Modern Stochastics: Theory and Applications

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 stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $t\to \infty $ is studied.

A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.

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.