Infinite-dimensional stochastic differential equations related to random matrices

Abstract

We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via twodimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson's measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with twodimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle's class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs. IntroductionConsider infinitely many Brownian particles X = (X i ) i∈N moving in R d interacting via the two-dimensional (2D) Coulomb potentials Ψ β :Then the stochastic dynamics X = (X i ) i∈N is described by the following infinitedimensional stochastic differential equation (ISDE):Here {B i } i∈N is a sequence of independent copies of d-dimensional Brownian motions and X = (X i ) i∈N is a continuous (R d ) N -valued process.

Infinite dimensional stochastic differential equations for Dyson’s model

“…In Sect. 8, we prove that near-equilibrium solutions (defined therein) are X rg T (α, ρ, p)-valued, to unify the construction of [17] with ours.…”

“…With v : W → U, (x i ) i∈Z → i∈Z δ x i denoting the map from labeled configurations to unlabeled configurations, we say a weak solution X of (1.1) is near-equilibrium if there exists S sine ⊂ U such that P(N ∈ S sine ) = 1 and that P(v(X(t)) ∈ S sine ) = 1, for all t ≥ 0. The motivation is to relate the solutions constructed in [17] to that of this paper. In [17], a near-equilibrium solution is constructed for each initial condition x in ∈ S sine .…”

“…The motivation is to relate the solutions constructed in [17] to that of this paper. In [17], a near-equilibrium solution is constructed for each initial condition x in ∈ S sine .…”

“…This starts with [23] constructing the equilibrium process as an L 2 Markovian semigroup. Combining the theory of Dirichlet form and the theory of determinantal or Pfaffian point processes, [17,18] obtain the weak existence for near-equilibrium configurations. The recent work of [20] further shows the strong existence and pathwise uniqueness at equilibrium.…”

Determinantal Martingales and Correlations of Noncolliding Random Walks

“…where W j (·), 1 ≤ j ≤ N are independent one-dimensional standard BMs [7,38,51,18,21,27,32,52,41,19,42,43]. (From now on BM stands for one-dimensional standard Brownian motion and Dyson's BM model with β = 2 is simply called the Dyson model in this paper.)…”