Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 57 min ago
PreprintarXiv licencearXiv

Wave equation with unbounded terms driven by a stochastic measure

  • Mykhailo Debeliak
  • Vadym Radchenko
Open at sourcePDFDOICite

Abstract

The one-dimensional wave equation with additive noise driven by a stochastic measure is considered. For the stochastic integrator, we assume only the sigma-additivity in probability and that |y|τ is integrable on ℝ for some τ>1/2. The class of stochastic measures includes many important cases of random processes. In the equation, the stochastic measure depends only on the spatial variable. The existence, uniqueness, and Hölder regularity of the mild solution are proved. In the previous publications, similar results were obtained for the case of bounded terms of the wave equation driven by a stochastic measure. In our paper, one term may be unbounded, satisfying the linear growth condition. Nevertheless, we prove that the solution is bounded in this case.

Related

  1. Global Existence and Stability of 3D Stochastic NSEs in Bounded and Unbounded Domains Driven by a Special Multiplicative Wiener Process
    Zdzisław Brzeźniak, Shijia Zhang, Guoli Zhou
    arXivPreprintarXiv licenceFull textProbabilityarXiv
  2. Aleksandrov-Bakelman-Pucci maximum principle for Ln-viscosity solutions of uniformly elliptic equations with unbounded terms: The critical case
    Andrzej Święch
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. Fundamental solutions to Kolmogorov-Fokker-Planck equations with unbounded lower order terms
    Pascal Auscher
    HALPreprintCC BY-NC 4.0Full textAnalysis & PDEsHALarXiv
  4. Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms
    Sallah Eddine Boutiah
    arXivPreprintCC BY-NC-ND 4.0Full textAnalysis & PDEsMathematical physicsarXiv
  5. Density Evolution for Stochastic Dynamical Systems Driven by α-Stable Processes: An Equation-Free Numerical Approach
    Yan Liu, Xu Sun
    arXivPreprintCC BY-NC-ND 4.0Full textNumerical analysis & scientific computingProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{debeliak2026wave,
  title = {{Wave equation with unbounded terms driven by a stochastic measure}},
  author = {Mykhailo Debeliak and Vadym Radchenko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29648},
  eprint = {2610.29648},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29648},
}
RIS
TY  - UNPB
TI  - Wave equation with unbounded terms driven by a stochastic measure
AU  - Mykhailo Debeliak
AU  - Vadym Radchenko
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29648
UR  - https://arxiv.org/abs/2610.29648
AB  - The one-dimensional wave equation with additive noise driven by a stochastic measure is considered. For the stochastic integrator, we assume only the sigma-additivity in probability and that |y|^τ is integrable on ℝ for some τ>1/2. The class of stochastic measures includes many important cases of random processes. In the equation, the stochastic measure depends only on the spatial variable. The existence, uniqueness, and Hölder regularity of the mild solution are proved. In the previous publications, similar results were obtained for the case of bounded terms of the wave equation driven by a stochastic measure. In our paper, one term may be unbounded, satisfying the linear growth condition. Nevertheless, we prove that the solution is bounded in this case.
ER  -