Stochastic flows and sticky Brownian motion

Doctoral thesis English OPEN
Howitt, Christopher John
  • Subject: QA
    arxiv: Mathematics::Probability

Sticky Brownian\ud motion\ud is\ud a one-dimensional\ud diffusion\ud with the\ud property that\ud the\ud amount of\ud time the process spends at zero\ud is\ud of positive\ud Lebesgue\ud measure\ud and yet\ud the\ud process\ud does\ud not stay at zero\ud for\ud any positive interval\ud of time. Sticky\ud Brownian\ud motion can\ud be\ud considered as qualitatively\ud between\ud standard\ud Brownian\ud motion and\ud Brownian\ud motion absorbed at zero.\ud A\ud system of coalescing\ud Brownian\ud motions\ud is\ud a collection of paths, where\ud each path\ud behaves as a\ud Brownian\ud motion\ud independent\ud of all other paths until\ud the\ud first\ud time two paths meet, at which point the two\ud paths that have just\ud met\ud behave is\ud a single\ud Brownian\ud motion\ud independent\ud of all remaining paths.\ud Thus the\ud difference between\ud any two paths of a system of coalescing\ud Brownian\ud motion\ud behaves\ud as a\ud Brownian\ud motion absorbed at zero.\ud In\ud this thesis\ud we\ud consider systems of\ud Brownian\ud paths, where the difference between\ud any two\ud paths\ud behaves as a sticky\ud Brownian\ud motion rather than a coalescing Brownian\ud motion.\ud We\ud consider systems of sticky\ud Brownian\ud motions starting\ud from\ud points\ud in\ud continuous\ud time and space.\ud The\ud evolution of systems of this type\ud may\ud be\ud described by\ud means of a stochastic\ud flow\ud of\ud kernels. A\ud stochastic\ud flow\ud of\ud kernels is\ud characterised\ud by its N-point\ud motions which\ud form\ud a consistent\ud family\ud of\ud Brownian\ud motions.\ud We\ud characterise such a consistent\ud family\ud such that the difference\ud between\ud any pair of coordinates\ud behaves as a sticky\ud Brownian\ud motion.\ud The Brownian\ud web\ud is\ud a way of\ud describing\ud a system of coalescing Brownian\ud motions starting\ud in\ud any point\ud in\ud space and time. We describe\ud a coupling of\ud Brownian\ud webs such\ud that the difference between one path\ud in\ud each web\ud behaves\ud as a sticky\ud Brownian motion.\ud Then by\ud conditioning one\ud Brownian\ud web on the\ud other we can construct a stochastic\ud flow\ud of\ud kernels.\ud Finally\ud we\ud discuss the\ud concept of\ud duality in\ud relation to flows\ud and we prove\ud some minor results relating\ud to these\ud dualities.\ud
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