LFE–MLFE asymptotic equivalence conjecture

Let $\rho_t$ and $\mu_t$ denote, respectively, the laws at time $t \geq 0$ of solutions to the local-field equation (LFE) and the Markov local-field equation (MLFE). Let $d$ be a suitable metric on $\mathcal{P}(\mathbb{R}^3)$. LFE–MLFE asymptotic equivalence conjecture. For ergodic interacting diffusions with drift of gradient form,

$$\lim_{t \rightarrow \infty} d(\rho_t,\mu_t)=0.$$

This conjecture makes precise the expected agreement between the long-time behavior of the non-Markovian LFE and its Markovian analogue. The paper verifies the claim in the quadratic-potential case, but leaves the stated general ergodic gradient-drift setting open.