Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion
- 1Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Abstract
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.
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BibTeX
@article{almualim2026stochastic,
title = {{Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion}},
author = {Anwar Almualim and Fethi Bouzeffour and Mhamed Eddahbi},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100782},
url = {https://www.mdpi.com/2075-1680/15/10/782},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Stochastic Euler–Poisson–Darboux Equations and a Sharp Dalang Criterion
AU - Anwar Almualim
AU - Fethi Bouzeffour
AU - Mhamed Eddahbi
PY - 2026
DA - 2026/10/18
JO - Axioms
DO - 10.3390/axioms15100782
UR - https://www.mdpi.com/2075-1680/15/10/782
AB - 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.
SN - 2075-1680
ER -