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On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes. / Sandra Landwehr

Swansea University Author: Sandra Landwehr

Abstract

In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn → R which satisfies psi(xi) ≥ 0 with psi(xi) = 0. T...

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Published: 2010
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42253
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spelling 2018-08-02T16:24:28.5733830 v2 42253 2018-08-02 On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes. 262bce38e565cf362aaab25e319f7a81 NULL Sandra Landwehr Sandra Landwehr true true 2018-08-02 In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn &rarr; R which satisfies psi(xi) &ge; 0 with psi(xi) = 0. This study is motivated by the investigation of a transition density estimate for pure jump processes on a general metric measure space. To gain a better insight into the behaviour of transition functions of symmetric Levy processes in this general setting, it seems desirable to understand geometrical properties of their underlying state spaces. More precisely, we show completeness of the metric spaces (Rn, psi1/2) and study under which circumstances open balls Bpsi(x,r), x &isin; Rn, r > 0, with respect to this metric are convex. Moreover, we focus on conditions of the metric measure spaces (Rn,psi1/2 ,mu) for the balls to satisfy the volume growth property [equation] for mu-almost all x &isin; Rn, 0 < r < R and a constant Cpsi(x,R)&ge;1. Finally, we show that the homogeneity property of a metric measure space can be applied to our case and provide some results associated with the construction of a Hajlasz-Sobolc space over (Rn,psi1/2, lambda(n)),where lambda(n) denotes the n-dirnensional Lebesgue measure. E-Thesis Mathematics. 31 12 2010 2010-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:28.5733830 2018-08-02T16:24:28.5733830 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Sandra Landwehr NULL 1 0042253-02082018162440.pdf 10797961.pdf 2018-08-02T16:24:40.0070000 Output 5815969 application/pdf E-Thesis true 2018-08-02T16:24:40.0070000 false
title On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
spellingShingle On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
Sandra Landwehr
title_short On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
title_full On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
title_fullStr On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
title_full_unstemmed On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
title_sort On the geometry related to jump processes: Investigating transition functions of Levy and Levy-type processes.
author_id_str_mv 262bce38e565cf362aaab25e319f7a81
author_id_fullname_str_mv 262bce38e565cf362aaab25e319f7a81_***_Sandra Landwehr
author Sandra Landwehr
author2 Sandra Landwehr
format E-Thesis
publishDate 2010
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn &rarr; R which satisfies psi(xi) &ge; 0 with psi(xi) = 0. This study is motivated by the investigation of a transition density estimate for pure jump processes on a general metric measure space. To gain a better insight into the behaviour of transition functions of symmetric Levy processes in this general setting, it seems desirable to understand geometrical properties of their underlying state spaces. More precisely, we show completeness of the metric spaces (Rn, psi1/2) and study under which circumstances open balls Bpsi(x,r), x &isin; Rn, r > 0, with respect to this metric are convex. Moreover, we focus on conditions of the metric measure spaces (Rn,psi1/2 ,mu) for the balls to satisfy the volume growth property [equation] for mu-almost all x &isin; Rn, 0 < r < R and a constant Cpsi(x,R)&ge;1. Finally, we show that the homogeneity property of a metric measure space can be applied to our case and provide some results associated with the construction of a Hajlasz-Sobolc space over (Rn,psi1/2, lambda(n)),where lambda(n) denotes the n-dirnensional Lebesgue measure.
published_date 2010-12-31T03:52:36Z
_version_ 1763752601516507136
score 11.035874

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