The Tricomi equation in the hyperbolic half plane under additive space-time Gaussian White Noise perturbation

Abstract

We study the Cauchy problem for the Tricomi equation perturbed by space-time Gaussian White Noise. To prove existence and uniqueness of the solution, we employ a Fourier transform approach that allows to obtain its representation in terms of certain integrals of the Airy functions. Then, via a careful analysis of the asymptotic behaviour of those integrals, we obtain all the desired properties of the solution, such as square integrability, continuity of its sample paths and stationarity with respect to the space variable. In relation to that stationarity, we also provide the precise description of how the correlation function behaves for small values of the space-lag. We also remark that, in contrast to the findings of the recent paper [11], the properties of the solution to our stochastic Tricomi equation are equivalent to those derived in studying the corresponding problem for the wave operator.