By Jean-Christophe Mourrat, Felix Otto

http://www.sciencedirect.com/science/article/pii/S0022123615003900

We introduce anchored types of the Nash inequality. they enable to manage the L2 norm of a functionality by way of Dirichlet types that aren't uniformly elliptic. We then use them to supply warmth kernel top bounds for diffusions in degenerate static and dynamic random environments. to illustrate, we practice our effects to the case of a random stroll with degenerate leap charges that depend upon an underlying exclusion strategy at equilibrium.

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

**Example text**

3 The graph of a simple function f (x) and its decreasing rearrangement f ∗ (t). 2. 2, N f (x) = ∑ a j χE j (x) , j=1 where the sets E j have finite measure and are pairwise disjoint and a1 > · · · > aN . 2 that N d f (α) = ∑ B j χ[a j+1 ,a j ) (α) , j=0 where j B j = ∑ µ(Ei ) i=1 and aN+1 = B0 = 0 and a0 = ∞. Observe that for B0 ≤ t < B1 , the smallest s > 0 with d f (s) ≤ t is a1 . Similarly, for B1 ≤ t < B2 , the smallest s > 0 with d f (s) ≤ t is a2 . Arguing this way, it is not difficult to see that f ∗ (t) = N ∑ a j χ[B j−1 ,B j ) (t) .

L p,∞ 1≤ j≤N The preceding expression for f L p,q is also valid when p = ∞, but in this case it is equal to infinity if at least one a j is strictly positive. We conclude that the only simple function with finite L∞,q norm is the zero function. For this reason we have that L∞,q = {0} for every 0 < q < ∞. 9. For 0 < p < ∞ and 0 < q ≤ ∞, we have the identity f L p,q =p ∞ 1 q 0 1 p d f (s) s q ds s 1 q . 5) Proof. 5, and we may therefore concentrate on the case q < ∞. 2, then N d f (s) = ∑ B j χ[a j+1 ,a j ) (s) j=1 with the understanding that aN+1 = 0.

38) We now change variables. On the interval [−π, 0) we use the change of variables iy = h(eiϕ ) or, equivalently, eiϕ = − tanh(πy) − i sech(πy). Observe that as ϕ ranges from −π to 0, y ranges from +∞ to −∞. Furthermore, dϕ = −π sech(πy) dy. We have 0 1 2π sin(πx) log |F(h(eiϕ ))| dϕ −π 1 + cos(πx) sin(ϕ) 1 ∞ sin(πx) = log |F(iy)| dy . 39) On the interval (0, π] we use the change of variables 1 + iy = h(eiϕ ) or, equivalently, eiϕ = − tanh(πy) + i sech(πy). Observe that as ϕ ranges from 0 to π, y ranges from −∞ to +∞.

### Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments by Jean-Christophe Mourrat, Felix Otto

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