Benchmarks

for the Dirichlet form associated to sticky Brownian motion. Denoting $a\_0=\backslash frac\{1\}\{2+\backslash omega\; L\}$ and $b\_0=\backslash omega\; a\_0$ , for all $s>0$ we have

$$\backslash int\; f^2\backslash mathrm\{d\}\backslash mu=a\_0(f(0)^2+f(L)^2)+b\_0\backslash int\_0^L\; f^2\backslash mathrm\{d\}x,$$ $$\backslash le\; a\_0(f(0)^2+f(L)^2)+b\_0s\backslash int\_0^L(f\text{'})^2\backslash mathrm\{d\}x+b\_0\backslash beta(s)\backslash left(\backslash int\_0^L|f|\backslash mathrm\{d\}x\backslash right)^2,$$ $$\backslash le\; b\_0s\backslash int\_0^L(f\text{'})^2\backslash mathrm\{d\}x+b\_0\backslash max(b\_0^\{-2\},a\_0^\{-1\})\backslash beta(s)\backslash left(\backslash int|f|\backslash mathrm\{d\}\backslash mu\backslash right)^2.$$

Therefore the sticky Brownian satisfies a super Poincaré inequality. Then by [Wan00, Th. 5.1], it has an empty essential spectrum. Now, by [BGL14, Th. A.6.4], the resolvent is compact and thus the generator has discrete spectrum. $\backslash square$

Corollary 19. Choosing $T=m^\{-1/2\}$ , the transition semigroup of the RTP process is exponentially contractive in $T$ -average with rate

$$\backslash nu=\backslash Omega\backslash left(\backslash frac\{\backslash omega\}\{1+(\backslash omega\; L)^2\}\backslash right).$$

Note that the relaxation time corresponding to this decay rate is of the same order as the mixing time obtained in [GHM24]. It reveals the existence of two regimes controlled by the parameter $\backslash omega\; L$ . In the ballistic regime $\backslash omega\; L\backslash ll\; 1$ , velocity flips are rare, leading to a fast exploration of the position space $\backslash mathcal\{S\}$ and a comparatively slow exploration of the velocity space $\backslash mathcal\{V\}$ . This results in the scaling $\backslash nu\backslash propto\backslash omega$ . On the contrary, in the diffusive regime $\backslash omega\; L\backslash gg\; 1$ , the high frequency of velocity flips makes the exploration of $\backslash mathcal\{V\}$ faster than the exploration of $\backslash mathcal\{S\}$ . This leads to the scaling $\backslash nu\backslash propto\backslash omega^\{-1\}L^\{-2\}$ .

Proof. We begin by verifying Assumption (A). Recall that $\backslash mathrm\{Dom\}(\backslash mathcal\{L\}\_\{C^0\})$ is a core of $\backslash mathcal\{L\}$ by Theorem 7. For all $f\backslash in\backslash mathrm\{Dom\}(\backslash mathcal\{L\}\_\{C^0\})$ we have $\backslash hat\{\backslash mathcal\{L\}\}\_v(f\backslash circ\backslash pi)=0$ hence $\backslash hat\{\backslash mathcal\{L\}\}\_\{\backslash mathrm\{tr\}\}$ is a lift of $\backslash mathcal\{L\}$ by Remark 8. Furthermore, for $f\backslash in\backslash mathrm\{Dom\}(\backslash mathcal\{L\}\_\{C^0\})$ one has

A straightforward computation yields

$$\backslash int\_\backslash mathcal\{V\}\backslash hat\{\backslash mathcal\{L\}\}\_vf(x,v)\backslash mathrm\{d\}\backslash kappa\_x(v)=0\backslash text\{\; for\; all\; \}x\backslash in\backslash mathcal\{S\}\backslash text\{\; and\; \}f\backslash in\backslash mathrm\{Dom\}(\backslash hat\{\backslash mathcal\{L\}\}).$$

Finally, we prove $\backslash |f-\backslash Pi\_vf\backslash |\_\{L^2(\backslash hat\{\backslash mu\})\}^2\backslash le\backslash frac\{1\}\{m\_v\}\backslash mathcal\{E\}\_v(f)$ with $m\_v=2$ . Define the matrices

as well as the scalar product $\backslash langle\; x,y\backslash rangle\_S=x^\backslash top\; Sy$ and let $\backslash Pi$ be the orthogonal projection on the kernel of $Q$ with respect to $\backslash langle\backslash cdot,\backslash cdot\backslash rangle\_S$ . The matrix $Q$ is symmetric w.r.t. the scalar

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