The space-time conductance IIP conjecture

Let $d\ge2$. For each edge $e\in E(\mathbb Z^d)$, let $t\mapsto a_t(e)\in(0,\infty)$ be locally Lebesgue integrable, and define the space-time shift by

$$[\tau_{t,x}(a)]_s(y,z)=a_{t+s}(y+x,z+x).$$

Assume that $t,e\mapsto a_t(e)$ is averaging in the stated sense, that the law $\mathbb P$ obtained from these limits is ergodic under space-time translations, and that, for some $p,q>1$ satisfying

$$\frac1{p-1}+\frac1{q(p-1)}+\frac1q<\frac2d,$$

one has

$$\sup_{n\ge1}\frac1{n^{d+2}}\int_0^{n^2}\sum_{e\in E(\Lambda_n)}a_t(e)^p\,\mathrm dt<\infty$$

and

$$\sup_{n\ge1}\frac1{n^{d+2}}\int_0^{n^2}\sum_{e\in E(\Lambda_n)}a_t(e)^{-q}\,\mathrm dt<\infty.$$

The space-time conductance IIP conjecture. An IIP holds for the continuous-time Markov chain on $\mathbb Z^d$ with generator

$$L_t f(x)=\sum_{y:(x,y)\in E(\mathbb Z^d)}a_t(x,y)[f(y)-f(x)].$$

This extends the paper's deterministic homogenization framework to time-dependent conductances and complements known results for almost every sample of a space-time ergodic random environment.