We introduce anchored versions of the Nash inequality. They allow to control the L2 norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments. As an example, we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium.
- Diffusion in dynamic random medium
- Heat kernel
- Nash inequality
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