By Jean-Christophe Mourrat, Felix Otto
We introduce anchored types of the Nash inequality. they permit to manage the L2 norm of a functionality via Dirichlet varieties that aren't uniformly elliptic. We then use them to supply warmth kernel higher bounds for diffusions in degenerate static and dynamic random environments. for instance, we follow our effects to the case of a random stroll with degenerate bounce charges that depend upon an underlying exclusion method at equilibrium.
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Additional info for Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments
10. Souslin sets and the A-operation obtain (A2 \A1 ) ∩ B = (A2 ∩ B)\(A1 ∩ B) ∈ S. Similarly, it is veriﬁed that ∞ n=1 Bn ∈ S1 for any sequence of disjoint sets in S1 . Since E ⊂ S1 as proved above, one has S1 = S. Therefore, A ∩ B ∈ S for all A, B ∈ S. Thus, S is a σ-algebra. 3 we prove the following useful result. 4. Lemma. If two probability measures µ and ν on a measurable space (X, A) coincide on some class of sets E ⊂ A that is closed with respect to ﬁnite intersections, then they coincide on the σ-algebra generated by E.
1) λn U (K) = λn (K). , λn U (K) = rλn (K), where r = 1. 2) λn U (Q) = rλn (Q) if U (Q) ⊂ I. Let d be the length of the edge of K. , may have in common only parts of faces). The cubes U (Kj ) are translations of each other and have equal measures as proved above. It is readily seen that faces of any cube have measure zero. Hence λn U (K) = pn λn U (K1 ) . Therefore, λn U (K1 ) = rλn (K1 ). 2) is true for any cube of the form qK + h, where q is a rational number. 2) for the ball Q. Indeed, by additivity this equality extends to ﬁnite unions of cubes with edges parallel to the coordinate axes.
It is clear that β is a bijection of IN × IN onto IN, since, for any l ∈ IN, there exists a unique pair of natural numbers (m, n) with l = 2m−1 (2n − 1). Set also ϕ(l) := m, ψ(l) := n, where ∞ β(m, n) = l. Let σ = (σi ) ∈ IN∞ and (τ i ) ∈ IN∞ , where τ i = (τji ) ∈ IN∞ . Finally, set ϕ(1) ϕ(l) Ψ σ, (τ i ) = β σ1 , τψ(1) , . . , β σl , τψ(l) , . . For every η = (ηi ) ∈ IN∞ , the equation Ψ σ, (τ i ) = η has a unique solution σi = ϕ(ηi ), τji = ψ(ηβ(i,j) ). Hence Ψ is bijective. Since m ≤ β(m, n) and β(m, k) ≤ β(m, n) whenever k ≤ n, it follows from the form of the solution that the ﬁrst β(m, n) components of Ψ σ, (τ i ) uniquely determine the ﬁrst m components of σ and the ﬁrst n components of τ m .