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We introduce anchored versions of the Nash inequality. They allow to controlthe $L^2$ norm of a function by Dirichlet forms that are not uniformlyelliptic. We then use them to provide heat kernel upper bounds for diffusionsin degenerate static and dynamic random environments. As an example, we applyour results to the case of a random walk with degenerate jump rates that dependon an underlying exclusion process at equilibrium.

Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments