arXiv subjects

Dimitra C. Antonopoulou

Publications and source records attributed to Dimitra C. Antonopoulou.

7 recordsLinked to original sources

arXiv2025

Malliavin Calculus for the one-dimensional Stochastic Stefan Problem

We consider the one-dimensional outer stochastic Stefan problem with reflection. The problem admits maximal solutions as long as the velocity of the moving boundary remains bounded, [3,9,10]. We apply Malliavin calculus to the transformed equation and first prove that its maximal solution u has continuous paths a.s. In the case of the unreflected problem, the previous enables the localization of a proper approximating sequence of the maximal solution. Then, we derive there locally the differentiability of maximal u in the Malliavin sense. The novelty of this work, apart from the derivation of continuity of the paths for the maximal solution with reflection, is that for the unreflected case we introduce a localization argument on maximal solutions and define efficiently the relevant sample space. More precisely, we prove the local (in the sample space) existence of the Malliavin derivative and, under a non-degeneracy condition on the noise coefficient, the absolute continuity of the law of the solution with respect to the Lebesgue measure.

math.PR↗

arXiv2020

The multi-dimensional Stochastic Stefan Financial Model for a portfolio of assets

The financial model proposed involves the liquidation process of a portfolio of $n$ assets through sell or (and) buy orders with volatility. We present the rigorous mathematical formulation of this model in a financial setting resulting to an $n$-dimensional outer parabolic Stefan problem with noise. In particular, our aim is to estimate for a short time period the areas of zero trading, and their diameter which approximates the minimum of the $n$ spreads of the portfolio assets for orders from the $n$ limit order books of each asset respectively. In dimensions $n=3$, and for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer. Therein, when the initial moving boundary consists of well separated spheres, a first order approximation system of odes had been rigorously derived for the dynamics of the interfaces and the asymptotic profile of the solution. In our financial case, we propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices, while each sphere may correspond to a different market; the relevant radii represent the half of the minimum spread. We apply Itô calculus and provide second order formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential equations for the radii evolution in time. A second order approximation seems to disconnect the financial model from the large diffusion assumption for the trading density. Moreover, we solve the approximating systems numerically.

math.PR↗

arXiv2016

The Sharp interface limit for the stochastic Cahn-Hilliard equation

We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter $ε$ tends to zero, which measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.

math.AP↗

arXiv2016

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Non-zero Initial Data

Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence that the Fokas method can be applied. This is an addendum to a previous paper. We consider the case of non-zero initial data, thus completing the discussion of the previous paper.

math.AP↗

arXiv2016

On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Zero Initial Data

Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data suffices to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also sufficiently decaying and hence that the Fokas method can be applied. For simplicity we considered here the case of zero initial data. An addendum will follow, discussing the case of non-zero initial data.

math.AP↗

arXiv2015

Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded two-dimensional domain with additive spatially smooth space-time noise. This equation associated with a small positive parameter describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply Itô calculus to derive the stochastic dynamics of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of droplets in $L^2$ and $H^1$, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales.

math.DS↗

arXiv2013

Existence and regularity of solution for a Stochastic Cahn-Hilliard/Allen-Cahn equation with unbounded noise diffusion

The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we consider a multiplicative space-time white noise with diffusion coefficient of sub-linear growth. Using technics from semigroup theory, we prove existence, and path regularity of stochastic solution depending on that of the initial condition. Our results are also valid for the stochastic Cahn-Hilliard equation with unbounded noise diffusion, for which previous results were established only in the framework of a bounded diffusion coefficient. We prove that the path regularity of stochastic solution depends on that of the initial condition, and are identical to those proved for the stochastic Cahn-Hilliard equation and a bounded noise diffusion coefficient. If the initial condition vanishes, they are strictly less than 2-d/2 in space and 1/2-d/8 in time. As expected from the theory of parabolic operators in the sense of Petrovski, the bi-Laplacian operator seems to be dominant in the combined model.

math.PR↗