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Uniqueness problems of diffusion operators on Euclidean space and on abstract Wiener space : 유클리드 공간과 추상적인 위너 공간 위에서의 확산 작용소들의 유일성에 관한 여러 문제들
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | Gerald Trutnau | - |
dc.contributor.author | 강승현 | - |
dc.date.accessioned | 2018-11-12T01:01:34Z | - |
dc.date.available | 2018-11-12T01:01:34Z | - |
dc.date.issued | 2018-08 | - |
dc.identifier.other | 000000152452 | - |
dc.identifier.uri | https://hdl.handle.net/10371/143329 | - |
dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 8. Gerald Trutnau. | - |
dc.description.abstract | The central question discussed in this thesis is whether a given diffusion operators,
i.e., a second order linear elliptic differential operator without zeroth order term, which is a priori only defined on test functions over some (finite or infinite dimensional ) state space, uniquely determines a strongly continuous semigroup on a corresponding weighted L^p space. On the first part of the thesis, we are mainly focus on equivalence of different definitions of capacities, and removability of singularities. More precisely, let L be either a fractional powers of Laplacian of order less than one whose domain is smooth compactly supported functions on R^d ∖ Σ of a given compact set Σ ⊂ R^d of zero Lebesgue measure or integral powers of Ornstein-Uhlenbeck operator defined on suitable algebras of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian measure in abstract Wiener space. Depending on the size of Σ, the operator under consideration, may or may not be L^p unique. We give descriptions for the critical size of Σ in terms of capacities and Hausdorff measures. In addition, we collect some known results for certain multi-parameter stochastic processes. On the second part of this thesis, we are mainly focus on Neumann problems on L^p(U, µ), where U ⊂ R^d is an open set. More precisely, let L be a nonsymmetric operator of type Lu = ∑ aij∂i∂ju+∑ bi∂iu, whose domain is C^2_0,Neu(U). We give some results about Markov uniqueness, L^p-uniqueness, relation of L^1-uniqueness and conservativeness, uniqueness of invariant measures, elliptic regularity, etc under certain assumption on µ and on the coefficients of L. | - |
dc.description.tableofcontents | Abstract i
Chapter 1 General Introduction 1 I Equivalence of capacities and removability of singularities 3 Chapter 2 Probabilistic characterizations of essential self-adjointness and removability of singularities 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Bessel potential spaces, capacities and kernels . . . . . . . . . . . . . . 8 2.3 Markov uniqueness, essential self-adjointness and capacities . . . . . . 12 2.4 Riesz capacities and Hausdorff measures . . . . . . . . . . . . . . . . . . 15 2.5 Additive processes and a probabilistic characterization . . . . . . . . . 18 Chapter 3 Capacities, removable sets and L^p-uniqueness on Wiener spaces 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Capacities and their equivalence . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Smooth truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 L^p-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.6 Comments on Gaussian Hausdorff measures . . . . . . . . . . . . . . . . 52 iii 3.7 Comments on stochastic processes . . . . . . . . . . . . . . . . . . . . . 54 II Markov uniqueness, L^p uniqueness and elliptic regularity on reflected Dirichlet space 56 Chapter 4 Markov uniqueness and L^2-uniqueness on reflected Dirichlet space 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Functional analytic framework, preliminary results and notations . . . 59 4.3 Main result on Markov Uniqueness . . . . . . . . . . . . . . . . . . . . . 65 4.4 L^2-uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.5 Markov uniqueness of Robin boundary condition . . . . . . . . . . . . . 92 Chapter 5 L^1-uniqueness and conservativeness on reflected Dirichlet space 94 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Functional analytic framework and notations . . . . . . . . . . . . . . . 95 5.3 Elliptic regularity and L^2-uniqueness . . . . . . . . . . . . . . . . . . . . 124 5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Reference 133 국문초록 144 Acknowledgement 146 | - |
dc.language.iso | en | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject.ddc | 510 | - |
dc.title | Uniqueness problems of diffusion operators on Euclidean space and on abstract Wiener space | - |
dc.title.alternative | 유클리드 공간과 추상적인 위너 공간 위에서의 확산 작용소들의 유일성에 관한 여러 문제들 | - |
dc.type | Thesis | - |
dc.contributor.AlternativeAuthor | Seunghyun Kang | - |
dc.description.degree | Doctor | - |
dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
dc.date.awarded | 2018-08 | - |
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