Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Ergodicity of Interacting Particle Systems under Number Rigidity   In this paper, we prove the equivalence among the convergence to the
equilibrium of infinite interacting diffusion processes and two different
concepts stemming from the optimal transport theory and the theory of point
processes. Namely, under the assumption of the rigidity in number of point
processes $\mu$ in the sense of Ghosh-Peres (Duke Math. J. 166 (10) 2017), we
discuss the equivalence of the following three concepts: ${\bf (a)}$ the
ergodicity of interacting diffusion processes with invariant measure $\mu$;
${\bf (b)}$ the finiteness of $L^2$-transportation distance ${\sf
d}_{\boldsymbol\Upsilon}$ between sets of $\mu$-positive measures; ${\bf (c)}$
the tail-triviality of $\mu$. As an application, we obtain the convergence to
the equilibrium of infinite particle diffusions with long-range interaction:
$\mathrm{Sine}_{2}$, $\mathrm{Airy}_{2}$, $\mathrm{Bessel}_{\alpha, 2}$
($\alpha \ge 1$), and $\mathrm{Ginibre}$ point processes, in particular, the
case of Dyson Brownian motion is covered. The key steps of the proofs are
twofold: First, we make full use of results regarding the extended metric
measure geometry on ${\boldsymbol\Upsilon}$ developed in the recent paper
arXiv:2109.03192. Second, we establish the Sobolev-to-Lipschitz property for
Dirichlet forms with respect to ${\sf d}_{{\boldsymbol\Upsilon}}$, which was
conjectured originally by R\"ockner-Schied (J. Func. Anal. 169} (2) 1999).

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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg