Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion

We developed a random-field theory for the Euler–Poisson–Darboux equation on Euclidean space under additive Gaussian forcing that was white in the evolution variable and spatially homogeneous. An explicit two-parameter Bessel Green kernel provided the basic deterministic propagator. Sharp high-frequency bounds for its integrated energy yielded a necessary and sufficient Dalang criterion for the existence of the stochastic convolution and showed that the singular Darboux coefficient did not change the wave-type spectral threshold. A dissipative energy identity for the Green kernel provided temporal increment estimates which, together with the spatial Fourier bounds, led to joint space–time Hölder regularity under a natural strengthened spectral moment. We also distinguished between the two initial traces: the Dalang condition guaranteed the mean-square displacement trace, whereas a mean-square zero-velocity trace required additional spectral regularity; finite total spectral mass provided a simple sufficient condition. Space–time white noise and Riesz-colored noise were treated explicitly. Finally, the mean solution was identified with the Darboux spherical mean, Brownian first-exit sampling, and the Pizzetti expansion.