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

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

Problem Statement
This paper investigates the Cauchy problem for the Tricomi equation perturbed by additive space-time Gaussian White Noise. The Tricomi equation, a second-order partial differential equation of mixed elliptic-hyperbolic type, is given by $\partial_{tt}u = t\partial_{xx}u$. It is hyperbolic for $t > 0$, elliptic for $t < 0$, and degenerate at $t = 0$. The specific stochastic problem addressed is:
$$\begin{cases} \partial_{tt}U(t, x) = t\partial_{xx}U(t, x) + W(t, x), & t > 0, x \in \mathbb{R}; \\ U(0, x) = 0, \quad \partial_t U(0, x) = 0, & x \in \mathbb{R}, \end{cases}$$
where $\{W(t, x)\}$ represents space-time Gaussian White Noise. The primary objective is to establish the existence and uniqueness of a square-integrable solution and to analyze its probabilistic and analytic properties, specifically focusing on the behavior in the hyperbolic half-plane ($t > 0$).

Methodology
The authors adopt a Fourier transform approach rather than relying on the explicit fundamental solution of the Tricomi operator, which is known but complex. The methodology proceeds as follows:

1. Fourier Transformation: The spatial variable is transformed, converting the stochastic PDE into a family of ordinary differential equations (ODEs) parameterized by the frequency $\xi$. The homogeneous part of the transformed equation, $\partial_{tt}\hat{v} = -t\xi^2 \hat{v}$, is reduced via the change of variable $\eta = |\xi|^{2/3}t$ to the Airy equation $\partial_{\eta\eta}v = -\eta v$.
2. Representation via Airy Functions: The solution to the transformed ODE is constructed using the linearly independent Airy functions, $Ai$ and $Bi$. The authors derive an integral representation for the solution involving a kernel $C(t, s, \rho)$ defined by these Airy functions.
3. Noise Approximation: The space-time White Noise is approximated by a finite-rank smooth process using a series expansion involving Laguerre functions (in time) and Hermite functions (in space) multiplied by independent standard Gaussian variables.
4. Asymptotic Analysis: The core of the analysis involves a careful study of the asymptotic behavior of the integrals containing the Airy functions, particularly for large arguments (negative real axis) and near zero. This analysis is crucial for establishing the convergence of the series solution and the regularity of the sample paths.

Key Results

• Existence and Uniqueness (Theorem 1.1): The paper proves that the Cauchy problem admits a unique mild random-field solution. This solution is a Gaussian random field represented as a convergent series in $L^2(\Omega, \mathcal{F}, P)$ and almost surely. The solution is given explicitly in terms of integrals of the Airy functions and the noise coefficients. The authors establish that the second moment of the solution is finite.
• Stationarity and Correlation (Corollary 1.2): For any fixed time $t > 0$, the stochastic process $x \mapsto U(t, x)$ is shown to be second-order stationary with zero mean. The covariance function is derived explicitly. Crucially, the authors analyze the behavior of the correlation function $r_t(h)$ for small spatial lags $h$. They prove that $r_t(h) = 1 - \alpha_t |h| + o(|h|)$ as $|h| \to 0$. This linear behavior near the origin implies that the process admits a continuous modification.
• Time Regularity (Corollary 1.3): For any fixed spatial point $x$, the process $t \mapsto U(t, x)$ admits a continuous modification. Using the Gaussian nature of the solution and the Kolmogorov continuity theorem, the authors establish that the time paths possess Hölder regularity of any order less than $1/2$.

Significance and Claims
The paper contributes to the literature on stochastic partial differential equations (SPDEs) with non-strictly hyperbolic operators, a field less explored than standard wave equations.

• Comparison with Wave Equation: A significant finding is that the properties of the solution to this stochastic Tricomi equation (specifically existence, uniqueness, and path continuity) are equivalent to those derived for the corresponding stochastic wave operator. This contrasts with recent findings in [11], where the introduction of lower-order terms in Tricomi-type operators led to the loss of well-defined random field solutions.
• Robustness: The results suggest that the degeneracy of the Tricomi operator at $t=0$ does not prevent the existence of a well-behaved solution under additive white noise perturbation, provided the analysis is conducted via the Fourier-Airy method.
• Technical Contribution: The paper provides a rigorous derivation of the solution's representation and detailed estimates on the asymptotic behavior of integrals involving Airy functions, which are necessary to prove the continuity of sample paths in both space and time.

The authors do not propose new applications or future experimental directions but focus strictly on the mathematical well-posedness and regularity properties of the stochastic Tricomi equation.