Motivated by Applied Physics and Photonics studies of random resonators, we study in the stochastic part of this paper random acoustic operators in non-smooth bounded domains $G \subset \mathbb{ R}^d$ and introduce m-dissipative impedance boundary conditions containing (eigenfunction) fractional Gaussian fields. The deterministic part of the paper constructs and studies the spaces of pointwise multipliers on Lipschitz boundaries $\partial G$, as well as the spaces of Sobolev (distribution-type) multipliers on boundaries $\partial G$ of better regularity. These multipliers are used as generalized impedance coefficients $\zeta$ in impedance boundary conditions $\zeta p = \mathbf{ n} \cdot \mathbf{ v}$ accompanying the 1st order acoustic system. We study the m-dissipativity of associated acoustic operators and the discreteness of their spectra aiming the main efforts on the weakest possible assumptions on the regularity of $\zeta$. In order to pass from deterministic results to the randomization, we introduce fractional Gaussian fields (FGFs) on Lipschitz boundaries $\partial G$ and study their regularity. To this end, we prove that a rough Weyl-type asymptotics takes place for the Laplace-Beltrami eigenvalues on arbitrary compact Lipschitz boundary $\partial G$.
Categories: math.PR math.AP math.SP