By Jean-Christophe Mourrat, Felix Otto
We introduce anchored models of the Nash inequality. they permit to regulate 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. as an instance, we follow our effects to the case of a random stroll with degenerate leap premiums that rely on an underlying exclusion technique at equilibrium.
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Extra info for Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments
1 To get (13) from (11) an integration has been performed; the vanishing of w ( ~ ) ( xt ), as x-+ w, (14) w ( ~ )w(, t ) =0, m=O, 1, clearly implied by the definition (12a), as well as the asymptotic vanishing of w i m ) ( x ,t ) implied by (12b), have moreover been invoked to set to zero the integration constant. Note that (13) is, if one assumes do) to be known, a Riccati equation for d ' ) ;and vice versa. The relationship (10) corresponding to the simplest Backlund transformation (1 1) or (13) reads simply (15) + R c l )k, ( t ) = -R (O)( k , t ) [ ( k i p )/ ( k - i p )] .
Let us end t h s section by mentioning two directions in which instead such a close correspondence does not yet seem to exist. First and most important, is the extension of the approach to more (space) variables. 1). Returning to the simplest case of one space and one time variable, there is another kind of extension that can be done very simply in the linear case but still has no simple counterpart in the nonlinear context: the inclusion of certain classes of integro-differential equations. Consider for instance, in place of (l), the evolution equation +m (27) dyK(x-y, t ) u ( y ,t ) .
N } . The motivation for such a definition is, that there is a one-to-one correspondence between functions u ( x ) (in an appropriate functional class, as indicated above), and the spectral transform (7). 2 indicates how S is determined (clearly uniquely) by u; this is the direct spectral problem. In the following subsection the inverse spectral problem is discussed, namely the determination of u from a given S. 2. , N } , the following procedure yields the corresponding function u( x). Define first of all the function N fa, (2) M(x)=(2~)-'/ dkexp(ikx)R(k)+ -a, p,exp(-p,x).
Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments by Jean-Christophe Mourrat, Felix Otto