Stationary Fokker–Planck equation on noncompact manifolds and in unbounded domains

We investigate the Fokker–Planck equation on an infinite cylindrical surface and in an infinite strip with reflecting boundary conditions, prove the existence of a positive (not necessarily integrable) solution, and derive various conditions on the vector field f that are sufficient for the existence of a solution that is the probability density. In particular, these conditions are satisfied for some vector fields f with integral trajectories going to infinity.     

Poisson’s equation for discrete-time single-birth processes

We consider Poisson’s equation for discrete-time single-birth processes, and we derive its solutions by solving a linear system of infinitely many equations. We apply the solution of Poisson’s equation to obtain the asymptotic variance. The results are further applied to birth–death processes and the scalar-valued GI/M/1-type Markov chains.     

The complex Monge–Ampère equation on weakly pseudoconvex domains

We show here a “weak” Hölder regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge–Ampère equation with data in the $L^p$ space and Ω satisfying an f-property. The f-property is a potential-theoretical condition that holds for all pseudoconvex domains of finite type and many examples of infinite-type ones.     

Equivalence transformations of the Euler–Bernoulli equation

We give a determination of the equivalence group of the Euler–Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation.     

A dynamic Euler–Bernoulli beam equation frictionally damped on an elastic foundation

This paper deals with a dynamic Euler–Bernoulli beam equation. The beam relies on a foundation composed of a continuous distribution of linear elastic springs. In addition to this time dependent uniformly distributed force, the model includes a continuous distribution of Coulomb frictional dampers, formalized by a partial differential inclusion. Under appropriate regularity assumptions on the initial data, the existence of a weak solution is obtained as a limit of a sequence of solutions associated with some physically relevant regularized problems.     

Balanced metrics on Cartan and Cartan–Hartogs domains

This paper consists of two results dealing with balanced metrics (in Donaldson terminology) on noncompact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan–Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in Loi and Zedda (Mathematische Annalen, 2011, to appear) we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that α g is not balanced for all α > 0.     

Cauchy transform and Poisson’s equation

Let $u\in W^{1,p}\cap W_0^{1,p}$, $1?p?\infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. We prove ${6?u6}_{L^p}?a_p{6h6}_{L^p}$ and ${6?u6}_{L^p}?b_p{6h6}_{L^p}$ with sharp constants $a_p$ and $b_p$, for $p=1$, $p=2$, and $p=\infty$. In addition, for $p>2$, with sharp constants $c_p$ and $C_p$, we show ${6?u6}_{L^{\infty }}?c_p{6h6}_{L^p}$ and ${6?u6}_{L^{\infty }}?C_p{6h6}_{L^p}$. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the $L^p$ norm of the Cauchy transform of Dirichlet’s problem.     

Elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation

Theoretical and Mathematical Physics - We study elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation and derive the equations of motion of their poles. The...     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Abstract Euler–Poisson–Darboux equation with nonlocal condition

We find conditions for the unique solvability of nonlocal problems for abstract differential equation of the Euler–Poisson–Darboux equation. Nonlocal conditions contain either Erdelyi–Kober operator or Riemann–Liouville fractional integration operator.     

Dissipative martingale solutions of the stochastically forced Navier–Stokes–Poisson system on domains without boundary

We construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of Dafermos (1979) and satisfy a renormalized form of the continuity equation in the sense of DiPerna and Lions (1989).     

Singular ergodic control for multidimensional Gaussian–Poisson processes

Singular control for multidimensional Gaussian–Poisson processes with a long-run (or ergodic) and a discounted criteria is discussed. The corresponding Hamilton–Jacobi–Bellman equations are discussed. Complete details on the proofs and further extensions are left for future works.     

Martin–Löf random generalized Poisson processes

Martin–Löf randomness was originally defined and studied in the context of the Cantor space $2^{\omega }$. In [2] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin–Löf randomness in spaces of closed sets. We use that framework to explore Martin–Löf randomness for the space of closed subsets of $\mathbb{R}$ and a particular family of measures on this space, the generalized Poisson processes. This gives a novel class of Martin–Löf random closed subsets of $\mathbb{R}$. We describe some of the properties of these Martin–Löf random closed sets; one result establishes that a real number is Martin–Löf random if and only if it is contained in some Martin–Löf random closed set.     

Quasineutral limit of the pressureless Euler–Poisson equation

In this short paper, we consider the quasineutral limit for the pressureless Euler–Poisson system for ions. By applying the modulated energy method, it shows that the weak solutions for the Euler–Poisson system converge weakly to the strong solutions of the compressible Euler equation as the Debye length tends to zero.     

Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process

The two parameter Poisson–Dirichlet distribution PD(α, θ) is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman’s Poisson–Dirichlet distribution. The two parameter Dirichlet process ${\Pi_{\alpha,\theta,\nu_0}}$ is the law of a pure atomic random measure with masses following the two parameter Poisson–Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures PD(α, θ) and ${\Pi_{\alpha,\theta,\nu_0}}$ . The methods used come from the theory of Dirichlet forms.     

Multivariate Bernoulli and Euler polynomials via Lévy processes

By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate Lévy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate polynomials is provided.     

A singular Cauchy problem for the Euler–Poisson–Darboux equation

We consider a singular Cauchy problem for the Euler–Poisson–Darboux equation of Fuchsian type in the time variable with ramified Cauchy data. In this paper we establish an expansion of the solutions in a series of hypergeometric functions and then investigate the nature of the singularities of the solutions.