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I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but this is probably not the right term and my quest for references would be more successful if I knew what they are called.

What I mean is that there is an $n$-dimensional process $(Y_t)_{t\geq 0}$ that is, say, ergodic, but not necessarily stochastically stable. It then drives the main system $(X_t)_{t\geq 0}$, which is a solution to an $m$-dimensional Itô SDE

$$dX_t=f(X_t,Y_t,t)dt+g(X_t,Y_t,t)dW_t,$$

where $Y_t$ is possibly driven by a Wiener process $V_t$ that may be correlated with $W_t$. Let us assume that the standard conditions for existence and uniqueness of solutions to the SDE hold.

Suppose that $f(0,z,t)=g(0,z,t)=0$ for all $z\in\mathbb{R}^n$ and $t\geq 0$. What can be said about the stochastic stability of the equilibrium solution $X_t\equiv 0$?


One way to deal with the system $(X_t)_{t\geq 0}$ that I can see is to include the dependence on $Y_t$ into the explicit time dependence of the coefficients, and thereby reduce it to the standard non-autonomous case studied in the standard literature on stochastic stability. However, I don't know whether this is the standard way to do it, and it might be hard to find a suitable (time-dependent) Lyapunov function.

Another way would be to study the stability of $X_t$ by studying the system

$$dX^y_t=f(X^y_t,y,t)dt+g(X^y_t,y,t)dW_t$$

that treats $y$ as a constant parameter, and there might be ways to exploit the ergodicity of $(Y_t)_{t\geq 0}$. Let $S=\{y\in\mathbb{R}^n|X^y_t\equiv 0 \text{ is stochastically stable}\}$. Is there a relation between $\mu(S)$, where $\mu$ is the ergodic probability measure of $Y_t$, and the stability of the "open" system above? Intuitively, if $Y_t$ spends most of its time in $S$, it seems that the "open" system should be stable.

Yet another scenario could be if there is a separation of time-scales such that $(Y_t)_{t\geq 0}$ changes very slowly compared to $(X_t)_{t\geq 0}$, for some choice of $y$, $X^y_t$ has similar behavior as $X_t$, and it seems that the stability of $X_t$ should be inherited from $X^y_t$.

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