Heat-kernel spectral bound $|H_t(x,y)-\pi(y)|\le\sqrt{\pi(y)/\pi(x)}\,e^{-\gamma t}$

Let $P$ be an irreducible Markov chain on a finite state space $V$ with stationary distribution $\pi$, reversible with respect to it (detailed balance: $\pi(x)P(x,y)=\pi(y)P(y,x)$). The heat kernel at real time $t\ge0$ is $H_t(x,y)=\sum_ke^{-t}\tfrac{t^k}{k!}P^k(x,y)$ — the continuous-time chain driven by a rate-one Poisson clock. Among the eigenvalues of $P$ (real $\lambda$ with $Pf=\lambda f$ for some nonzero $f$), $\lambda_2$ is the largest eigenvalue different from $1$, and $\gamma=1-\lambda_2$ is the spectral gap (Mission VII).

The theorem (Theorem 20.6 of Levin–Peres–Wilmer) asserts the pointwise estimate: for all states $x,y$ and all $t\ge0$,

In continuous time the convergence rate is governed by the spectral gap $\gamma$ itself — not the absolute gap $\gamma_\star$ of the discrete theory: negative eigenvalues, which cause the discrete chain to oscillate, are killed by the Poisson smoothing, since $P$'s eigenvalue $\lambda$ becomes $e^{-(1-\lambda)t}$ for the heat kernel. The prefactor $\sqrt{\pi(y)/\pi(x)}$ comes from expanding the transition kernel in the $\ell^2(\pi)$-orthonormal eigenbasis of Mission VII's spectral representation. This estimate is the continuous-time engine behind the product-chain theorem of this mission.