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... Some other equations driven by Wiener process are considered in [4,6] and [20]. Different types of equations with general stochastic measures are investigated in [2,3,14,19] and [16]. ...

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.

... The averaging principle for equations driven by general stochastic measures is considered in [6,22,23,26,30]. ...

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.

This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases.
General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed.
The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.

We study the one-dimensional stochastic heat equation in the mild form driven by a general stochastic measure $\mu$, for $\mu$ we assume only $\sigma$-additivity in probability. The time averaging of the equation is considered, uniform a. s. convergence to the solution of the averaged equation is obtained.

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

The stochastic heat equation on [0, T] x R driven by a general stochastic measure is investigated. Existence and uniqueness of the solution is established. Hölder regularity of the solution in time and space variables is proved.

We consider the solution {u(t, x); t ≥ 0, x ∈ R) of a system of d linear stochastic wave equations driven by a d-dimensional symmetric space-time Lévy noise. We provide a necessary and sufficient condition on the characteristic exponent of the Lévy noise, which describes exactly when the zero set of u is non-void. We also compute the Hausdorff dimension of that zero set when it is non-empty. These results will follow from more general potential-theoretic theorems on the level sets of Lévy sheets.

In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path space. Some Ito formulae (or change of variables formulae) are given for smooth functions of a fractional Brownian motion or some processes related to a fractional Brownian motion. A stochastic integral of Stratonovich type is defined and the two types of stochastic integrals are explicitly related. A square integrable functional of a fractional Brownian motion is expressed as an infinite series of orthogonal multiple integrals. Research partially funded by NSF Grant DMS 9623439

Averaging principle is a powerful tool for studying qualitative analysis of nonlinear dynamical systems. In this paper, we will establish an averaging principle for stochastic Korteweg-de Vries equation under a general averaging condition. With the help of this averaging principle, we can establish an effective approximation for the solution of stochastic Korteweg-de Vries equation, this can tell us the asymptotic behavior of the solution and make the interaction between nonlinearity, uncertainty and multiple scales more clear. In order to obtain this averaging principle, we need to establish the smoothing effect for the third order operator.

The Fourier series and Fourier–Haar series are introduced for general stochastic measures. The convergence of partial sums of these series and the absolute continuity of a stochastic measure are studied. An application is given for the convergence of solutions of the stochastic heat equation.

The Cauchy problem for the wave equation on the line driven by a general stochastic measure is studied. The existence, uniqueness, and Hölder regularity of the mild solution are proved. The continuous dependence of the solution on the data is established.

This article deals with averaging principle for stochastic hyperbolic-parabolic equations with slow and fast time-scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved. As a consequence, an effective dynamics for slow variable which takes the form of stochastic wave equation is derived. Also, the rate of strong convergence for the slow component towards the solution of the averaging equation is obtained as a byproduct.

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.

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.

Wave equation for a homogeneous string with fixed ends driven by a stable random noise

- L I Rusaniuk
- G M Shevchenko

L. I. Rusaniuk and G. M. Shevchenko, Wave equation for a homogeneous string with fixed
ends driven by a stable random noise, Teor. Imovirnost. Matem. Statyst., 98 (2018), 163-172;

English transl. in Theory Probab

English transl. in Theory Probab. Math. Statist. Theory
Probab. Math. Statist. 96 (2018), no. 1, 145-157. MR3666878

Wave equation with stable noise

- L I Rusaniuk
- G M Shevchenko

L. I. Rusaniuk and G. M. Shevchenko, Wave equation with stable noise, Teor. Imovirnost.
Matem. Statyst. 96 (2017), 142-154;

- J Duan
- W Wang

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations,
Birkhäuser, Boston, 1992. MR3289240