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Analytic Approach to Invariant Measures and Itô-Stochastic Differential Equations
불변 측도와 이토 확률미분방정식에 대한 해석적 접근

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dc.contributor.advisorGerald Trutnau-
dc.contributor.author이해성-
dc.date.accessioned2019-10-21T03:37:16Z-
dc.date.available2019-10-21T03:37:16Z-
dc.date.issued2019-08-
dc.identifier.other000000157514-
dc.identifier.urihttps://hdl.handle.net/10371/162408-
dc.identifier.urihttp://dcollection.snu.ac.kr/common/orgView/000000157514ko_KR
dc.description학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. Gerald Trutnau.-
dc.description.abstractIn this thesis, we study an analytic approach to global well-posedness and long-time behavior for weak solutions to Itô-SDEs with rough coefficients. Using elliptic and parabolic regularity theory and generalized Dirichlet form theory, we show existence of a pre-invariant measure for a large class of elliptic second order partial differential operators and show that these are in fact infinitesimal generators of a Hunt process. Subsequently, this Hunt process is identified for every starting point as a weak solution to an Itô-SDE in R^d up to its explosion time. The Hunt process has continuous sample paths on the one-point compactification of R^d and by a known local well-posedness result, it is a pathwise unique and strong solution up to its explosion time to the SDE that it weakly solves. Using analytic and probabilistic methods, we derive general strong Feller properties, including the classical strong Feller property, Krylov type estimates, moment inequalities and various non-explosion criteria. Using a parabolic Harnack inequality, we show irreducibility and strict irreducibility of the process and derive explicit conditions for recurrence and ergodic behavior. Moreover, we investigate well-posedness of weak solutions to Itô-SDEs with degenerate and rough diffusion coefficients whose points of degeneracy form a set of Lebesegue measure zero. In the final part we consider the case where the pre-invariant density is explicitly given. In contrast to the previous case, where we only knew its existence with a certain regularity, we investigate how far our previous methods can be extended and applied in case of a non-degenerate, possibly non-symmetric and discontinuous diffusion matrix. For this, we develop some variational approach to regularity theory for linear parabolic PDEs involving divergence form operators with weight in the term where time derivative appear.-
dc.description.abstract이 논문에서 우리는 거친 계수를 갖는 이토 확률미분방정식의 약한 해들의 대역적인 존재 및 유일성, 장시간의 행동에 대한 해석학적 접근을 연구한다. 타원형 및 포물형 정칙성 이론과 일반화된 디리클레 형식 이론을 사용함으로써, 우리는 넓은 유형의 타원형 2계 편미분 작용소의 예비 불변측도의 존재성을 보이고, 그 작용소는 사실은 어느 헌트 과정의 극소 생성자가 됨을 보인다. 그 후, 이 헌트 과정은 R^d 위의 모든 점들을 시작점으로 갖는 이토 확률미분방정식의 폭발 시간 안에서 약한 해로 동일시된다. 그 헌트 프로세스는 R^d의 한점 컴팩트화된 공간에서 연속인 샘플 경로를 갖고 알려진 존재 및 유일성 정리에 의해 그것은 폭발 시간 안에서 경로마다 유일한 강한 해가 된다. 해석학적, 확률론적 방법을 사용하여 우리는 고전적인 강한 펠러 성질을 포함하는 일반화된 강한 펠러 성질들, 크릴로프 유형의 가늠, 모먼트 부등식, 다양한 비폭발 판정법을 유도한다. 포물형 하르낙 부등식을 이용하여 우리는 프로세스의 기약성과 강한 기약성을 보이고 재귀성과 에르고딕 행동들에 대한 명확한 조건들을 이끌어 낸다. 더 나아가서 우리는 퇴화된 정도의 점들이 르벡 측도 0을 만족하는 퇴화된 거친 확산 계수에 관한 이토 확률미분방정식의 약한 해의 존재성과 유일성을 조사한다. 마지막으로 우리는 예비 불변측도의 밀도함수가 명확히 주어졌을 때를 고려한다. 단순히 그 예비측도의 존재성과 어떤 정칙성만 알았던 이전의 경우와는 달리, 우리는 퇴화되지 않은 비대칭, 불연속 확산 행렬인 경우에 얼마나 우리의 방법들이 확장되고 적용될 수 있는지 조사한다. 이를 위해 우리는 시간 텀에 무게를 갖는 발산 형식 선형 포물형 편미분방정식의 정칙이론에 대한 변분적 접근을 발전시킨다.-
dc.description.tableofcontentsAbstract

1 Introduction . . . . . . 1

2 Notations . . . . . . 15


I Existence, uniqueness and ergodic properties for time-homogeneous Itô-SDEs with locally integrable drifts and Sobolev diffusion coefficients . . . . . . 19

3 Weak solutions via analytic theory . . . . . . 20
3.1 Analytic theory of generalized Dirichlet forms . . . . . . 20
3.2 Construction of a weak solution . . . . . . 37

4 Conservativeness and ergodic properties . . . . . . 47
4.1 Non-explosion criteria and moment inequalities . . . . . . 47
4.1.1 Non-explosion criteria and moment inequalities without involving the density . . . . . . 47
4.1.2 Non-explosion criteria involving the density . . . . . . 54
4.2 Recurrence criteria and other ergodic properties involving and not involving the density . . . . . . 55
4.2.1 Explicit recurrence criteria for possibly infinite m . . . . . . 60
4.2.2 Uniqueness of invariant measures and ergodic properties in case m is a probability measure . . . . . . 63
4.3 An application to pathwise uniqueness and strong solutions . . . . . . 68


II Existence and regularity of pre-invariant measures, transition functions and time homogeneous Itô-SDEs . . . . . . 69

5 Analytic results . . . . . . 70
5.1 Elliptic H^{1,p}-regularity and estimates . . . . . . 70
5.2 Existence of a pre-invariant measure and construction of a generalized Dirichlet form . . . . . . 77
5.3 Regularity results for resolvent and semigroup . . . . . . 83

6 Probabilistic results . . . . . . 88
6.1 The underlying SDE . . . . . . 88
6.2 Uniqueness in law under low regularity . . . . . . 93


III Well-posedness for Itô-SDEs with degenerate and rough diffusion coefficients . . . . . . 96

7 Regularity of solutions . . . . . . 97
7.1 Regularity results for linear parabolic equation with singular weight in the time derivative term . . . . . . 97
7.2 Elliptic Hölder regularity and estimates . . . . . . 105

8 Analytic theory for degenerate second order partial differential operators . . . . . . 108
8.1 Framework . . . . . . 108
8.2 L^1-existence results . . . . . . 111
8.3 Existence of a pre-invariant measure and general strong Feller properties . . . . . . 131
8.4 Some auxiliary results . . . . . . 140

9 Well-posedness and irreducibility for degenerate Itô-SDEs . . . . . . 148
9.1 Weak existence of degenerate Itô-SDEs with rough coefficients . . . . . . 148
9.2 Strict irreducibility for special weight functions . . . . . . 151
9.3 Uniqueness in law for degenerate Itô-SDEs with discontinuous dispersion coefficients . . . . . . 156


IV Existence and regularity of transition functions with general pre-invariant measures
and corresponding Itô-SDEs . . . . . . 171

10 Regularity results for weighted parabolic PDEs . . . . . . 172
10.1 L^1-estimate in terms of the L^2-norm . . . . . . 175
10.2 Parabolic Harnack inequality . . . . . . 179
11 Analytic and probabilistic results . . . . . . 193
11.1 Strong Feller property and irreducibility with general pre-invariant measures . . . . . . 193
11.2 Application to weak existence of Itô-SDEs . . . . . . 202
11.3 Explicit conditions for global well-posedness and ergodic properties . . . . . . 204


Abstract (in Korean) . . . . . . 215

Acknowledgement . . . . . . 216
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dc.language.isoeng-
dc.publisher서울대학교 대학원-
dc.subjectgeneralized Dirichlet form-
dc.subjectinvariant measure-
dc.subjectHunt process-
dc.subjectItô-SDE-
dc.subjectelliptic and parabolic regularity-
dc.subjectstrong Feller property-
dc.subjectnon-explosion-
dc.subjectconservativeness-
dc.subjectirreducibility-
dc.subjectstrict irreducibility-
dc.subjectrecurrence-
dc.subjecttransience-
dc.subjectergodicity-
dc.subjectweak uniqueness-
dc.subjectKrylov type estimate-
dc.subject.ddc510-
dc.titleAnalytic Approach to Invariant Measures and Itô-Stochastic Differential Equations-
dc.title.alternative불변 측도와 이토 확률미분방정식에 대한 해석적 접근-
dc.typeThesis-
dc.typeDissertation-
dc.contributor.department자연과학대학 수리과학부-
dc.description.degreeDoctor-
dc.date.awarded2019-08-
dc.identifier.uciI804:11032-000000157514-
dc.identifier.holdings000000000040▲000000000041▲000000157514▲-
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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