ﻻ يوجد ملخص باللغة العربية
The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d W[0,t] for t large and d >= 3. In particular, we show that the largest regions in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^d W[0,t] for t large and the epsilon-cover time of T^d for epsilon small. Our results, which generalise laws of large numbers proved by Dembo, Peres and Rosen, are based on a large deviation principle for the shape of the component with largest capacity in T^d W_rho[0,t], where W_rho[0,t] is the Wiener sausage of radius rho = rho(t), with rho(t) chosen much smaller than phi but not too small. The idea behind this choice is that T^d W[0,t] consists of lakes, whose linear size is of order phi, connected by narrow channels. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of T^d W_rho[0,t] for t large. Our results give a complete picture of the extremal geometry of T^d W[0,t] and of the optimal strategy for W[0,t] to realise the extremes.
Flows through porous media can carry suspended and dissolved materials. These sediments may deposit inside the pore-space and alter its geometry. In turn, the changing pore structure modifies the preferential flow paths, resulting in a strong couplin
We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, th
The paper identifies families of quasi-stationary initial conditions for infinite Brownian particle systems within a large class and provides a construction of the particle systems themselves started from such initial conditions. Examples of particle systems falling into our framework include Browni
We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distri
We derive the asymptotic winding law of a Brownian particle in the plane subjected to a tangential drift due to a point vortex. For winding around a point, the normalized winding angle converges to an inverse Gamma distribution. For winding around a