Assume we are given two densities, $p_0$ and $p_1$ on $\mathbb{R}^d$, and define (up to the normalization constant) the interpolation $p_t \propto p_0^{1-t} p_1^t$, which interpolates between $p_0$ and $p_1$.

I am looking for bounds of the form (under mild conditions, such as finite second moment of $p_0$ and $p_1$) of $W_2(p_t, p_s) \leq C \vert t - s \vert$, where $W_2$ denotes the Wasserstein-2 distance.

The Wasserstein-2 distance between measures $\mu$ and $\nu$ is defined via $W_2^2(\mu, \nu) = \inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^{2d}} \Vert x - y \Vert^2 d\pi(x,y) $, where $\Pi(\mu,\nu)$ is the set of all probability measures on $\mathbb{R}^{2d}$ with first marginal $\mu$ and second marginal $\nu$.

If that is not possible, a bound of the form $W_2(p_t, p_s) \leq \int_s^t m(r)dr$ for an $L^1((0,1))$ function $m$ would also be appreciated.