arXiv subjects

Larysa Pryhara

Publications and source records attributed to Larysa Pryhara.

3 recordsLinked to original sources

arXiv2017

Wave equation with a coloured stable noise

We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $α\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $σ$-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 Hölder 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.

math.PR↗

arXiv2016

Stochastic wave equation in a plane driven by spatial stable noise

The main object of this paper is the planar wave equation \[\bigg(\frac{\partial^2}{\partial t^2}-a^2\varDelta\bigg)U(x,t)=f(x,t),\quad t\ge0, x\in \mathbb {R}^2,\] with random source $f$. The latter is, in certain sense, a symmetric $α$-stable spatial white noise multiplied by some regular function $σ$. We define a candidate solution $U$ to the equation via Poisson's formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that $U$ is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where $σ$ does not vanish. Finally, we prove that $U$ is a generalized solution to the equation.

math.PR↗

arXiv2016

Approximations for a solution to stochastic heat equation with stable noise

We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process $Z$ with Hurst parameter $H>1/2$ and stability index $α>1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for $Z$, converge to the solution.

math.PR↗