Consider piecewise \(C^r\) domains \(\Omega,\Omega_1, \dots, \Omega_n\) in \(\mathbb{R}^d\), such that \(\overline\Omega\) is the closure of the disjoint union of the \(\Omega_j\)’s. Let \(K\) be the maximum number of faces of \(\Omega_1,\dots,\Omega_n\) which meet at the same point.
Consider then a \(C^r\) embedding from each \(\Omega_j\) into \(\Omega\), and its expansion constant \[ \lambda(f): =\inf_{1\leq j\leq n}\inf_{x\in\Omega_j} \inf_{v\in T_x\Omega} {\bigl |Df(x) \cdot v \bigr|\over |v|}. \] Assuming that \(\lambda(f) >K\), the author proves that \(f\) admits an invariant density, and that any Markov process resulting from a sufficiently small random perturbation of \(f\) also does; moreover, if the invariant density of \(f\) is unique, then it is stochastically stable in \(L^1\) under small random perturbations of \(f\).