4,525 research outputs found
Invariance Principle for the Random Conductance Model with dynamic bounded Conductances
We study a continuous time random walk X in an environment of dynamic random
conductances. We assume that the conductances are stationary ergodic, uniformly
bounded and bounded away from zero and polynomially mixing in space and time.
We prove a quenched invariance principle for X, and obtain Green's functions
bounds and a local limit theorem. We also discuss a connection to stochastic
interface models
Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions
The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes
and Vargas and in a weaker form also by Berestycki, is a diffusion process
evolving in a planar random geometry induced by the Liouville measure
, formally written as , , for a (massive) Gaussian free
field . It is an -symmetric diffusion defined as the time change
of the two-dimensional Brownian motion by the positive continuous additive
functional with Revuz measure .
In this paper we provide a detailed analysis of the heat kernel of
the LBM. Specifically, we prove its joint continuity, a locally uniform
sub-Gaussian upper bound of the form for
for each , and an
on-diagonal lower bound of the form for , with
heavily dependent on , for each
for -almost every . As applications, we deduce that the
pointwise spectral dimension equals -a.e.\ and that the global
spectral dimension is also .Comment: 36 page
Diffusion processes on branching Brownian motion
We construct a class of one-dimensional diffusion processes on the particles
of branching Brownian motion that are symmetric with respect to the limits of
random martingale measures. These measures are associated with the extended
extremal process of branching Brownian motion and are supported on a
Cantor-like set. The processes are obtained via a time-change of a standard
one-dimensional reflected Brownian motion on in terms of the
associated positive continuous additive functionals. The processes introduced
in this paper may be regarded as an analogue of the Liouville Brownian motion
which has been recently constructed in the context of a Gaussian free field.Comment: 25 pages, 1 figure, published versio
Energy inequalities for cutoff functions and some applications
We consider a metric measure space with a local regular Dirichlet form. We
establish necessary and sufficient conditions for upper heat kernel bounds with
sub-diffusive space-time exponent to hold. This characterization is stable
under rough isometries, that is it is preserved under bounded perturbations of
the Dirichlet form. Further, we give a criterion for stochastic completeness in
terms of a Sobolev inequality for cutoff functions. As an example we show that
this criterion applies to an anomalous diffusion on a geodesically incomplete
fractal space, where the well-established criterion in terms of volume growth
fails
The origin of compression influences geometric instabilities in bilayers
Geometric instabilities in bilayered structures control the surface morphology in a wide range of biological and technical systems. Depending on the application, different mechanisms induce compressive stresses in the bilayer. However, the impact of the chosen origin of compression on the critical conditions, post-buckling evolution and higher-order pattern selection remains insufficiently understood. Here, we conduct a numerical study on a finite-element set-up and systematically vary well-known factors contributing to pattern selection under the four main origins of compression: film growth, substrate shrinkage and whole-domain compression with and without pre-stretch. We find that the origin of compression determines the substrate stretch state at the primary instability point and thus significantly affects the critical buckling conditions. Similarly, it leads to different post-buckling evolutions and secondary instability patterns when the load further increases. Our results emphasize that future phase diagrams of geometric instabilities should incorporate not only the film thickness but also the origin of compression. Thoroughly understanding the influence of the origin of compression on geometric instabilities is crucial to solving real-life problems such as the engineering of smart surfaces or the diagnosis of neuronal disorders, which typically involve temporally or spatially combined origins of compression
Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances
We establish heat kernel upper bounds for a continuous-time random walk under
unbounded conductances satisfying an integrability assumption, where we correct
and extend recent results by the authors to a general class of speed measures.
The resulting heat kernel estimates are governed by the intrinsic metric
induced by the speed measure. We also provide a comparison result of this
metric with the usual graph distance, which is optimal in the context of the
random conductance model with ergodic conductances.Comment: 19 pages; accepted version, to appear in Electron. Commun. Proba
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