%0 Journal Article %1 delarue2008hitting %A Delarue, F. %D 2008 %I Institute of Mathematical Statistics %J Ann. Inst. H. Poincar&\#x00E9$\mathsemicolon$ Probab. Statist. %K Brownian_motion Brownian_motion_in_a_wedge diffusions hitting_times local_time two_dimensional_diffusion %N 5 %P 946--961 %R 10.1214/07-aihp128 %T Hitting time of a corner for a reflected diffusion in the square %U http://dx.doi.org/10.1214/07-AIHP128 %V 44 %X We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish in the different cases the attainability or nonattainability of the origin for the reflected process. From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems.

@article{delarue2008hitting, abstract = {We discuss the long time behavior of a two-dimensional reflected diffusion in the unit square and investigate more specifically the hitting time of a neighborhood of the origin. We distinguish three different regimes depending on the sign of the correlation coefficient of the diffusion matrix at the point 0. For a positive correlation coefficient, the expectation of the hitting time is uniformly bounded as the neighborhood shrinks. For a negative one, the expectation explodes in a polynomial way as the diameter of the neighborhood vanishes. In the null case, the expectation explodes at a logarithmic rate. As a by-product, we establish in the different cases the attainability or nonattainability of the origin for the reflected process. From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems. }, added-at = {2016-02-09T22:04:39.000+0100}, author = {Delarue, F.}, biburl = {https://www.bibsonomy.org/bibtex/2ac7bb2496950e66421f92f5c8b861f72/peter.ralph}, doi = {10.1214/07-aihp128}, interhash = {9b81e727ff211a1c48fff1875e0022a3}, intrahash = {ac7bb2496950e66421f92f5c8b861f72}, journal = {Ann. Inst. H. Poincar{\&}{\#}x00E9$\mathsemicolon$ Probab. Statist.}, keywords = {Brownian_motion Brownian_motion_in_a_wedge diffusions hitting_times local_time two_dimensional_diffusion}, month = oct, number = 5, pages = {946--961}, publisher = {Institute of Mathematical Statistics}, timestamp = {2016-02-09T22:04:39.000+0100}, title = {Hitting time of a corner for a reflected diffusion in the square}, url = {http://dx.doi.org/10.1214/07-AIHP128}, volume = 44, year = 2008 }