works 143
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Bibliography 145

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