###### tags: `one-offs` `markov chains` `functional inequalities`

# Logarithmic Sobolev Inequalities and Logarithmic Lipschitz Regularity

**Overview**: In this note, I describe some ideas which arose in the study of Logarithmic Sobolev Inequalities in two separate works, and offer some rough interpretations of what the two notions have in common.

## Ollivier's Log-Sobolev Inequality under Ricci Curvature

A little while ago, I was curious about a proposed definition for a "generalised gradient norm"-type object in the work of Yann Ollivier on Ricci curvature of Markov chains, given for $\lambda\geqslant0$ as:

\begin{align}
\left(\mathrm{D}^{\lambda}f\right)\left(x\right)=\sup\left\{ \frac{\left|f\left(y\right)-f\left(z\right)\right|}{\mathsf{d}\left(y,z\right)}\cdot\exp\left(-\lambda\cdot\mathsf{d}\left(x,y\right)-\lambda\cdot\mathsf{d}\left(y,z\right)\right):y,z\in\mathcal{X},y\neq x\right\}.
\end{align}

While the motivation for such an object was reasonably clear, at the time, I couldn't totally follow why this definition is a priori the right one (the later derivations make clear that it works out as they need it to). Ollivier goes on to use this definition to prove a functional inequality (akin to the Logarithmic Sobolev Inequality) for Markov chains which satisfy only a Ricci curvature condition and some additional stability conditions.

## Salez-Tikhomirov-Youssef and Upgrading MLSI to LSE

Separately, I've recently come across a similar notion in a separate, recently-uploaded paper of Salez-Tikhomirov-Youssef, entitled 'Upgrading MLSI to LSI for reversible Markov chains'. In this paper, the authors are interested in translating between two notions of Logarithmic Sobolev Inequality for Markov chains on discrete spaces. They observe that the relevant calculations are particularly nice if one is working with functions which are Lipschitz at a logarithmic scale, in the following sense: say that a positive function $f$ is $r$-regular if for all adjacent pairs $\left(x,y\right)$, it holds that $f\left(x\right)\leqslant r\cdot f\left(y\right)$. Then, any $r$-regular function $f$ satisfies

\begin{align}
\frac{\mathcal{E}\left(f,\log f\right)}{\mathcal{E}\left(f^{1/2},f^{1/2}\right)}\in\left[4,\frac{r^{1/2}-1}{r^{1/2}+1}\cdot\log r\right],
\end{align}

noting that the lower bound holds for arbitrary $f>0$.

Building on this observation, the authors introduce a construction for converting a given positive function $f$ into a function $f_{*}$ which obeys this logarithmic regularity property, and is not too far from the original $f$. In particular, they define

\begin{align}
f_{*,r}\left(x\right):=\max\left\{ r^{-\mathsf{d}\left(x,y\right)}\cdot f\left(y\right):y\in\mathcal{X}\right\} ,
\end{align}

which is not exactly the same as the construction of Ollivier, but nevertheless exhibits some harmony with it. The authors then show that by taking $r$ to have a certain value (depending on the transition properties of the Markov chain in question, but independently of $f$), one can ensure that

\begin{align}
\mathcal{E}\left(f_{*,r}^{1/2},f_{*,r}^{1/2}\right) &\leqslant \frac{4}{3}\cdot\mathcal{E}\left(f^{1/2},f^{1/2}\right) \\
\mathcal{E}\left(f_{*,r},\log f_{*,r}\right) &\leqslant \frac{4}{3}\cdot\mathcal{E}\left(f,\log f\right) \\
\mathrm{Ent}\left(f\right) &\leqslant 2\cdot\mathrm{Ent}\left(f_{*,r}\right).
\end{align}

From here, the authors go on to show that the a priori weaker Modified Logarithmic Sobolev Inequality implies a 'genuine' Logarithmic Sobolev Inequality, with explicit control of the constants which are involved.

For me, the combined story thus appears to be a bit clearer now:

1. When studying Logarithmic Sobolev Inequalities (and their relatives), it is relevant to think about log-Lipschitz functions (or perhaps, densities), and
2. Given a positive function f, there are 'natural' ways in which to construct a regularised approximation to f which has favourable regularity on the logarithmic scale.

Both of the cited works then seem to be engaging with these observations in order to prove more substantial results. It will be interesting to understand this connection more clearly.