arXiv subjects

Sebastian Andres

Publications and source records attributed to Sebastian Andres.

25 records · Page 2Linked to original sources

arXiv2018

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.

math.PR↗

arXiv2018

Berry-Esseen Theorem and Quantitative homogenization for the Random Conductance Model with degenerate Conductances

We study the random conductance model on the lattice $\mathbb{Z}^d$, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension $d\geq 3$ quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed $t^{-\frac 1 5+\varepsilon}$ for $d\geq 4$ and $t^{-\frac{1}{10}+\varepsilon}$ for $d=3$. Additionally, in the uniformly elliptic case in low dimensions $d=2,3$ we improve the rate in a quantitative Berry-Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for $d\geq 3$ we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.

math.PR↗

arXiv2015

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 $M_γ$, formally written as $M_γ(dz)=e^{γX(z)-{γ^2} \mathbb{E}[X(z)^2]/2}\, dz$, $γ\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_γ$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_γ$. 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 $p_t(x,y)\leq C_{1} t^{-1} \log(t^{-1}) \exp\bigl(-C_{2}((|x-y|^β\wedge 1)/t)^{\frac{1}{β-1}}\bigr)$ for $t\in(0,\frac{1}{2}]$ for each $β>\frac{1}{2}(γ+2)^2$, and an on-diagonal lower bound of the form $p_{t}(x,x)\geq C_{3}t^{-1}\bigl(\log(t^{-1})\bigr)^{-η}$ for $t\in(0,t_η(x)]$, with $t_η(x)\in(0,\frac{1}{2}]$ heavily dependent on $x$, for each $η>18$ for $M_γ$-almost every $x$. As applications, we deduce that the pointwise spectral dimension equals $2$ $M_γ$-a.e.\ and that the global spectral dimension is also $2$.

math.PR↗

arXiv2013

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.

math.PR↗

arXiv2012

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.

math.PR↗

arXiv2009

Regularity Properties for a System of Interacting Bessel Processes

We study the regularity of a diffusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with Coulomb interaction and reflection between nearest neighbors. As our main result we establish the Feller property for the process in both cases of repulsion and attraction. In particular the system can be started from any initial state, including multiple point configurations. Moreover we show that the process is a Euclidean semi-martingale if and only if the interaction is repulsive. Hence, contrary to classical results about reflecting Brownian motion in smooth domains, in the attractive regime a construction via a system of Skorokhod SDEs is impossible. Finally, we establish exponential heat kernel gradient estimates in the repulsive regime. The main proof for the attractive case is based on potential theory in Sobolev spaceswith Muckenhoupt weights.

math.PR↗

arXiv2007

Particle Approximation of the Wasserstein Diffusion

We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein Diffusion, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya.

math.PR↗