On the eigenfunctions of the complex Ornstein–Uhlenbeck operators

Abstract

Starting from the 1-dimensional complex-valued Ornstein-Uhlenbeck process, we present two natural ways to imply the associated eigenfunctions of the 2-dimensional normal Ornstein-Uhlenbeck operators in the complex Hilbert space L 2 C (µ). We call the eigenfunctions Hermite-Laguerre-Ito polynomials. In addition, the Mehler summation formula for the complex process are shown. MSC: 60H10,60H07,60G15.

On the fourth moment theorem for complex multiple Wiener–Itô integrals

“…The following proposition gives an integration by parts formula of complex Gaussian random variables, whose proof is straight forward. Please refer to Lemma 3.2 of [5] or Lemma 2.3 of [2]. Proposition 2.2 (integration by parts formula).…”

“…We define the divergence operators δ andδ as the adjoint of D andD respectively, with the domains Dom(δ) and Dom(δ) the subsets of L 2 (Ω, H) composed of those elements u such that there exists a constant c > 0 verifying for all F ∈ S, Proof. First, we claim that a complex Hermite polynomials J m,n (z, ρ) [7,5] satisfies that 1) partial derivatives:…”

Complex Wiener-Itô Chaos Decomposition Revisited

“…Especially, when x is a complex Brownian motion, the right hand side of the above equality is equal to the complex multiple Wiener-Itô integral: [9] t 0 t 0 · · · t 0 dx t1 · · · dx tm dx tm+1 · · · dx tm+n .…”

One Dimensional Complex Ornstein-Uhlenbeck Operator

“…− e iθ z ∂ ∂z − e −iθz ∂ ∂z , (1.2) where θ ∈ (− π 2 , π 2 ) is fixed and ∂f ∂z = 1 2 ( ∂f ∂x − i ∂f ∂y ), ∂f ∂z = 1 2 ( ∂f ∂x + i ∂f ∂y ) are the Wirtinger derivatives of f at point z = x + iy with x, y ∈ R. In [8], it is shown that the eigenfunctions are the complex Hermite polynomials and form an orthonormal basis of L 2 (γ) where dγ = 1 2π e − x 2 +y 2 2 dxdy (see Proposition 2.2 below). In this paper, we will firstly show thatL θ can be realized as an unbounded normal operator (see [19, p. 368]) in L 2 (γ) but nonsymmetric when θ ̸ = 0.…”