Stochastic flows and sticky Brownian motion

Sticky Brownian \nmotion \nis \na one-dimensional \ndiffusion \nwith the \nproperty that \nthe \namount of \ntime the process spends at zero \nis \nof positive \nLebesgue \nmeasure \nand yet \nthe \nprocess \ndoes \nnot stay at zero \nfor \nany positive interval \nof time. Sticky \nBrownian \nmotion can \nbe \nconsidered as qualitatively \nbetween \nstandard \nBrownian \nmotion and \nBrownian \nmotion absorbed at zero. \nA \nsystem of coalescing \nBrownian \nmotions \nis \na collection of paths, where \neach path \nbehaves as a \nBrownian \nmotion \nindependent \nof all other paths until \nthe \nfirst \ntime two paths meet, at which point the two \npaths that have just \nmet \nbehave is \na single \nBrownian \nmotion \nindependent \nof all remaining paths. \nThus the \ndifference between \nany two paths of a system of coalescing \nBrownian \nmotion \nbehaves \nas a \nBrownian \nmotion absorbed at zero. \nIn \nthis thesis \nwe \nconsider systems of \nBrownian \npaths, where the difference between \nany two \npaths \nbehaves as a sticky \nBrownian \nmotion rather than a coalescing Brownian \nmotion. \nWe \nconsider systems of sticky \nBrownian \nmotions starting \nfrom \npoints \nin \ncontinuous \ntime and space. \nThe \nevolution of systems of this type \nmay \nbe \ndescribed by \nmeans of a stochastic \nflow \nof \nkernels. A \nstochastic \nflow \nof \nkernels is \ncharacterised \nby its N-point \nmotions which \nform \na consistent \nfamily \nof \nBrownian \nmotions. \nWe \ncharacterise such a consistent \nfamily \nsuch that the difference \nbetween \nany pair of coordinates \nbehaves as a sticky \nBrownian \nmotion. \nThe Brownian \nweb \nis \na way of \ndescribing \na system of coalescing Brownian \nmotions starting \nin \nany point \nin \nspace and time. We describe \na coupling of \nBrownian \nwebs such \nthat the difference between one path \nin \neach web \nbehaves \nas a sticky \nBrownian motion. \nThen by \nconditioning one \nBrownian \nweb on the \nother we can construct a stochastic \nflow \nof \nkernels. \nFinally \nwe \ndiscuss the \nconcept of \nduality in \nrelation to flows \nand we prove \nsome minor results relating \nto these \ndualities. \n