1. A PDE for non-intersecting Brownian motions and applications
- Author
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Adler, Mark, Delépine, Jonathan, van Moerbeke, Pierre, and Vanhaecke, Pol
- Subjects
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PARTIAL differential equations , *WIENER processes , *DIFFUSION processes , *PROBABILITY theory , *MATHEMATICAL variables , *DETERMINANTS (Mathematics) , *BIFURCATION theory , *PERTURBATION theory - Abstract
Abstract: Consider non-intersecting Brownian motions on the real line, starting from the origin at , with particles forced to reach p distinct target points at time , with . This can be viewed as a diffusion process in a sector of . This work shows that the transition probability, that is the probability for the particles to pass through windows at times , satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size , with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view. As an application of these equations, let the number of particles forced to the extreme points and tend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to depart from those going to . These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture. [Copyright &y& Elsevier]
- Published
- 2011
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