Wave equation with a coloured stable noise

Larysa Pryhara; Georgiy Shevchenko

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arXiv · 1707.08415

Wave equation with a coloured stable noise

Larysa Pryhara·Georgiy Shevchenko

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Abstract

We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $\alpha\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $\sigma$-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in $\mathbb R^3$ with a random source generated by $Z^H$. We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is H\"older continuous of any order up to $(3H-1)\wedge 1$. In the case where $H\in(2/3,1)$, we show further that the modification is absolutely continuous.

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Larysa Pryhara, Georgiy Shevchenko. 2017-07-26. Wave equation with a coloured stable noise. https://arxiv.org/abs/1707.08415

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