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.
The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.
The class of one-dimensional equations driven by a stochastic measure μ is studied. For μ only σ-additivity in probability is assumed. This class of equations includes the Burgers equation and the heat equation. The existence and uniqueness of the solution are proved, and the averaging principle for the equation is studied.
The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.
A one-dimensional stochastic wave equation driven by a general stochastic measure is studied in this paper. The Fourier series expansion of stochastic measures is considered. It is proved that changing the integrator by the corresponding partial sums or by Fejèr sums we obtain the approximations of mild solution of the equation.
A stochastic heat equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure $d\mu (t)$ is investigated in this paper. For the integrator μ, we assume the σ-additivity in probability only. The existence, uniqueness, and Hölder regularity of the solution are proved.