%0 Book Section %1 warren1997branching %A Warren, Jonathan %B Séminaire de Probabilités XXXI %D 1997 %E Azéma, Jacques %E Yor, Marc %E Emery, Michel %I Springer Berlin Heidelberg %K Ray-Knight_theorem branching_processes continuous-state_branching_processes decomposition sticky_Brownian_motion %P 1-15 %R 10.1007/BFb0119287 %T Branching processes, the Ray-Knight theorem, and sticky Brownian motion %U http://dx.doi.org/10.1007/BFb0119287 %V 1655 %X Ikeda and Watanabe show that (1.1) admits a weak solution and enjoys the uniqueness-in-law property. In 2, Chitashvili shows that, indeed, the joint law of X and W is unique (modulo the initial value of W), and that X is not measurable with respect to W, so verifying a conjecture of Skorokhod that (1.1) does not have a strong solution. The filtration (her) cannot be the (augmented) natural filtration of W and the process X contains some 'extra randomness'. It is our purpose to identify this extra randomness in terms of killing in a branching process. To this end we will study the squared Bessel process, which can be thought of as a continuous-state branching process, and a simple decomposition of it induced by introducing a killing term. We will then be able to realise this decomposition in terms of the local-time processes of X and W. Finally we will prove the following result which essentially determines the conditional law of sticky Brownian motion given the driving Wiener process. %@ 978-3-540-62634-3

@incollection{warren1997branching, abstract = {Ikeda and Watanabe show that (1.1) admits a weak solution and enjoys the uniqueness-in-law property. In [2], Chitashvili shows that, indeed, the joint law of X and W is unique (modulo the initial value of W), and that X is not measurable with respect to W, so verifying a conjecture of Skorokhod that (1.1) does not have a strong solution. The filtration (her) cannot be the (augmented) natural filtration of W and the process X contains some 'extra randomness'. It is our purpose to identify this extra randomness in terms of killing in a branching process. To this end we will study the squared Bessel process, which can be thought of as a continuous-state branching process, and a simple decomposition of it induced by introducing a killing term. We will then be able to realise this decomposition in terms of the local-time processes of X and W. Finally we will prove the following result which essentially determines the conditional law of sticky Brownian motion given the driving Wiener process. }, added-at = {2013-02-03T18:32:33.000+0100}, author = {Warren, Jonathan}, biburl = {https://www.bibsonomy.org/bibtex/229fd1301adc9ca88c20137d06a66e04e/peter.ralph}, booktitle = {Séminaire de Probabilités XXXI}, description = {Branching processes, the Ray-Knight theorem, and sticky Brownian motion - Springer}, doi = {10.1007/BFb0119287}, editor = {Azéma, Jacques and Yor, Marc and Emery, Michel}, interhash = {96c8bcbce4a9e93d606ab07098e04eb4}, intrahash = {29fd1301adc9ca88c20137d06a66e04e}, isbn = {978-3-540-62634-3}, keywords = {Ray-Knight_theorem branching_processes continuous-state_branching_processes decomposition sticky_Brownian_motion}, pages = {1-15}, publisher = {Springer Berlin Heidelberg}, series = {Lecture Notes in Mathematics}, timestamp = {2013-02-03T18:32:33.000+0100}, title = {Branching processes, the Ray-Knight theorem, and sticky Brownian motion}, url = {http://dx.doi.org/10.1007/BFb0119287}, volume = 1655, year = 1997 }