Linear bounds for stochastic dispersion
It has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid. We define a stochastic flow by the equations Φ(x) = x, dΦt(x) = F(dt, Φt(x)), where F(t, x) is a field of semimartingales on x ∈ ℝd for d ≥ 2 whose local characteristics are bounded...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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2000
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| _version_ | 1826300152971788288 |
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| author | Cranston, M Scheutzow, M Steinsaltz, D |
| author_facet | Cranston, M Scheutzow, M Steinsaltz, D |
| author_sort | Cranston, M |
| collection | OXFORD |
| description | It has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid. We define a stochastic flow by the equations Φ(x) = x, dΦt(x) = F(dt, Φt(x)), where F(t, x) is a field of semimartingales on x ∈ ℝd for d ≥ 2 whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set script X sign, and we ask how far the substance has spread in a time T. That is, we define Φ*T = supx∈script T sign 0≤t≤Tsup ∥Φt(x)∥, and seek to bound P {Φ*T > z}. Without drift, when F(·, x) are required to be martingales, although single points move on the order of √T, it is easy to construct examples in which the supremum Φ*T still grows linearly in time - that is, lim inf T→∞ Φ*T/T > 0 almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant K0, depending on the bounds for the local characteristics, such that lim supT→∞ Φ*T/T ≤ K0 almost surely. A linear bound on growth holds even when the field itself includes a drift term. |
| first_indexed | 2024-03-07T05:12:50Z |
| format | Journal article |
| id | oxford-uuid:dc253f75-3826-4283-b24c-d9e8532e4514 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T05:12:50Z |
| publishDate | 2000 |
| record_format | dspace |
| spelling | oxford-uuid:dc253f75-3826-4283-b24c-d9e8532e45142022-03-27T09:15:40ZLinear bounds for stochastic dispersionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:dc253f75-3826-4283-b24c-d9e8532e4514EnglishSymplectic Elements at Oxford2000Cranston, MScheutzow, MSteinsaltz, DIt has been suggested that stochastic flows might be used to model the spread of passive tracers in a turbulent fluid. We define a stochastic flow by the equations Φ(x) = x, dΦt(x) = F(dt, Φt(x)), where F(t, x) is a field of semimartingales on x ∈ ℝd for d ≥ 2 whose local characteristics are bounded and Lipschitz. The particles are points in a bounded set script X sign, and we ask how far the substance has spread in a time T. That is, we define Φ*T = supx∈script T sign 0≤t≤Tsup ∥Φt(x)∥, and seek to bound P {Φ*T > z}. Without drift, when F(·, x) are required to be martingales, although single points move on the order of √T, it is easy to construct examples in which the supremum Φ*T still grows linearly in time - that is, lim inf T→∞ Φ*T/T > 0 almost surely. We show that this is an upper bound for the growth; that is, we compute a finite constant K0, depending on the bounds for the local characteristics, such that lim supT→∞ Φ*T/T ≤ K0 almost surely. A linear bound on growth holds even when the field itself includes a drift term. |
| spellingShingle | Cranston, M Scheutzow, M Steinsaltz, D Linear bounds for stochastic dispersion |
| title | Linear bounds for stochastic dispersion |
| title_full | Linear bounds for stochastic dispersion |
| title_fullStr | Linear bounds for stochastic dispersion |
| title_full_unstemmed | Linear bounds for stochastic dispersion |
| title_short | Linear bounds for stochastic dispersion |
| title_sort | linear bounds for stochastic dispersion |
| work_keys_str_mv | AT cranstonm linearboundsforstochasticdispersion AT scheutzowm linearboundsforstochasticdispersion AT steinsaltzd linearboundsforstochasticdispersion |