Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf\_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int\_0^1v\left(\gamma(t)\right)\left\|\gamma'(t)\right\|\_E\:{\rm d}t\;\;\;\text{for }x,y\in E,$$ $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space, $X:\Omega\times[0,\infty)\times E\to E$ be a stochastic flow, $$X^x\_t:=X(\;\cdot\;,t,x)\;\;\;\text{for }(t,x)\in[0,\infty)\times E$$ and $$\kappa\_t(x,B):=\operatorname P\left[X^x\_t\in B\right]\;\;\;\text{for }(x,B)\in E\times\mathcal B(E)\text{ and }t\ge0.$$


Assume $$\operatorname E[v(X^x\_t)]\le cv^{\lambda(t)}(x)\;\;\;\text{for all }(t,x)\in[0,\infty)\times E\tag1$$ for some $c>0$ and decreasing $\lambda:[0,\infty)\to[0,1]$. By $(1)$, $$\operatorname E[\rho(X^x\_t,X^y\_t)]\le c\rho(x,y)\tag2$$ for all $x,y\in E$ and $t\in[0,1]$.


Let, $\mathcal M\_1$ denote the set of probability measures on $(E,\mathcal B(E))$, $\operatorname W\_\rho$ denote the Wasserstein metric associated with $\rho$ and $$\mathcal S^1:=\{\mu\in\mathcal M\_1\mid\exists y\in E:(\mu\otimes\delta\_y)\rho<\infty\}.$$ By $(2)$, $$\operatorname W\_\rho(\delta\_x\kappa\_t,\delta\_y\kappa\_t)\le c\operatorname W\_\rho(\delta\_x,\delta\_y)\tag3$$ for all $x,y\in E$ and $t\in[0,1]$.



> 
> Let $t\ge0$. Can we show that $\kappa\_t^\ast$ is $\mathcal S^1$-preserving? Or even that $\kappa\_t^\ast\mathcal M\_1\subseteq\mathcal S^1$?
> 
> 
> 


I'm quite sure that at least the $\mathcal S^1$-preserving claim is true. If $\mu\in\mathcal M\_1$, then we need to show that there is a $y\in E$ with $(\mu\kappa\_t\otimes\delta\_y)\rho\_r<\infty$. Maybe we can pick $y=0$.


**EDIT 1**: Assume $\delta\_x\kappa\_t\in S^1$ for all $x\in E$ and $t\ge0$.


**EDIT 2**: Assume there are nondecreasing $v\_i:[0,\infty)\to(1,\infty)$ with $v\_1(\left\|x\right\|\_E)\le v(x)\le v\_2(\left\|x\right\|\_E)$ for all $x\in E$ and $rv\_2(r)\le \alpha v\_1^\beta(r)$ for all $r>0$ for some $\alpha\ge0$ and $\beta\ge1$. Assume further that $\operatorname E[V^\theta(X^x\_t)]\le\eta v^{\beta\lambda(t)}(x)$ for all $x\in E$ and $t\ge0$.


Then we easily see $\rho(0,x)\le\alpha v^\beta(x)$ for all $x\in E$. Now, since $\lambda$ is decreasing, it must hold $\lambda(t)\to0$ as $t\to\infty$ and hence $$\operatorname W\_\rho(\mu\kappa\_t,\delta\_0)=\int\mu({\rm d}x)\operatorname E[\rho(0,X^x\_t)]\le\alpha\eta\int\mu({\rm d}x)v^{\beta\lambda(t)}(x)\xrightarrow{t\to\infty}1\tag4$$ by monotone convergence for all $\mu\in\mathcal M\_1$ and $t\ge0$. This should yield that $\kappa\_t^\ast$ maps $\mathcal M\_1$ to $\mathcal S^1$ for all $t\ge0$.



Can we show that this transition semigroup preserves a certain Wasserstein space?
There are some issues that I point out in comments, but assuming (3) you would get $\mathcal{S}^1$-preservation easily by convexity of Wasserstein distance, assuming that for at least one $x\in E$ you have $\delta\_x\kappa\_t\in\mathcal{S}^1$.


**1.** Convexity of $\mathrm{W}\_\rho$ enables us to turn (3) into
$$\mathrm{W}\_\rho(\mu\kappa\_t,\nu\kappa\_t) \le c\mathrm{W}\_\rho(\mu,\nu).$$
(Let indeed $\mu,\nu\in\mathcal{S}^1$, and for each $t$ and each $(x,y)$ choose (measurably) optimal transport plans $\eta\_{x,y}^t$ between $\delta\_x\kappa\_t$ and $\delta\_y\kappa\_t$. Let $\zeta$ be an optimal transport plan from $\mu$ to $\nu$; Then $\int \eta\_{x,y}^t d\zeta(x,y)$ is a transport plan from $\mu\kappa\_t$ to $\nu\kappa\_t$, so that
\begin{align\*}
\mathrm{W}\_\rho(\mu\kappa\_t,\nu\kappa\_t) &\le \iint \rho(x',y') d\eta\_{x,y}^t(x',y') d\zeta(x,y) \\
 &\le \int \mathrm{W}\_\rho(\delta\_x\kappa\_t,\delta\_y\kappa\_t) d\zeta(x,y) \\
 &\le \int c\mathrm{W}\_\rho(\delta\_x,\delta\_y) d\zeta(x,y) = c\int \rho(x,y) d\zeta(x,y) = c\mathrm{W}\_\rho(\mu,\nu)
\end{align\*}
as claimed.)


**2.** Then if for some $x\in E$, $\delta\_x\kappa\_t\in \mathcal{S}^1$, for all $\mu\in\mathcal{S}^1$ we have
$$ \mathrm{W}\_\rho(\mu\kappa\_t,\delta\_x\kappa\_t)\le c\mathrm{W}\_\rho(\mu,\delta\_x) <\infty$$
and thus $\mu\kappa\_t\in\mathcal{S}^1$, as you wished.


**3.** You do need the additional assumption on $\delta\_x\kappa\_t$, (1) or (2) are not enough. Take $v\equiv 1$ and let $\kappa\_t$ be a Markov kernel sending $\delta\_x$ to some distribution with infinite first moment, translated by $x$. Then you have obviously (1) and (2), but you do not have $\mathcal{S}^1$ preservation.


**4.** You cannot expect to have $\mathcal{M}\_1$ sent into $\mathcal{S}^1$ without additional assumption: the trivial dynamics $\delta\_x\kappa\_t=\delta\_x$ satisfies your assumption.


