Parabolic Anderson Model on R^2

For my thesis project we have been studying the analysis of the parabolic Anderson model in 2 spatial dimensions on the whole plane, performed by Hairer and Labbe in early 2015. This problem is a nice example as it requires renormalization to control the singularities and weighted spaces to control the divergence at infinity. After adding the necessary logarithmic counter term and posing the problem in the correct space we are then able to prove existence and uniqueness of the solution. Our main contribution is to offer a more explicit account than was previously available, and to correct some typos in the original work. This work is of importance because the parabolic Anderson model, which models a random walk driven by a random potential, can be used to study several topics such as spectral theory and some variational problems. Moreover, this analysis is of interest because it presents a particularly clean example, in that there is no need for any complicated (though more general) renormalization procedures. Rather, we use a trick from the analysis of smooth partial differential equations to identify the diverging terms and then add an appropriate counter term.Mathematic

A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation

We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the $n$-th order annihilation reaction $A+A+A+...+A\rightarrow 0$, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for $n>\alpha$) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for $n<\alpha<n+1$) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.Comment: Article, 10 pages, 5 figure

Average variance bounds for integer points on the sphere

Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we consider the variance of the number of points in a spherical cap. An asymptotic for this variance was conjectured by Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the average (over radii) of these variances.Comment: 16 page

Full Poissonian Local Statistics of Slowly Growing Sequences

Fix $\alpha>0$, then by Fej\'er's theorem $(\alpha(\log n)^{A}\,\mathrm{mod}\,1)_{n\geq1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided $A>1$. This is the first example of a deterministic sequence modulo one whose gap distribution, and all of whose correlations are proven to be Poissonian. The range of $A$ is optimal and complements a result of Marklof and Str\"{o}mbergsson who found the limiting gap distribution of $(\log(n)\, \mathrm{mod}\,1)$, which is necessarily not Poissonian.Comment: Minor edits, 23 pages, 1 figur

Hyperbolic lattice point counting in unbounded rank

We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank

Pair Correlation of the Fractional Parts of αnθ\alpha n^\theta

Fix $\alpha,\theta >0$, and consider the sequence $(\alpha n^{\theta} \mod 1)_{n\ge 1}$. Since the seminal work of Rudnick--Sarnak (1998), and due to the Berry--Tabor conjecture in quantum chaos, the fine-scale properties of these dilated mononomial sequences have been intensively studied. In this paper we show that for $\theta \le 1/3$, and $\alpha>0$, the pair correlation function is Poissonian. While (for a given $\theta \neq 1$) this strong pseudo-randomness property has been proven for almost all values of $\alpha$, there are next-to-no instances where this has been proven for explicit $\alpha$. Our result holds for all $\alpha>0$ and relies solely on classical Fourier analytic techniques. This addresses (in the sharpest possible way) a problem posed by Aistleitner--El-Baz--Munsch (2021).Comment: 10 pages. Revised version with Athanasios Sourmelidis added as an author and an improved range in the main theore