Determinantal process starting from an orthogonal symmetry is a Pfaffian process

When the number of particles $N$ is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index $\nu > -1$ (BESQ$^{(\nu)}$) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The $2 \times 2$ skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, $N \delta_0$, and by the equivalence between the noncolliding BESQ$^{(\nu)}$ and that of the noncolliding squared generalized meander starting from $N \delta_0$.

💡 Research Summary

The paper studies one‑dimensional non‑colliding diffusion processes – the non‑colliding Brownian motion (BM) and the non‑colliding squared Bessel process with index ν > −1 (BESQ^{(ν)}) – when the number of particles N is finite. It is already known that for any fixed deterministic initial configuration ξ = ∑{j=1}^{N}δ{x_j} these processes are determinantal: all multi‑time correlation functions can be written as determinants built from a scalar kernel K(s,x; t,y) given by contour integrals (formulas (1.2) and (1.3)).

The new contribution is to consider random initial configurations whose law possesses orthogonal symmetry (β = 1) rather than unitary symmetry (β = 2). The orthogonal‑symmetric initial measures are denoted μ^{(1)}{N,σ²} for the BM case and μ^{(1,a)}{N,σ²} for the BESQ^{(ν)} case (the latter depends on a parameter a with −1 < a ≤ ν). The main result is that, under these orthogonal‑symmetric initial laws, the processes cease to be determinantal and become Pfaffian: every multi‑time correlation function is a Pfaffian of a 2 × 2 matrix‑valued kernel.

The key technical device is an equivalence (Lemma 2.1) that relates the orthogonal‑symmetric processes to time‑reversed, dilated versions of the same processes started from the deterministic configuration N δ₀ (all particles at the origin). Define the time‑dependent scaling factor c_{σ²}(t)=σ²/(σ²+t). Then for the BM case

 (Ξ(t), P_{μ^{(1)}{N,σ²}}) = { c{σ²}(t) ∘ Ξ_{σ²}(σ² c_{σ²}(t)) },

where Ξ_{σ²} denotes the non‑colliding BM started from N δ₀ with a finite duration σ². For the BESQ^{(ν)} case, if a and κ satisfy a = ν − κ/2, an analogous identity holds with the non‑colliding BESQ^{(ν,κ)} process (a temporally inhomogeneous generalised meander). These identities show that the orthogonal‑symmetric processes are precisely time‑reversals of the processes for which explicit determinantal kernels are known, but after the reversal the scalar kernel becomes antisymmetric in the time arguments.

Using the known determinantal kernels for the origin‑started processes and applying the time‑reversal and scaling, the authors derive explicit expressions for the 2 × 2 skew‑symmetric kernels A(s,x; t,y; σ²) (formula (1.11)) and A^{(ν,κ)}(s,x; t,y; σ²) (formula (1.12)). The diagonal entries A_{11} and A_{22} are essentially the extended Hermite or Laguerre kernels with a time‑dependent scaling, while the off‑diagonal entry A_{12} encodes the antisymmetry required for a Pfaffian.

Consequently, for any collection of observation times 0 < t₁ < … < t_M and any subsets of particles, the multi‑time correlation functions satisfy

 ρ_{μ^{(1)}_{N,σ²}}(t₁,…,t_M) = Pf