A PDE for non-intersecting Brownian motions and applications
(2011) Advances in mathematics — Vol. 226, n° 2, p. 1715-1755 (2011)
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- Consider N=n<inf>1</inf>+n<inf>2</inf>+...+n<inf>p</inf> non-intersecting Brownian motions on the real line, starting from the origin at t=0, with ni particles forced to reach p distinct target points β<inf>i</inf> at time t=1, with β<inf>1</inf><β<inf>2</inf><β<β<inf>p</inf>. This can be viewed as a diffusion process in a sector of R<sup>N</sup>. This work shows that the transition probability, that is the probability for the particles to pass through windows Ẽ<inf>k</inf> at times t<inf>k</inf>, 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 p+1, 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 β<inf>1</inf> and β<inf>p</inf> 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 -√n depart from those going to √n. 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. © 2010 Elsevier Inc.
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Adler, M., Delepine, J., Van Moerbeke, P., & Vanhaecke, P. (2011). A PDE for non-intersecting Brownian motions and applications. Advances in mathematics, 226(2), 1715-1755. https://doi.org/10.1016/j.aim.2010.09.004 (Original work published 2011)