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arXiv · 2606.27869

Walk dimension and vanishing curve modulus in metric measure spaces

Aobo Chen
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Abstract

On a regular local $p$-Dirichlet space supporting a Poincar\'{e} inequality and a capacity upper bound, we characterize the existence of curve families of positive $p$-modulus in terms of the small-scale behaviour of the scaling function. As an application, we show that, on an Ahlfors-regular metric measure space carrying a strongly local regular Dirichlet form with sub-Gaussian heat kernel estimates, the comparison between the walk dimension and $2$ determines whether the given metric attains its Ahlfors-regular conformal dimension.

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Aobo Chen. 2026-06-26. Walk dimension and vanishing curve modulus in metric measure spaces. https://arxiv.org/abs/2606.27869

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