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Martin boundary of Brownian motion on hyperbolic graphs : 쌍대 그래프 위에서의 브라운 운동의 마틴 경계
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | 임선희 | - |
| dc.contributor.author | 홍순기 | - |
| dc.date.accessioned | 2018-11-12T00:58:03Z | - |
| dc.date.available | 2018-11-12T00:58:03Z | - |
| dc.date.issued | 2018-08 | - |
| dc.identifier.other | 000000152856 | - |
| dc.identifier.uri | https://hdl.handle.net/10371/143178 | - |
| dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 8. 임선희. | - |
| dc.description.abstract | Let X be a hyperbolic graph with infinitely many geometric boundary points. Suppose that there exists a discrete group Γ that acts geometrically on the graph X. We study the relation between the geometric boundary and the λ- Martin boundary of the hyperbolic graph X.
First, we show the properties of the Laplacian on the graph. We observe the eigenfunctions of the Laplacian. The bottom of the spectrum of the Laplacian is positive. Using the properties, we construct the heat kernel of the graph. After we construct the heat kernel, we show the convergence of λ-Green functions and the Ancona inequality. Using the Ancona inequality, we show that the geometric boundary coincides with the λ-Martin boundary. | - |
| dc.description.tableofcontents | 1 Introduction 1
1.1 Introduction. 1 2 Laplacian and weak solution 5 2.1 Dirichlet forms . 6 2.2 Laplacianonthegraph. 12 2.2.1 Dirichletformonthegraph 12 2.2.2 Laplacianonthegraph. 14 2.3 Propertiesofeigenfunctions 15 2.3.1 Smoothness of eigenfunctions of Laplacian . . . . . . . . 15 2.3.2 Thediscretenessofthespectrum . . . . . . . . . . . . . 18 2.3.3 Theorthonormalbasis . 22 2.4 ThebottomofthespectrumofLaplacian. . . . . . . . . . . . . 27 3 The Brownian motion on the graph 34 3.1 Theconstructionofheatkernel 34 3.1.1 Themaximalprinciple . 35 3.1.2 Theheatkernels 36 3.2 ConvergenceofGreenfunction. 43 4 Martin boundary of hyperbolic graph 52 4.1 ThegraphversionofHarnackprinciple . . . . . . . . . . . . . . 52 4.2 ProofofAncona-Gou ezelinequality 56 4.3 Martin boundary 70 Abstract (in Korean) 79 Acknowledgement (in Korean) 80 | - |
| dc.language.iso | en | - |
| dc.publisher | 서울대학교 대학원 | - |
| dc.subject.ddc | 510 | - |
| dc.title | Martin boundary of Brownian motion on hyperbolic graphs | - |
| dc.title.alternative | 쌍대 그래프 위에서의 브라운 운동의 마틴 경계 | - |
| dc.type | Thesis | - |
| dc.contributor.AlternativeAuthor | Soonki Hong | - |
| dc.description.degree | Doctor | - |
| dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
| dc.date.awarded | 2018-08 | - |
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