Searcharxiv⌕ Search

arXiv subjects

Konstantinos Dareiotis

Publications and source records attributed to Konstantinos Dareiotis.

At least 19 recordsLinked to original sources
arXiv2026

Itô perspective on variance renormalisation

We show that the Itô solutions of the nonlinear stochastic heat equation $$ \partial_t u^\varepsilon- Δu^\varepsilon =\varepsilon^{3/4} g (u^\varepsilon) \nabla ξ_\varepsilon, $$ where $ ξ_\varepsilon$ denotes the mollification in space at scale $\varepsilon>0$ of a space-time white noise $ξ$, converge in law, as $\varepsilon\to 0$, to the solution of the stochastic heat equation with right-hand side $cg'g(u)ξ$ with a constant $c>0$. Since the noise $\nablaξ$ is supercritical, the small prefactor is not unexpected to obtain a limit, but the exponent $3/4$ is not predicted by naive scaling arguments. The case $g(u)=u$, modulo a Cole-Hopf transform, corresponds to the result of [Hai25] for the KPZ equation. Our argument is relatively short and relies solely on stochastic analytic techniques.

math.PR↗
arXiv2025

Uniform pathwise stability of additive singular SDEs driven by fractional Brownian motion

We study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a singular term of regularity $γ>1-1/(2H)$ in Besov-Hölder scales. We establish well-posedness and, through a Markovian enhancement, existence of an invariant measure. If the singular contribution is sufficiently small, we prove exponential contraction of solutions, and thereby, uniqueness of the invariant measure. Our methods rely on uniform pathwise estimates which utilise together the dissipativity of the drift and the regularisation effect of the noise.

math.PR↗
arXiv2024

Regularisation by Gaussian rough path lifts of fractional Brownian motions

The aim of the paper is to show the probabilistically strong well-posedness of rough differential equations with distributional drifts driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter $H\in(1/3,1/2)$. We assume that the noise is nondegenerate and the drift lies in the Besov-Hölder space $\mathcal{C}^α$ for some $α>1-1/(2H)$. The latter condition matches the one of the additive noise case, thereby providing a multiplicative analogue of Catellier-Gubinelli in the regime $H\in(1/3,1/2)$.

math.PR↗
arXiv2024

Solutions to the stochastic thin-film equation for initial values with non-full support

The stochastic thin-film equation with mobility exponent $n\in [\frac{8}{3},3)$ on the one-dimensional torus with multiplicative Stratonovich noise is considered. We show that martingale solutions exist for non-negative initial values. This advances on existing results in three aspects: (1) Non-quadratic mobility with not necessarily strictly positive initial data, (2) Measure-valued initial data, (3) Less spatial regularity of the noise. This is achieved by carrying out a compactness argument based solely on the control of the $α$-entropy dissipation and the conservation of mass.

math.AP↗
arXiv2024

Regularisation by multiplicative noise for reaction-diffusion equations

We consider the stochastic reaction-diffusion equation in $1+1$ dimensions driven by multiplicative space-time white noise, with a distributional drift belonging to a Besov-Hölder space with any regularity index larger than $-1$. We assume that the diffusion coefficient is a regular function which is bounded away from zero. By using a combination of stochastic sewing techniques and Malliavin calculus, we show that the equation admits a unique solution.

math.PR↗
arXiv2024

Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt + dL_t, $$ with irregular $β$-Hölder drift, $β> 0$, driven by a Lévy process with exponent $α\in (0, 2]$. For $α\in [2/3, 2]$, we obtain strong $L_p$ and almost sure convergence rates in the entire range $β> 1 - α/2$, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of $α$. Notably, the obtained convergence rate does not depend on $p$, which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong $L_p$ convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality.

math.PR↗
arXiv2023

A central limit theorem for the Euler method for SDEs with irregular drifts

The goal of this article is to establish a central limit theorem for the Euler-Maruyama scheme approximating multidimensional SDEs with elliptic Brownian diffusion, under very mild regularity requirements on the drift coefficients. When the drift is Hölder continuous, we show that the limiting law of the rescaled fluctuations around the true solution is characterised as the unique solution of a hybrid Young-Itô differential equation. When the drift has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our result is an extension of the results of Jacod-Kurtz-Protter (1991, 1998) in which SDEs with differentiable coefficients were considered. To compensate for the lack of regularity of the drifts, we utilize the regularisation effect from the non-degenerate noise.

math.PR↗
arXiv2023

Optimal rate of convergence for approximations of SPDEs with non-regular drift

A fully discrete finite difference scheme for stochastic reaction-diffusion equations driven by a $1+1$-dimensional white noise is studied. The optimal strong rate of convergence is proved without posing any regularity assumption on the non-linear reaction term. The proof relies on stochastic sewing techniques.

math.PR↗
arXiv2022

Quantifying a convergence theorem of Gyöngy and Krylov

We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order $α\in (0,1)$ lead to rate $(1+α)/2$.

math.PR↗
arXiv2022

Path-by-path regularisation through multiplicative noise in rough, Young, and ordinary differential equations

Differential equations perturbed by multiplicative fractional Brownian motions are considered. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ODE, YDE, or RDE. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness with irregular drift: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case [CG16, Ger22]. In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong well-posedness, while for distributional drift we obtain weak existence.

math.PR↗
arXiv2021

Approximation of SDEs -- a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (2020). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler-Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $H\in(0,1)$ and the drift is $\mathcal{C}^α$, $α\in[0,1]$ and $α>1-1/(2H)$, we show the strong $L_p$ and almost sure rates of convergence to be $((1/2+αH)\wedge 1) -\varepsilon$, for any $\varepsilon>0$. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier, Gubinelli (2016). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $1/2-\varepsilon$ of the Euler-Maruyama scheme for $\mathcal{C}^α$ drift, for any $\varepsilon,α>0$.

math.PR↗
arXiv2021

Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

We prove the existence of nonnegative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grün, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.

math.PR↗
arXiv2020

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $Δ(|u|^{m-1}u)$ for all $m\in(1,\infty)$, and Hölder continuous diffusion nonlinearity with exponent $1/2$.

math.PR↗
arXiv2020

Nonlinear diffusion equations with nonlinear gradient noise

We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.

math.PR↗
arXiv2020

On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $α$-Hölder drift in the recent literature the rate $α/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $α>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

math.PR↗
arXiv2020

Porous media equations with multiplicative space-time white noise

The existence of martingale solutions for stochastic porous media equations driven by nonlinear multiplicative space-time white noise is established in spatial dimension one. The Stroock-Varopoulos inequality is identified as a key tool in the derivation of the corresponding estimates.

math.PR↗
arXiv2019

Ergodicity for Stochastic Porous Media Equations

The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-estimates the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way the existence and uniqueness of entropy solutions is shown.

math.PR↗