Kernel representation formula: From complex to real Wiener-It? integrals and vice versa

We characterize the relation between the real and complex Wiener-Ito integrals. Given a complex multiple Wiener-Ito integral, we get explicit expressions for the kernels of its real and imaginary parts. Conversely, considering a two-dimensional real Wiener-Ito integral, we obtain the representation formula by a finite sum of complex Wiener-Ito integrals. The main tools are a recursion technique and Malliavin derivative operators. As an application to stochastic processes, we investigate the regularity of the stationary solution of the stochastic heat equation with dispersion.(c) 2023 Elsevier B.V. All rights reserved.