date: '2020-04-11'
title: Pathwise Unique Solutions and Stochastic Averaging for Mixed Stochastic Partial Differential Equations Driven by Fractional Brownian Motion and Brownian Motion
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Introduction {#se-1}
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It is widely known that there are many phenomena the well-studied theory of semimartingales cannot describe. For example, telecommunication connections, climate, weather derivatives and other objects have long memory [@decreusefond1998fractional; @Mandelbrot1968; @Taqqu2003fractional]. Brownian motion (Bm) with independent increments which has no memory turns out to be insufficient to describe this effect. Another example could be that the concept of turbulence in hydrodynamics can be described with the help of stationary (dependent) increments [@Monin1967Statistical]. Thus, the long-range dependence properties of fracional Brownian motion (fBm) make this process a suitable candidate to describe this kind of phenomena.

For $H \in  (0, 1)$, a continuous centered Gaussian process $\beta^H =(\beta^H (t))_{t \ge 0}$ with the covariance function $$\mathbb{E} [\beta^{H}(t) \beta^{H}(s)]=\frac{1}{2}\left(|t|^{2 H}+|s|^{2 H}-|t-s|^{2 H}\right), \qquad t, s \ge 0,$$ is called one-dimensional fBm [@Mandelbrot1968] with the Hurst parameter $H$. An fBm differs significantly from an Bm and semimartingales. It is characterized by the stationarity of its (dependent) increments and long-memory property only for $H\in(\frac{1}{2},1)$. In the case $H\in(0,\frac{1}{2})$ it is a process with short 