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Register a new userA one-dimensional non-local singular SPDE
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We examine in this article the one-dimensional, non-local, singular SPDE begin{equation*} partial_t u ;=; -, (-Delta)^{1/2} u ,-, sinh(gamma u) ,+, xi;, end{equation*} where $gammain mathbb{R}$, $(-Delta)^{1/2}$ is the fractional Laplacian of order $1/2$, $xi$ the space-time white noise in $mathbb{R} times mathbb{T}$, and $mathbb{T}$ the one-dimensional torus. We show that for $0<gamma^2<pi/7$ the Da Prato--Debussche method applies. One of the main difficulties lies in the derivation of a Schauder estimate for the semi-group associated to the fractional Laplacian due to the lack of smoothness resulting from the long range interaction.
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We introduce the $Ntimes N$ random matrices $$ X_{j,k}=expleft(2pi i sum_{q=1}^d omega_{j,q} k^qright) quad text{with } {omega_{j,q}}_{substack{1leq jleq N 1leq qleq d}} text{ i.i.d. random variables}, $$ and $d$ a fixed integer. We prove that the distribution of their singular values converges to the local Marchenko-Pastur law at scales $N^{-theta_d}$ for an explicit, small $theta_d>0$, as long as $dgeq 18$. To our knowledge, this is the first instance of a random matrix ensemble that is explicitly defined in terms of only $O(N)$ random variables exhibiting a universal local spectral law. Our main technical contribution is to derive concentration bounds for the Stieltjes transform that simultaneously take into account stochastic and oscillatory cancellations. Important ingredients in our proof are strong estimates on the number of solutions to Diophantine equations (in the form of Vinogradovs main conjecture recently proved by Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward identity with an algebraic uniqueness condition for Diophantine equations derived from the Newton-Girard identities.
General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {cal B}(S), mu)$, with $S$ Fr{e}chet spaces such that $S subset {mathbb R}^{mathbb N}$, ${cal B}(S)$ is the Borel $sigma$-field of $S$, and $mu$ is a Borel probability measure on $S$, are introduced. Firstly, a family of non-local Markovian symmetric forms ${cal E}_{(alpha)}$, $0 < alpha < 2$, acting in each given $L^2(S; mu)$ is defined, the index $alpha$ characterizing the order of the non-locality. Then, it is shown that all the forms ${cal E}_{(alpha)}$ defined on $bigcup_{n in {mathbb N}} C^{infty}_0({mathbb R}^n)$ are closable in $L^2(S;mu)$. Moreover, sufficient conditions under which the closure of the closable forms, that are Dirichlet forms, become strictly quasi-regular, are given. Finally, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given. The application of the above theorems to the problem of stochastic quantizations of Euclidean $Phi^4_d$ fields, for $d =2, 3$, by means of these Hunt processes is indicated.
Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $ u$, or a voter model on $mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $M u/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.
A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.
We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the systems stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.
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