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Register a new userExplicit formulas for heat kernels on diamond fractals
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Authors
Patricia Alonso Ruiz
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This paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a $(mathbb{N}timesmathbb{N})$-parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property. The formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free Schrodinger equation. The results also open up the possibility to approach infinite dimensional spaces based on this model.
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This paper presents a detailed analysis of the heat kernel on an $(mathbb{N}timesmathbb{N})$-parameter family of compact metric measure spaces, which do not satisfy the volume doubling property. In particular, uniform bounds of the heat kernel and its Lipschitz continuity, as well as the continuity of the corresponding heat semigroup are studied; a specific example is presented revealing a logarithmic correction. The estimates are further applied to derive several functional inequalities of interest in describing the convergence to equilibrium of the diffusion process.
We construct and study a family of continuum random polymer measures $mathbf{M}_{r}$ corresponding to limiting partition function laws recently derived in a weak-coupling regime of polymer models on hierarchical graphs with marginally relevant disorder. The continuum polymers are identified with isometric embeddings of the unit interval $[0,1]$ into a compact diamond fractal with Hausdorff dimension two, and there is a natural probability measure, $mu$, identifiable as being `uniform over the space of continuum polymers, $Gamma$. Realizations of the random measures $mathbf{M}_{r}$ exhibit strong localization properties in comparison to their subcritical counterparts when the diamond fractal has dimension less than two. Whereas two directed paths $p,qin Gamma$ chosen independently according to the pure measure $mu$ have only finitely many intersections with probability one, a realization of the disordered product measure $ mathbf{M}_{r}times mathbf{M}_{r}$ a.s. assigns positive weight to the set of pairs of paths $(p,q)$ whose intersection sets are uncountable but with Hausdorff dimension zero. We give a more refined characterization of the size of these dimension zero sets using generalized (logarithmic) Hausdorff measures. The law of the random measure $mathbf{M}_{r}$ cannot be constructed as a subcritical Gaussian multiplicative chaos because the coupling strength to the Gaussian field would, in a formal sense, have to be infinite.
The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques. A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup $P_t$ on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an $varepsilon^2$ term in the variation allowing greater control. We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time, furthermore the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to bypass the lack of continuity of the exit time of the Brownian motions on its initial position.
Ganter and Kapranov associated a 2-character to 2-representations of a finite group. Elgueta classified 2-representations in the category of 2-vector spaces 2Vect_k in terms of cohomological data. We give an explicit formula for the 2-character in terms of this cohomological data and derive some consequences.
We study strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators which are defined on a noncompact Riemannian manifold.
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