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 $M_\gamma$, formally written as $M_\gamma(dz)=e^{\gamma X(z)-{\gamma^2} \mathbb{E}[X(z)^2]/2}\, dz$, $\gamma\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_\gamma$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_\gamma$. In this paper we provide a detailed analysis of the heat kernel $p_t(x,y)$ of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^{\beta}\wedge 1)/t)^{\frac{1}{\beta -1}}\bigr)$ for $t\in(0,\frac{1}{2}]$ for each $\beta>\frac{1}{2}(\gamma+2)^2$, and an on-diagonal lower bound of the form $p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-\eta}$ for $t\in(0,t_{\eta}(x)]$, with $t_{\eta}(x)\in(0,\frac{1}{2}]$ heavily dependent on $x$, for each $\eta>18$ for $M_{\gamma}$-almost every $x$. As applications, we deduce that the pointwise spectral dimension equals $2$ $M_\gamma$-a.e.\ and that the global spectral dimension is also $2$.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 $\mathbb{R}_+$ 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