Continuity in κ in SLEκ theory using a constructive method and rough path theory
Questions regarding the continuity in κ of the SLEκ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probabi...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Institut Henri Poincaré
2021
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| _version_ | 1797068404048265216 |
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| author | Beliaev, D Lyons, TJ Margarint, V |
| author_facet | Beliaev, D Lyons, TJ Margarint, V |
| author_sort | Beliaev, D |
| collection | OXFORD |
| description | Questions regarding the continuity in κ of the SLEκ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLEκ depends continuously on κ. It is rather easy to show that SLE is continuous in the Carath´eodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κj → κ ∈ (0, 8/3), for almost every Brownian motion SLEκ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLEκ traces for varying parameter κ ∈ (0, 8/3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √κBt when started away from the origin are continuous in the p-variation topology in the parameter κ, for all κ ∈ R+. |
| first_indexed | 2024-03-06T22:10:17Z |
| format | Journal article |
| id | oxford-uuid:5192bbd2-9ac5-4ce3-97d3-8328bcebcea6 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:10:17Z |
| publishDate | 2021 |
| publisher | Institut Henri Poincaré |
| record_format | dspace |
| spelling | oxford-uuid:5192bbd2-9ac5-4ce3-97d3-8328bcebcea62022-03-26T16:20:23ZContinuity in κ in SLEκ theory using a constructive method and rough path theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5192bbd2-9ac5-4ce3-97d3-8328bcebcea6EnglishSymplectic ElementsInstitut Henri Poincaré2021Beliaev, DLyons, TJMargarint, VQuestions regarding the continuity in κ of the SLEκ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of κ we use the same Brownian motion. It is very natural to assume that with probability one, SLEκ depends continuously on κ. It is rather easy to show that SLE is continuous in the Carath´eodory sense, but showing that SLE traces are continuous in the uniform sense is much harder. In this note we show that for a given sequence κj → κ ∈ (0, 8/3), for almost every Brownian motion SLEκ traces converge locally uniformly. This result was also recently obtained by Friz, Tran and Yuan using different methods. In our analysis, we provide a constructive way to study the SLEκ traces for varying parameter κ ∈ (0, 8/3). The argument is based on a new dynamical view on the approximation of SLE curves by curves driven by a piecewise square root approximation of the Brownian motion. The second question can be answered naturally in the framework of Rough Path Theory. Using this theory, we prove that the solutions of the backward Loewner Differential Equation driven by √κBt when started away from the origin are continuous in the p-variation topology in the parameter κ, for all κ ∈ R+. |
| spellingShingle | Beliaev, D Lyons, TJ Margarint, V Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title | Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title_full | Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title_fullStr | Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title_full_unstemmed | Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title_short | Continuity in κ in SLEκ theory using a constructive method and rough path theory |
| title_sort | continuity in κ in sleκ theory using a constructive method and rough path theory |
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