arXiv subjects

Colin Piernot

Publications and source records attributed to Colin Piernot.

2 recordsLinked to original sources

arXiv2026

Central limit theorem for the Allen-Cahn equation with supercritical random initial conditions

We study the large-scale behavior of solutions to the Allen-Cahn reaction-diffusion equation with Gaussian initial data. We consider the case of short-range dependence in the associated supercritical regime with spatial dimension $d \ge 3$. Under diffusive rescaling, the non-linearity formally vanishes on large scales in this case. Accordingly, we prove a central limit theorem for the rescaled solution, more precisely, that it converges to the solution of the heat equation started from a white noise. These initial conditions for the limit depend non-trivially both on the source of randomness and on the non-linearity. Our proof uses estimates obtained by a combination of comparison principles and Malliavin calculus, initiated by Castillo and Dunlap in arXiv:2509.06260 in the critical case. However, the result there is not a fluctuation result but rather an $L^2_\mathbf{P}$ comparison to a McKean-Vlasov problem with Gaussian solutions. Hence the mechanism behind the Gaussianity of the limit differs, and the proof requires new ideas that should be further applicable to other supercritical problems.

math.PR↗

arXiv2025

Fluctuations from a random fractional averaging limit

We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space.

math.PR↗