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On a closed convex set $Z$ in ${ℝ}^N$ with sufficiently smooth (${𝒲}^{2,\infty }$) boundary, the stop operator is locally Lipschitz continuous from ${𝐖}^{1,1}([0,T],{ℝ}^N)\times Z$ into ${𝐖}^{1,1}([0,T],{ℝ}^N)$. The smoothness of the boundary is essential: A counterexample shows that ${𝒞}^1$-smoothness is not sufficient.

Let $(X,d_X)$, $(\Omega ,d_{\Omega })$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_p$ the p-Wasserstein metric with some p ∈ [1,∞), and by ${}_p\left(X\right)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f:\Omega {\to }_p\left(X\right)$, $\omega \mapsto f_{\omega }$, be a Borel map such that f is a contraction from $(\Omega ,d_{\Omega })$ into ${(}_p\left(X\right),W_p)$. Let ν₁,ν₂ be probability measures on Ω with $W_p(\nu ₁,\nu ₂)$ finite. On X we consider the subordinated measures ${\mu }_i={\int }_{\Omega }{f}_{\omega }d{\nu }_i\left(\omega \right)$. Then $W_p(\mu ₁,\mu ₂)\le W_p(\nu ₁,\nu ₂)$. As an application we show that the solution measures ${\varrho }_{\alpha }\left(t\right)$ to the partial...