Consider the diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ on $\mathbb R^d$. Denote the solution starting at $x\in\mathbb R^d$ by $X^x$. Let $$\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]\;\;\;\text{for }(x,B)\in\mathbb R^d\times\mathcal B(\mathbb R^d)\text{ and }t\ge0$$ denote the transition semigroup.
Now consider the wrap around $$\iota:\mathbb R\to[0,1)\;,\;\;\;x\mapsto x-\lfloor x\rfloor=x\operatorname{mod}1$$ and let $$(\iota(x))_i:=\iota(x_i)\;\;\;\text{for }x\in\mathbb R^d.$$ Assume $b(x+k)=b(x)$ and $\sigma(x+k)=\sigma(x)$ for $k\in\mathbb Z^d$ and $x\in\mathbb R^d$.
Now let $$\tilde X^x_t:=\iota(X^x_t)\;\;\;\text{for }(t,x)\in[0,\infty)\times[0,1)^d$$ and $$\tilde\kappa_t(x,B):=\operatorname P\left[\tilde X^x_t\in B\right]\;\;\;\text{for }(x,B)\in[0,1)^d\times\mathcal B([0,1)^d)\text{ and }t\ge0.$$ We easily see that $$\tilde\kappa_t(x,\;\cdot\;)=\iota(\kappa_t(x,\;\cdot\;))\tag2$$ for all $x\in[0,1)^d$ and $t\ge0$.
Assume $\mu$ is a probability measure on $\mathbb R^d$ and $\mu$ is invariant with respect to $(\kappa_t)_{t\ge0}$.
Question: Can we show that the pushforward $\iota(\mu)$ is invariant with respect to $(\tilde\kappa_t)_{t\ge0}$?
This claim seems to be made in Proposition 1-i of this paper. However, I don't get why it holds.
Let $t\ge0$. Since $\mu$ is $\kappa_t$ invariant we have $\mu\kappa_t=\mu$. Building the pushforward under $\iota$ on both sides yields $$\int\mu({\rm d}x)\kappa_t(x,\iota^{-1}(B))=\iota(\mu\kappa_t)(B)=\iota(\mu)(B)\tag3$$ for all $B\in\mathcal B([0,1)^d)$. This is not precisely what we need. On the other hand, $$(\iota(\mu)\tilde\kappa_t)(B)=\int\mu({\rm d}x)\kappa_t(\iota(x),\iota^{-1}(B))\tag4$$ for all $B\in\mathcal B([0,1)^d)$. It seems like $\iota$ is occurring one time too often ... What am I doing wrong here?
EDIT: In the paper, they are assuming that $\kappa_t$ and $\mu$ both have densities with respect to the $d$-dimensional Lebesgue measure. However, I don't see how this assumption helps.
EDIT2: Maybe my definition of $\tilde\kappa_t$ is wrong. Maybe it should be $$\tilde\kappa_t(x,B):=\kappa_t(\iota^{-1}(x),\iota^{-1}(B))=\sum_{k,\:l\:\in\:\mathbb Z^d}\kappa_t(k+x,l+B)\tag5$$ for $(x,B)\in[0,1)^d\times\mathcal B([0,1)^d)$ instead ... However, even with this definition I don't see why the claim holds.
(I'm abusing notation a bit in $(5)$, since $\iota^{-1}(x)$ is actually the set $\bigcup_{k\in\mathbb Z^d}\{k+x\}$.)