Let $\mu_0$ be a log-concave distribution on $\mathbb R^d$ and let $f:\mathbb R^d \to \mathbb R$ be $C^2$. Let $x_0$ be sampled uniformly at random from a log-concave distribution $\mu_0$, meaning that $\mu_0$ has density of the form $p_0(x) = e^{-U(x)}dx$, where $U$ is of class $\mathcal C^2$ and $Hess U \succeq cI_d$ for some $c>0$. Let $x(t):=X^{(t)}(x_0)$ be the gradient-flow of $f$ started at $x_0$, i.e
$$ \begin{split} \dot x(t) &= -\nabla f(x(t)),\\ x(0) &= x_0. \end{split} $$
Question. Under what general conditions on $f$ can one hope for the distribution of $\mu_t$ to be log-concave ?
Observation. Let $\mu_t$ be the probability distribution of $x(t)$. Then $\mu_t = X^{(t)}|_{\#} \mu_0$.
One could ask a similar question on a general riemannian manifold (with an appropriate notion of "gradient") instead of $\mathbb R^d$.
