Abstract
We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion
Highlights
Introduction and ResultsDyson’s Brownian motion model of GUE (Gaussian unitary ensemble) is a stochastic process of positions of m particles, X(t) = (X1(t), . . . , Xm(t)) described by the stochastic differential equation, dt dXi = dBi + 1≤j≤m Xi − Xj, j=i 1 ≤ i ≤ m, (1.1)where Bi, 1 ≤ i ≤ m are independent one dimensional Brownian motions [5]
We remark that the process X can be started from the origin, i.e., one can take Xi(0) = 0, 1 ≤ i ≤ m
Some authors consider a process defined by the s.d.e.s (1.3) without the local time term
Summary
This process will be referred to as Dyson’s Brownian motion of type D. Some authors consider a process defined by the s.d.e.s (1.3) without the local time term. The process we consider with a reflecting wall is obtained from this by replacing the first component with its absolute value, with the local time term appearing as a consequence of Tanaka’s formula.
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