# Regularity estimates of the Fokker-Planck equation on the torus

The following comes from Pages 37-41 of the following paper:

https://arxiv.org/abs/1509.02505

Let $$\mathbb{T}^d$$ be a $$d$$-dimensional torus and let $$C^{n+ \alpha}$$ be the $$n$$th order $$\alpha$$-Holder space defined in the usual way. We define functions $$V \in C([0,T], (C^{n+ 1+\alpha}(\mathbb{T}^d))^d)$$ and $$c \in L^{\infty} \big([0,T], \big( C^{n+ \alpha}(\mathbb{T}^d) \big)' \big)$$. Let $$\rho$$ be a measure on $$\mathbb{T}^d$$. We consider, in the distribution sense, the PDE given by $$\partial_t \rho + \Delta \rho - \text{div} (\rho V)- \text{div} (c) = 0, \quad \quad \text{ on } [0,T],$$ with initial condition $$\rho(0)=0$$. Lemma 3.5 of the paper states that there exists a unique solution which satisfies $$\sup_{t \in [0,T]} \| \rho (t) \|_{( C^{n+ 1+\alpha}(\mathbb{T}^d) )'} \leq C \sup_{t \in [0,T]} \| c (t) \|_{( C^{n+ \alpha}(\mathbb{T}^d) )'} < \infty.$$

My question is whether or not this holds for $$\mathbb{R}^d$$ instead of $$\mathbb{T}^d$$. I am not able to check in the paper as most of the technical details are hidden. Thanks.