Equations driven by general stochastic measures

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.

Stochastic parabolic equation driven by a σ-finite stochastic measure in the interval [0,T] × R is studied. The only condition imposed on the stochastic integrator is its σ-additivity in probability on bounded Borel sets. The existence, uniqueness, and Hölder continuity of a mild solution are proved. These results generalize those known earlier for usual stochastic measures.

We consider the stochastic heat equation on [0; T] x R in the mild form driven by a general stochastic measure m, for m we assume only sigma-additivity in probability. The time-averaging of the equation is studied, we estimate the rate of uniform a. s. convergence to the solution of the averaged equation.

This paper considers stochastic measures, i.e., sets of functions given on the Borel sigma-algebra in [0, 1]d sigma-additive with respect to probability. It is shown that realizations of continuous random functions generated by stochastic measures belong to the Besov spaces under some general sufficiently assumptions.

We prove a theorem on the continuity with respect to a parameter and an analogue of Fubini's theorem for integrals with respect to a general stochastic measure defined on Borel subsets of $ \mathbb{R}$. These results are applied to study the stochastic heat equation considered in a mild as well as in a weak form.

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.