arXiv subjects

Carlos Escudero

Publications and source records attributed to Carlos Escudero.

At least 37 records · Page 2Linked to original sources

arXiv2019

Itô vs Stratonovich in the presence of absorbing states

It is widely assumed that there exists a simple transformation from the Itô interpretation to the one by Stratonovich and back for any stochastic differential equation of applied interest. While this transformation exists under suitable conditions, and transforms one interpretation into the other at the price of modifying the drift of the equation, it cannot be considered universal. We show that a class of stochastic differential equations, characterized by the presence of absorbing states and of interest in applications, does not admit such a transformation. In particular, formally applying this transformation may lead to the disappearance of some absorbing states. In turn, this modifies the long-time, and even the intermediate-time, behavior of the solutions. The number of solutions can also be modified by the unjustified application of the mentioned transformation, as well as by a change in the interpretation of the noise. We discuss how these facts affect the classical debate on the Itô vs Stratonovich dilemma.

math.PR↗

arXiv2019

Numerical solutions for a class of singular boundary value problems arising in the theory of epitaxial growth

The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value problem which is the exact result of the reduction of this fourth order singular boundary value problem. It turns out that the existence or nonexistence of numerical solutions fully depends on the value of a parameter. We show that numerical solutions exist for small positive values of this parameter. For large positive values of the parameter, we find nonexistence of solutions. We also observe existence of solutions for negative values of the parameter and determine the range of parameter values which separates existence and nonexistence of solutions. This parameter has a clear physical meaning as it describes the rate at which new material is deposited onto the system. This fact allows us to interpret the physical significance of our results.

math.NA↗

arXiv2018

A triple comparison between anticipating stochastic integrals in financial modeling

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. Our results give some indication that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability.

math.PR↗

arXiv2018

Polyharmonic $k-$Hessian equations in $\mathbb{R}^N$

This work is focused on the study of the nonlinear elliptic higher order equation \begin{equation}\nonumber \left( -Δ\right)^m u = S_k[-u] + λf, \qquad x \in \mathbb{R}^N, \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix of the solution and the datum $f$ belongs to a suitable functional space. This problem is posed in $\mathbb{R}^N$ and we prove the existence of at least one solution by means of topological fixed point methods for suitable values of $m \in \mathbb{N}$. Questions related to the regularity of the solutions and extensions of these results to the nonlocal setting are also addressed. On the way to construct these proofs, some technical results such as a fixed point theorem and a refinement of the critical Sobolev embedding, which could be of independent interest, are introduced.

math.AP↗

arXiv2018

A simple comparison between Skorokhod & Russo-Vallois integration for insider trading

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Skorokhod and the Russo-Vallois forward integrals within this context. We conclude that, while the forward integral yields results with a clear financial meaning, the Skorokhod integral does not provide a suitable formulation for this problem.

math.PR↗

arXiv2017

Radial biharmonic $k-$Hessian equations: The critical dimension

This work is devoted to the study of radial solutions to the elliptic problem \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf, \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$. We also study the existence of entire solutions to this partial differential equation in the case in which they are assumed to decay to zero at infinity and under analogous conditions of summability on the datum. Our results illustrate how, for $k=2$, the dimension $N=4$ plays the role of critical dimension separating two different phenomenologies below and above it.

math.CA↗

arXiv2015

On polyharmonic regularizations of $k-$Hessian equations: Variational methods

This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (-1)^αΔ^αu = (-1)^k S_k[u] + λf, \qquad x &\in& Ω\subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u = \cdots = \partial_n^{α-1} u = 0, \qquad x &\in& \partial Ω, \end{eqnarray} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f$ obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We look for the optimal values of $α\in \mathbb{N}$ that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results.

math.AP↗

arXiv2015

Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian

We consider a partial differential equation that arises in the coarse-grained description of epitaxial growth processes. This is a parabolic equation whose evolution is governed by the competition between the determinant of the Hessian matrix of the solution and the biharmonic operator. This model might present a gradient flow structure depending on the boundary conditions. We first extend previous results on the existence of stationary solutions to this model for Dirichlet boundary conditions. For the evolution problem we prove local existence of solutions for arbitrary data and global existence of solutions for small data. By exploiting the boundary conditions and the variational structure of the equation, according to the size of the data we prove finite time blow-up of the solution and/or convergence to a stationary solution for global solutions.

math.AP↗

arXiv2015

Existence of radial solutions to biharmonic $k-$Hessian equations

This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while $λ\in \mathbb{R}$. We prove the existence of a Carathéodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided $|λ|$ is small enough. Moreover, we prove that the solvability set of $λ$ is finite, giving an explicity bound of the extreme value.

math.CA↗

arXiv2015

Existence results for a fourth order partial differential equation arising in condensed matter physics

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.

math.AP↗

arXiv2014

Kinetic theory and numerical simulations of two-species coagulation

In this work we study the stochastic process of two-species coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called well-stirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays self-similar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.

math.AP↗

arXiv2014

Globular Clusters: DNA of Early-Type galaxies?

This paper explores if the mean properties of Early-Type Galaxies (ETG) can be reconstructed from "genetic" information stored in their GCs (i.e., in their chemical abundances, spatial distributions and ages). This approach implies that the formation of each globular occurs in very massive stellar environments, as suggested by some models that aim at explaining the presence of multi-populations in these systems. The assumption that the relative number of globular clusters to diffuse stellar mass depends exponentially on chemical abundance, [Z/H], and the presence of two dominant GC sub-populations blue and red, allows the mapping of low metallicity halos and of higher metallicity (and more heterogeneous) bulges. In particular, the masses of the low-metallicity halos seem to scale up with dark matter mass through a constant. We also find a dependence of the globular cluster formation efficiency with the mean projected stellar mass density of the galaxies within their effective radii. The analysis is based on a selected sub-sample of galaxies observed within the ACS Virgo Cluster Survey of the {\it Hubble Space Telescope}. These systems were grouped, according to their absolute magnitudes, in order to define composite fiducial galaxies and look for a quantitative connection with their (also composite) globular clusters systems. The results strengthen the idea that globular clusters are good quantitative tracers of both baryonic and dark matter in ETGs.

astro-ph.GA↗

arXiv2014

Invited review: KPZ. Recent developments via a variational formulation

Recently, a variational approach has been introduced for the paradigmatic Kardar--Parisi--Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation alpha+z=2, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.

cond-mat.stat-mech↗

arXiv2013

Some fourth order nonlinear elliptic problems related to epitaxial growth

This paper deals with some mathematical models arising in the theory of epitaxial growth of crystal. We focalize the study on a stationary problem which presents some analytical difficulties. We study the existence of solutions. The central model in this work is given by the following fourth order elliptic equation, $$\begin{array}{rclll} Δ^2 u=\text{det} \left(D^2 u \right) &+&λf, \quad & x\in Ω\subset\mathbb{R}^2\\ \hbox{conditions on} &\quad& & \partial Ω. \end{array} $$ The framework to study the problem deeply depends on the boundary conditions.

math.AP↗

arXiv2013

On radial stationary solutions to a model of nonequilibrium growth

We present the formal geometric derivation of a nonequilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of nonequilibrium statistical mechanics.

math.CA↗

arXiv2013

Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. Our results depend on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter we prove existence of solutions to this boundary value problem. For large values of the same parameter we prove nonexistence of solutions. We also provide rigorous bounds for the values of this parameter which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions.

math.CA↗

arXiv2013

Nonlinear field theories during homogeneous spatial dilation

The effect of a uniform dilation of space on stochastically driven nonlinear field theories is examined. This theoretical question serves as a model problem for examining the properties of nonlinear field theories embedded in expanding Euclidean Friedmann-Lema\^ıtre-Robertson-Walker metrics in the context of cosmology, as well as different systems in the disciplines of statistical mechanics and condensed matter physics. Field theories are characterized by the speed at which they propagate correlations within themselves. We show that for linear field theories correlations stop propagating if and only if the speed at which the space dilates is higher than the speed at which correlations propagate. The situation is in general different for nonlinear field theories. In this case correlations might stop propagating even if the velocity at which space dilates is lower than the velocity at which correlations propagate. In particular, these results imply that it is not possible to characterize the dynamics of a nonlinear field theory during homogeneous spatial dilation {\it a priori}. We illustrate our findings with the nonlinear Kardar-Parisi-Zhang equation.

cond-mat.stat-mech↗

arXiv2012

Origins of scaling relations in nonequilibrium growth

Scaling and hyperscaling laws provide exact relations among critical exponents describing the behavior of a system at criticality. For nonequilibrium growth models with a conserved drift there exist few of them. One such relation is $α+z=4$, found to be inexact in a renormalization group calculation for several classical models in this field. Herein we focus on the two-dimensional case and show that it is possible to construct conserved surface growth equations for which the relation $α+z=4$ is exact in the renormalization group sense. We explain the presence of this scaling law in terms of the existence of geometric principles dominating the dynamics.

cond-mat.stat-mech↗