Nadia Sidorova

A TWO CITIES THEOREM FOR THE PARABOLIC ANDERSON MODEL

The parabolic Anderson problem is the Cauchy problem for the heat equation partial derivative(t)u(t, z) = Delta u(t,z) + xi(z)u(t,z) on (0,infinity) x Z(d) with random potential (xi(z): z is an element of Z(d)). We consider independent and identically distributed potentials, such that the distribution function of (z) converges polynomially at infinity. If u is initially localized in the origin, that is, if u(0, z) = 1(0)(z), we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

Ageing in the parabolic Anderson model

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

Localisation and ageing in the parabolic Anderson model with Weibull potential

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential

Construction of surface measures for Brownian motion

\xi

Small deviations of a Galton–Watson process with immigration

. We consider the case when

Phase Transitions For Dilute Particle Systems with Lennard-Jones Potential

\{\xi(z):z\in\mathbb{Z}^d\}

Galton-Watson trees with vanishing martingale limit

is a collection of independent identically distributed random variables with Weibull distribution with parameter

Sound compression: a rough path approach

0<\gamma<2

Wiener surface measures on trajectories in Riemannian manifolds

, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing.

On the radius of convergence of the logarithmic signature

Let