Infinite-Dimensional Stochastic Differential Equations with Symmetry

Abstract

We review recent progress in the study of infinite-dimensional stochastic differential equations with symmetry. This paper contains examples arising from random matrix theory. AMC2010: 60H110, 60J60, 60K35, 60B20, 15B52

Dunkl jump processes: relaxation and a phase transition

“…The radial Dunkl processes of types A N −1 and B N correspond to the Dyson model [4] and the Wishart-Laguerre processes [15,16] respectively. In the particular case β = 2, some of their properties in the infinite-particle limit have been elucidated in [28,29,30], in particular in the bulk scaling limit, where the process time t is scaled linearly with the number of particles, N . Under these conditions, we have the following theorem.…”

The logarithmic derivative for point processes with equivalent Palm measures

“…is almost surely locally finite for every t ∈ R + , and the process X(t), considered as a process on the space Conf(R d ), preserves the measure P. For example, if P is the standard Poisson point process on R d , then ξ i (t) are independent Brownian motions. In the series of papers [6,[9][10][11][12][13][14][15] the third author with collaborators developed a general approach to constructing the process ξ. The key step is the computation of the logarithmic derivative d P of the measure P, d P : R d × Conf(R d ) → R d , introduced by the third author in [10].…”

“…where the configuration X i is defined by the formula X i (ξ(u)) := {ξ j (u)} j =i and B i are independent Brownian motions. In [6,10,15] logarithmic derivatives were calculated for determinantal processes arising in random matrix theory: sine 2 , Airy 2 , Bessel 2 and the Ginibre point processes. The computation was based on finite particle approximation and had to be adapted for each determinantal process separately.…”

A note on tail triviality for determinantal point processes

“…Osada and Osada relied on the earlier result of Lyons [3] that the conjecture held in the discrete case, as does the present short proof.In the discrete case and under the restrictive assumption that the spectrum of K is contained in the open interval (0, 1), Shirai and Takahashi [7] also proved that the tail σ-field is trivial. In the continuous setting, tail triviality is important in proving pathwise uniqueness of solutions of certain infinite-dimensional stochastic differential equations related to determinantal point processes [6].Our proof here relies on an extension of Goldman's transference principle, as elucidated in [4].…”

“…In the discrete case and under the restrictive assumption that the spectrum of K is contained in the open interval (0, 1), Shirai and Takahashi [7] also proved that the tail σ-field is trivial. In the continuous setting, tail triviality is important in proving pathwise uniqueness of solutions of certain infinite-dimensional stochastic differential equations related to determinantal point processes [6].…”