On some stochastic hyperbolic equations with symplectic characteristics
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913490791301120 |
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| author | Bernardi, Enrico Marconi, Leonardo |
| author_facet | Bernardi, Enrico Marconi, Leonardo |
| contents | We study the effect of Gaussian perturbations on
a class of model hyperbolic partial differential
equations with double symplectic characteristics in
low spatial dimensions, extending some recent work
in [5]. The coefficients of our
partial differential operators contain harmonic oscillators
in the space variables, while the noise is additive,
white in time and colored in space.
We provide sufficient conditions on the spectral
measure of the covariance functional describing the
noise that allows for the existence of a random field
solution for the resulting stochastic partial
differential equation.
Furthermore we show how the symplectic structure of the set of
multiple points affects the regularity of the noise
needed to build a measurable process solution.
Our approach is based on some explicit computations for the
fundamental solutions of several model partial
differential operators together with their explicit Fourier transforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some stochastic hyperbolic equations with symplectic characteristics Bernardi, Enrico Marconi, Leonardo Probability Analysis of PDEs We study the effect of Gaussian perturbations on a class of model hyperbolic partial differential equations with double symplectic characteristics in low spatial dimensions, extending some recent work in [5]. The coefficients of our partial differential operators contain harmonic oscillators in the space variables, while the noise is additive, white in time and colored in space. We provide sufficient conditions on the spectral measure of the covariance functional describing the noise that allows for the existence of a random field solution for the resulting stochastic partial differential equation. Furthermore we show how the symplectic structure of the set of multiple points affects the regularity of the noise needed to build a measurable process solution. Our approach is based on some explicit computations for the fundamental solutions of several model partial differential operators together with their explicit Fourier transforms. |
| title | On some stochastic hyperbolic equations with symplectic characteristics |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2409.01956 |