Searcharxiv⌕ Search

arXiv subjects

Sefika Kuzgun

Publications and source records attributed to Sefika Kuzgun.

7 recordsLinked to original sources
arXiv2026

Time-dependent averages of a critical long-range stochastic heat equation

We study the time-dependent spatial averages of a critical stochastic partial differential equation, namely the stochastic heat equation in dimension $d\geq 3$ with noise white in time and colored in space with covariance kernel $\|\cdot\|^{-2}$. The solution to this SPDE is a singular measure and was constructed by Mueller and Tribe in [MT04]. We show that the time-dependent spatial averages of this SPDE over a ball of radius $R$ at time $t$ have different limits under different space-time scales. In particular, when $t\ll R^2$, the central limit theorem holds; when $t=R^2$, the spatial average is a non-Gaussian random variable; when $t\gg R^2$, the spatial average becomes extinct.

math.PR↗
arXiv2026

Intermittency of geometric Brownian motion on $ \textbf{SL}(n) $

This short note is motivated by a recently discovered connection between a drift-diffusion process in $n$-dimensional Euclidean space with a divergence-free drift sampled from a stationary and isotropic Gaussian ensemble of critical scaling on the one hand, and a geometric Brownian motion on $\textbf{SL}(n)$ on the other hand. This can be seen as a tensorial form of a stochastic exponential; it thus is naturally intermittent, which transfers to the pair distance of the drift-diffusion process. In this note, we quantify the intermittency of the geometric Brownian motion $\{F_τ\}_{τ\ge0}$ on $\textbf{SL}(n)$ also in dimensions $n>2$. We do so in two (related) ways: 1) by identifying the exponential growth rate for the $2p$-th stochastic moment $\mathbb{E}|F_τ|^{2p}$ with its anomalous dependence on $p$ (and $n$), and 2) by quantifying a non-tightness of $|F_τ|^2/\mathbb{E}|F_τ|^2$ as $τ\uparrow\infty$. It is the second property that transmits to the drift-diffusion process. The arguments rely on stochastic analysis: We write $\{F_τ\}_{τ\geq 0}$ as the solution of $dF=F_τ\circ dB$ with $\{B_τ\}_{τ\geq 0}$ a Brownian motion on the Lie algebra $\mathfrak{sl}(n)$. The arguments leverage isotropy: The diffusion projects onto the spectrum of the Gram matrix $G=F^*F$, as captured by ${\rm tr}G^p$.

math.PR↗
arXiv2023

On the radius of self-repellent fractional Brownian motion

We study the radius $R_T$ of a self-repellent fractional Brownian motion $\left\{B^H_t\right\}_{0\le t\le T}$ taking values in $\mathbb{R}^d$. Our sharpest result is for $d=1$, where we find that with high probability, \begin{equation*} R_T \asymp T^ν, \quad \text{with $ν=\frac{2}{3}\left(1+H\right)$.} \end{equation*} For $d>1$, we provide upper and lower bounds for the exponent $ν$, but these bounds do not match.

math.PR↗
arXiv2022

Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.

math.PR↗
arXiv2021

Convergence of Densities of Spatial Averages of Stochastic Heat Equation

In this paper, we consider the one-dimensional stochastic heat equation driven by a space time white noise. In two different scenarios: {\it (i)} initial condition $u_0=1$ and general nonlinear coefficient $σ$ and {\it (ii)}: initial condition $u_0=δ_0$ and $σ(x)=x$ (Parabolic Anderson Model), we establish rates of convergence for the uniform distance between the density of (renormalized) spatial averages and the standard normal density. These results are based on the combination of Stein method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the $L^p$-norm of the second Malliavin derivative.

math.PR↗
arXiv2020

Feynman-Kac formula for iterated derivatives of the parabolic Anderson model

The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution.

math.PR↗