Differential Equations In Dimension Research Articles

Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach

Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic processes with arbitrary finitely regular paths. These results are proved by using the notion of \(\Delta \)-power variations, introduced herein, along with the Hölder-Zygmund norms. Consequently, we prove a new central limit theorem for \(\Delta \)-power variations of the iterated integrals of a fractional Brownian motion. These abstract results, besides being of independent interest, in the second part of the paper are applied to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations in dimension one, driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we solve the earlier conjecture from Cialenco et al. (Stat. Inference Stoch. Process. 23:83-103, 2020) about existence of a nontrivial bias in the estimators derived by naive approximations of derivatives by finite differences. We give an explicit formula for the bias and derive the convergence rates of the corresponding estimators. Theoretical results are illustrated by numerical examples.

Bounding the degree of solutions of differential equations

We present an algorithmic strategy to compute an upper bound for the degree of the algebraic solutions of non-degenerate polynomial differential equations in dimension two.

The Geometry of Almost Einstein (2,3,5) Distributions

We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures $\mathbf{c}$ that are induced by at least two distinct oriented (2,3,5) distributions; in this case there is a 1-parameter family of such distributions that induce $\mathbf{c}$. Second, they are characterized by the existence of a holonomy reduction to $SU(1,2)$, $SL(3,{\mathbb R})$, or a particular semidirect product $SL(2,{\mathbb R})\ltimes Q_+$, according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2,3,5) distributions and many other geometries - several classical geometries among them - including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; K\"ahler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional K\"ahler-Einstein or para-K\"ahler-Einstein structure a family of (2,3,5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2,3,5) conformal structures for which the Einstein constant is positive and negative.

STOCHASTIC CENTER OF SYSTEMS OF STOCHASTIC DIFFERENTIAL EQUATIONS ON THE PLANE

We study a stochastic analogy of the famous center problem of Dulac for quadratic differential equations in the plane. We introduce the concept of center for systems of stochastic differential equations of Ito's type on the plane, called stochastic center. We derive a criterion for the existence of such a center. We apply it to obtain necessary and sufficient conditions for quadratic stochastic differential equations in dimension 2.

Adams type inequalities and related elliptic partial differential equations in dimension four

Motivated by Ruf–Saniʼs recent work, we prove an Adams type inequality and a singular Adams type inequality in the whole four-dimensional Euclidean space. As applications of those inequalities, a class of elliptic partial differential equations are considered. Existence of nontrivial weak solutions and multiplicity results are obtained via the mountain-pass theorem and the Ekelandʼs variational principle. This is a continuation of our previous work about singular Trudinger–Moser type inequality.

Density estimates for a random noise propagating through a chain of differential equations

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the n − 1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on … , so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.

Local simplification of first-order systems of ordinary differential equations of dimension two

In this paper, we present an algorithm to simplify a first-order system of two ordinary differential equations in the neighborhood of singular points with evidence, we prove a theorem dealing with its convergence and compare it with already known methods and apply it to some particular systems.

Criteria for the stability of spatial extensions of fixed points and periodic orbits of differential equations in dimension 2

We consider partial differential equations that can be viewed as spatial extensions of two-dimensional differential equations with respect to a coupling matrix. We provide analytical and geometrical criteria governing the stability of spatially homogeneous stationary solutions and the phase stability of spatially homogeneous time periodic solutions. We distinguish cases where the differential equation has a conservative or a dissipative behavior. The geometrical criteria roughly speaking relate to the “sense of rotation” of, on one hand, the coupling matrix, and, on the other, the flow of the differential equation around the spatially homogeneous solution.

Self-dual Yang-Mills fields ind=4 and integrable systems in 1?d?3

The Ward correspondence between self-dual Yang-Mills fields and holomorphic vector bundles is used to develop a method for reducing the Lax pair for the self-duality equations of the Yang-Mills model ind=4 with respect to the action of continuous symmetry groups. It is well known that reductions of the self-duality equations lead to systems of nonlinear differential equations in dimension 1≤d≤3. For the integration of the reduced equations, it is necessary to find a Lax pair whose compatibility conditions is these equations. The method makes it possible to obtain systematically a Lax representation for the reduced self-duality equations. This is illustrated by a large number of examples.