2005 September, Osaka University

Infinite-dimensional SDE related to random matrix theory: a new method to construct interacting Brownian particles via

Dirichlet form theory

Graduate School of Mathematics, Kyushu University, Hirofumi OSADA

Let S be a connected closed set in ℝ^{d}. Interacting Brownian particles (IBPs) are (infinitely dimensional) diffusion processes with free potential Φ : S → ℝ and interaction potential Ψ : ℝ^{d} → ℝ ∪ {∞} (Ψ(x) = Ψ(−x)) . These diffusions are given by the following infinitely dimensional stchastic differential equations (SDEs):

dX_{t}^{i} = dB_{t}^{i} − \frac{1}{2}∇Φ(X_{t}^{i})dt − \frac{1}{2} ∑_{j∈ℤ, j≠i} ∇Ψ(X_{t}^{i} − X_{t}^{j})dt  (i ∈ ℤ)  (1)

In SDE(1) we label the particles and the state space of diffusion is (ℝ^{d})^{ℤ}.

The purpose of this talk is to present a method to solve SDEs describing IBPs. Our method is an application of Dirichlet form theory. We are in particular motivated by the following three SDEs, that will be solved by using our method. All of them are related to random matrix theory.

Dyson’s model: Let S = ℝ.

dX_{t}^{i} = dB_{t}^{i} + [ ∑_{j∈ℤ, j≠i}^{∞} \frac{1}{X_{t}^{i} − X_{t}^{j}}] dt  (i ∈ ℤ),  (Dy)

Bessel kernel: Let S = [0, ∞). Let α > −1 be a constant.

dX_{t}^{i} = dB_{t}^{i} + [ ∑_{j∈ℕ, j≠i}^{∞} \frac{1}{X_{t}^{i} − X_{t}^{j}}]dt + [−\frac{1}{2} + \frac{α}{2}(X_{t}^{i})^{α−1}] dt  (i ∈ ℕ),  (Be)

exponential kernel: Let S = ℂ. Let Z_{t}^{i} = X_{t}^{i} + \sqrt{−1}Y_{t}^{i} ∈ ℂ. Let B_{t}^{i} be a 2-D Brownian motion.

dZ_{t}^{i} = dB_{t}^{i} + [ ∑_{j∈ℤ, j≠i}^{∞} \frac{Z_{t}^{i} − Z_{t}^{j}}{|Z_{t}^{i} − Z_{t}^{j}|^{2}} ] dt − Z_{t}^{i}dt  (i ∈ ℤ),  (exp)

We remark interaction potentials of these diffusions have logarithmic fuctions. So the stationary measures are not Gibbs measures; they are Fermion meaures as we will introduce later. The difficulty of the problems is the presence of singular drifts coming from the log potentials.

When we use a Dirichlet form approach to IBPs, it is convenient to unlabel the particles. So we recall the definition of configurations on S. Let

Θ = {θ = ∑_{i} δ_{x_{i}} ; {x_{i}} are sequences in S such that θ({|x| ≤ r}) < ∞ for all r}.

The set Θ is called the configuration space on ℝ^{d} and equipped with the vague topology. From the labeled solution (X_{t}^{i}) we construct Θ-valued diffusion 𝕏 as follows;

𝕏_{t} = ∑_{i∈ℤ} δ_{X_{t}^{i}}  (2)

Then the diffusion can have invariant probability measures.

Let μ be a probability measures on Θ. We assume the bilinear form (E, D, L^{2}(Θ, μ)) is closable, and its closure, denoted by the same symbol (E, D, L^{2}(Θ, μ)), is a quasi regular Dirichlet form. Here

E(f, g) = ∫_{Θ} 𝔻[f, g]dμ,  𝔻[f, g](θ) = ∑_{i} \frac{1}{2} \frac{∂f}{∂x_{i}} · \frac{∂g}{∂x_{i}} ,  (3)

D = ⋃_{r=1}^{∞} D_{r}, D_{r} = {f ∈ L^{2}(Θ, μ); f is π_{r}-measurable, smooth E(f, f) < ∞}.

Moreover, we assume

Cap({θ({x}) ≥ 2 for some x}) = 0  (4)

This means particles never collide each other. Hence, one can derive a labeled dynamics {X_{t}^{i}}_{i∈ℤ} from the unlabeled dynamics 𝕏_{t} = ∑_{i∈ℤ} δ_{X_{t}^{i}} by giving a labeling at time zero.

Let μ_{x}^{∘} denote the Palm measure of μ at x ∈ S, that is, μ_{x}^{∘} = μ( · |θ({x}) = 1)ρ^{1}(x), where ρ^{1} is the 1-correlation function of μ. Let (ℙ_{θ}, 𝕏_{t}) denote the Θ-valued diffusion associated with the Dirichlet space (E, D, L^{2}(Θ, μ)). We write 𝕏_{t} = ∑_{i∈ℤ} δ_{X_{t}^{i}} as before. .

Theorem 1. Assume that there exists a b(θ, x) ∈ {L_{loc}^{1}(Θ × ℝ^{d})}^{d} such that

∫_{Θ×ℝ^{d}}{∇_{x}f(θ, x)}μ_{x}^{∘}(dθ)dx = − ∫_{Θ×ℝ^{d}} f(θ, x)b(θ, x)μ_{x}^{∘}(dθ)dx for all f ∈ D ⊗ C_{0}^{∞}(ℝ^{d}).  (A.1)

Let b̂((y^{j})_{j∈ℤ}, x) = b(θ, x), where θ = ∑_{j∈ℤ} δ_{y^{j}}. Then (X_{t}^{i})_{i∈ℤ} under ℙ_{ξ} is a (weak) solution of

dX_{t}^{i} = dB_{t}^{i} + ∑_{j∈ℤ, j≠i}^{∞} b̂((X_{t}^{j} − X_{t}^{i})_{j∈ℤ}, X_{t}^{i}) dt, X_{0}^{i} = x^{i} (i ∈ ℤ)  (5)

for q.e. ξ = ∑_{i} δ_{x^{i}}.

We say μ is a Fermion measure with kernel K if the correlation functions of μ is given by

ρ_{n}(x_{1}, …, x_{n}) = det(K(x_{i}, x_{j}))_{i,j=1,... ,n}  (6)

It is known that for given kernel there exists at most one associated Fermion measure. The invariant probability measures of unlabeled diffusions of our SDEs are Fermion measures with kernels as follows:

Example 1 (sine kernel). Let d = 1. Let 0 < ρ̄ < 1 and define the sine kernel K_{sin} by

K_{sin}(t) = \frac{1}{2π} ∫_{|k|≤πρ̄} e^{\sqrt{−1}kt} dk = \frac{ρ̄sin(πx)}{πx}.

Example 2 (Bessel kernel). Let us take a state space of particles as [0, ∞) in stead of ℝ. Let J_{α} be the Bessel function of order α. Define the Bessel kernel K_{Be} by

K_{Be}(x, y) = \frac{J_{α}(\sqrt{x}) · \sqrt{y} · J′_{α}(\sqrt{y}) − J_{α}(\sqrt{y}) · \sqrt{x} · J′_{α}(\sqrt{x})}{2(x − y)}.

Example 3 (exponential kernel in ℂ = ℝ^{2}). Let us take a state space of particles as ℂ = ℝ^{2}. Let the reference measure m to be m(dv) = (1/π)e^{−|v|^{2}}dv Define the exponential kernel K_{exp} by

K_{exp}(v, w) = e^{−v·w̄}.

Here v, w ∈ ℂ. It is known that the resulting Fermion measures are rotation and translation invariant although the kernel K_{exp} is not translation invariant (for any reference measures).

We remark these Fermion measures have logarithmic interaction potential if we regard them as Gibbs measures. So there is no chance that they satisfy the DRL equation. Indeed, the Hamiltonian in (5)

H_{r,η}(θ) = ∑_{x_{i}∈Q_{r}} Φ(x_{i}) + ∑_{x_{i},x_{j}∈Q_{r},i≠j} Ψ(x_{i} − x_{j}) + ∑_{x_{i}∈Q_{r},y_{j}∈Q_{r}^{c}} Ψ(x_{i} − y_{j}),

where π_{r}(θ) = ∑_{i} δ_{x_{i}}, π_{r}(η) = ∑_{i} δ_{y_{i}} and η is an outer condition, does not make sense in infinite volume.

Theorem 2. The Fermion measures above satisfy (A.1). Moreover, (X_{t}^{i})_{i∈ℤ} under ℙ_{ξ} is a (weak) solution of (Dy), (Be) and (exp), respectively.

Remark 1. As for the SDE (exp) we obtain a translation invariant solution in the sense that if the initial distribution comes from the Fermion measure μ_{exp} with the exponentioal kernel and the same reference measure as in Example 3. This is because μ_{exp} and the Dirichlet form are translation invariant. Nevertheless, the coefficients of SDE (exp) is not translation invariant.