In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1,$$ where $\sigma:\mathbb R^d\to\mathbb R^{d\times k}$, $$b:=\frac{\Sigma+U}2\nabla\ln p+\frac12\nabla\cdot\Sigma,\tag2$$ $\Sigma:=\sigma\sigma^\ast$ and $(W_t)_{t\ge0}$, $U\in\mathbb R^{d\times d}$ is any anti-symmetric matrix is a $\mathbb R^k$-valued standard Wiener process, has an unique invariant measure with density $p$ with respect to the Lebesgue measure on $\mathbb R^d$.
Now, in my application, I'm working on $[0,1)^d$ instead. So I have a given density $p:[0,1)^d\to[0,\infty)$. In my application, it works nicely to consider $[0,1)^d$ with toroidal boundary and the standard MCMC sampling approach is Metropolis-Hastings with proposal kernel given by the [wrapped normal distribution](https://en.wikipedia.org/wiki/Wrapped_normal_distribution).
What I would like to do now is defining a (continuous-time) Markov process, with values in $[0,1)^d$, with unique invariant measure with density $p$ with respect to the restriction of the Lebesgue measure on $\mathbb R^d$ to $[0,1)^d$. Is this possible?
One might think that we can simply use $(1)$ with toroidal wrapping, but as [this paper](https://arxiv.org/abs/1705.00296) shows, the resulting process is not even Markov anymore.
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EDIT1
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I think the question is trickier than one might think at first glance. It seems like the Markov property of $(1)$ will not be maintained under the transformation by $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor\tag3.$$ A counterxample is given [here](https://math.stackexchange.com/q/4925093/47771).
However, we got the following general result: Assume $(E,\mathcal E)$ is a measurable space and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$. Now, given another measurable space $(\tilde E,\tilde{\mathcal E})$ and an $(\mathcal E,\tilde{\mathcal E})$-measurable $\varphi:E\to\tilde E$ satisfying
1. $\varphi(E)=\tilde E$;
2. If $t\ge0$ and $\tilde f:\tilde E\to\mathbb R$ is bounded and $\tilde E$-measurable, then $$\kappa_t(\tilde f\circ\varphi)=\tilde g\circ\varphi\tag4$$ for some bounded $\tilde{\mathcal E}$-measurable $\tilde g:\tilde E\to\mathbb R$,
then $\tilde g$ is uniquely determined and hence $$\tilde\kappa_tf:=\tilde g\tag5$$ is well-defined. If now $(X_t)_{t\ge0}$ is a Markov process, with respect to some filtration $(\mathcal F_t)_{t\ge0}$, with transiton semigroup $(\kappa_t)_{t\ge0}$, then $$\tilde X_t:=\varphi\circ X_t\;\;\;\text{for }t\ge0$$ is a Markov process with respect to the same filtration $(\mathcal F_t)_{t\ge0}$ with transition semigroup $(\tilde\kappa_t)_{t\ge0}$.
We clearly would want to apply this for $E=\mathbb R^d$, $\tilde E=[0,1)^d$, $\varphi=\iota$ and $(X_t)_{t\ge0}$ given by $(1)$. However, with that I end up with $$\operatorname E\left[f\left(\tilde X_{s+t}\right)\mid\tilde X_s\right]=\operatorname E\left[\left(\kappa_t(f\circ\iota)\right)(X_s)\mid\tilde X_s\right]\tag6.$$ But this will not be possible in general.
**EDIT 2**:
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If $X$ is a Lévy process, we can actually show that $\tilde X$ is again a time-homogeneous Markov process. The fact that the increments of $X$ are independent in that case, is crucial for the proof. Now, a Langevin-like process $X$ - at least in the form $(1)$ + $(2)$ - will not be a Lévy process. So, the currently given answer does actually *not* answer my question. We still don't verified that there is a Markov process with prescribed distribution. While the paper referenced in the answer seems to claim the existence, it is unfortunately not accessible to me due to the heavy amount of differential geometry I'm not aware of. Also, while it's nice to have an expression for the generator, it is still not sufficient to simulate it on a computer, for which I would need to know the precise shape of the transition semigroup. I'm sure that in the present, way simpler setting, a direct construction is possible. Any advice and or answer on that would be highly appreciated.
**EDIT 3**
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[Meanwhile I was able to prove](https://math.stackexchange.com/a/4926007/47771) that the toroidal wrapping of *any* Lévy process is a Markov process and I could identify its transition semigroup in terms of the transition semigroup of the original process. However, the open question is whether the same can be shown for the toroidal wrapping of a general SDE (under the condition of a periodic drift and a periodic diffusion coefficient); at least for the one with Langevin drift and constant diffusion coefficient.