Large deviations for sticky Brownian motions

Abstract

We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability.These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

The Bethe Ansatz for Sticky Brownian Motions

“…In fact Barraquand and Rychnovsky conjectured in [2] that the Backwards equation for the system of sticky Brownian motions was the heat equation with the boundary conditions corresponding to b − a = 1 in (16), by looking at the Bethe ansatz answer for the system. It's important to note that for any other choice of characteristic measure ν with ν([0, 1]) = θ 2 the boundary conditions corresponding to b − a = 1 would be the same, so we do not expect these boundary conditions alone to give uniqueness of the PDE.…”

“…Hence 1 ε q(1 − q)µ (ε) ⇒ θ 2 dx; since we also have 1 0 (1 − 2q)µ (ε) (dq) = 0 for all ε > 0 the theorem implies the convergence of the n-point motions of the Beta random walk in a random environment to solutions of the Howitt-Warren martingale problem with characteristic measure θ 2 ½ [0,1] dx and zero drift. This is the key motivator for looking for exact solutions in the sticky Brownian motion case and was used by Barraquand and Rychnovsky in [2] to find Fredholm determinant expressions in the sticky Brownian motions case by taking limits of those found for the Beta random walk in a random environment in [1].…”

“…We would instead want an analogue of the conjectured formula (92) in [15]. In [2] Barraquand and Rychnovsky instead considered the tails of the Howitt-Warren process, ρ t ([tx, ∞]), and derived a Fredholm determinant formula for the Laplace transform via a scaling limit from the Beta random walk in a random environment, with which they were able to prove the existence of GUE fluctuations. We make the following conjecture for the fluctuations of the individual atoms.…”

“…A straightforward calculation, see Section 5.1, shows that the scaling limit of random walks in a Beta random environment corresponds to the sticky Brownian motions with a uniform characteristic measure. Barraquand and Rychnovsky [2], working independently of us, derived exact solutions for the point to halfline probabilities of sticky Brownian motions with uniform characteristic measure by taking an appropriate scaling limit of the RWRE case. An asymptotic analysis then led to the discovery of GUE Tracey-Widom fluctuations in the large deviations of sticky Brownian motions as well.…”

Universal out-of-equilibrium dynamics of 1D critical quantum systems perturbed by noise coupled to energy

“…Since θ > 0, this shows, interestingly, that the trajectories X t (y 1 ), X t (y 2 ) tend to get closer as time grows, a manifestation of the phenomenon of sticky particles observed in turbulent fluids [54][55][56][57]. On the other hand, r = 1 holds independently of time, which shows that although the typical value of r decreases to zero, the distribution of r is broadening with time.…”