applications [@Tanner2009; @Chappell2013; @CLST14].

Note that, without losses, the FP operator $\cal L$ has a leading eigenvalue 1 and the solution to Eq. (\[FPK\]) diverges. It is thus necessary - in order to arrive at mathematically and physically meaningful solutions - to account for losses such as naturally occur in wave systems due to wall absorption or absorption in the interior of $\Omega$ or due to radiation through apertures. Absorption can be included in a FP operator formalism by introducing, for example, $$\begin{aligned}
 \label{FP_damp}
{\cal L}_\mu[\rho_0](X) &=& \int e^{-\mu(X')} \delta\left(X - \varphi(X')\right) \rho_0(X')\\ 
&=& e^{-\mu(\varphi^{-1}(X))} \rho_0[\varphi^{-1}(X)],\end{aligned}$$ where we require that the damping coefficient $\mu \ge 0$ is additive along the flow, mimicking distributed losses in many real-life engineering systems. For interior damping, such as for acoustic waves in air, we typically have $\mu(X) = \mu_0\,  L(X)$, where $L$ is the length of a trajectory segment between two reflections on the boundary starting at $X$ and $\mu_0\ge 0 $ is a real constant.

Numerical illustration of averaged response using quantum map models {#sec:QKH}
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We will demonstrate the validity of the relations described in the last section with the help of a simple model in which the transfer operator is simulated 