Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$ $$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$ representing the Fokker-Planck evolution equation for the distribution $u_t=\text{Law}(X_t)$ of the SDE, where $W_t$ denotes the standard $d$-dimensional Brownian motion: $$dX_t = -\nabla V(X_t)\,dt+\sqrt{2} \,dW_t$$ with $\nabla V$ Lipschitz and initial distribution $u_0\in C^2$ so that existence & uniqueness of the flow $(u_t)_{t\geq 0}\in C^{1,2}(\mathbb{R}_{\geq 0}\times\mathbb{R}^d)$ for all time is given. We assume $V$ (and $\log u_0$) is strongly convex with a $L$-Lipschitz perturbation so that the stationary distribution $u_\infty = e^{-V}$ exists and satisfies the log-Sobolev inequality with a constant $\alpha>0$. This further ensures convergence of $u_t$ to $u_\infty$ in KL divergence, 1- and 2-Wasserstein distance (via Talagrand's inequality), weak topology, TV distance (via Pinsker's inequality), existence of spectral gap for $L$, etc. Relevant: this question.
However, I am in need of results that tell me more about what $u_t$ actually looks like. Concretely, I am looking for Lipschitzity guarantees for the ratios $\log(u_t/u_0)$ or $\log(u_t/u_\infty)$. My idea was that if both the initial and stationary distributions were of the form $\exp(-V+\text{Lip})$, then perhaps $u_t$ would be of the same form (maybe with a larger Lipschitz constant) for all time; but I haven't been able to resolve this hypothesis in any direction nor found any good related results - most regularity estimates are given on compact domains. Clearly this doesn't work if the original distribution doesn't share the $\exp(-V)$ part since the variance of even the standard Wiener process evolves over time, although even something like $\log(u_t/u_\infty)\leq C(1+|x|^N)$ for some $N$ would be enough for my purposes. I'm willing to add any assumptions.
