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.
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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.
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From a practical point of view, the considered hitting time appears as a deadlock time in various resource sharing problems.