Related papers: The parabolic Anderson model on the hypercube
We investigate the long-time behavior of phenotype-structured models describing evolutionary dynamics of asexual populations, and analyze the joint effects of nonlocal interactions and spatial resource distributions on the global dynamics…
The paper deals with the one-dimensional parabolic potential barrier $V(x)={V_0-m\gamma^2 x^2/2}$, as a model of an unstable system in quantum mechanics. The time-independent Schr\"{o}dinger equation for this model is set up as the…
We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that…
We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=\Delta u+\xi u$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{\xi(z):z\in{\mathbb{Z}}^d\}$ and localized…
Classical particles in random potentials typically experience a percolation phase transition, being trapped in clusters of mean size $\chi$ that diverges algebraically at a percolation threshold. In contrast, quantum transport in random…
We study the large scale behaviour of a population consisting of two types which evolve in dimension d = 1, 2 according to a spatial Lambda- Fleming-Viot process subject to random time-independent selection. If one of the two types is rare…
We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves…
The parallel mutation-selection evolutionary dynamics, in which mutation and replication are independent events, is solved exactly in the case that the Malthusian fitnesses associated to the genomes are described by the Random Energy Model…
Resource are often not uniformly distributed within a population. Spatial variations of concentration of a resource, change the fitness of competing strategies locally. The notion of fitness varying with respect to both genotype and…
In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\xi u$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete…
This work is devoted to the study of scaling limits in small mutations and large time of the solutions u^$\epsilon$ of two deterministic models of phenotypic adaptation, where the parameter $\epsilon$ > 0 scales the size of mutations. The…
Using the variational cluster approach (VCA), we study the transition from the antiferromagnetic to the superconducting phase of the two-dimensional Hubbard model at zero temperature. Our calculations are based on a new method to evaluate…
In this paper, we study a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We show that the resulting…
We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass…
We show that the Tangled Nature model can be interpreted as a general formulation of the quasi-species model by Eigen et al. in a frequency dependent fitness landscape. We present a detailed theoretical derivation of the mutation threshold,…
The $NK$ model, introduced by Kauffman, Levin, and Weinberger, is a random field used to describe the fitness landscape of certain species with $N$ genetic loci, each interacting with $K$ others. The model has wide applications in…
We study a non-local parabolic Lotka-Volterra type equation describing a population structured by a space variable x 2 Rd and a phenotypical trait 2 . Considering diffusion, mutations and space-local competition between the individuals, we…
Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the…
We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are…
We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary…