Modern Stochastics: Theory and Applications

For the heat equations driven by a stochastic measure on $[-L,L]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on $\mathbb{R}$ as $L\to \mathrm{\infty }$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the σ-additivity in probability only.

Random functions $\mu (x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu (x)$, $x\in {[0,1]^{d}}$, is proved. Fourier series expansion of $\mu (x)$, $x\in [0,2\pi ]$, is obtained. These results are proved under weaker conditions than similar results in previous papers.