Use Of Pseudo-differential Operators Research Articles

Markov projection of semimartingales — Application to comparison results

In this paper we derive generalizations of comparison results for semimartingales. Our results are based on Markov projections and on known comparison results for Markov processes. The first part of the paper is concerned with an alternative method for the construction of Markov projections of semimartingales. In comparison to the construction in Bentata and Cont (2009) which is based on the solution of a well-posed martingale problem, we make essential use of pseudo-differential operators as investigated in Böttcher (2008) and of fundamental solutions of related evolution problems. This approach allows to dismiss with some boundedness assumptions on the differential characteristics in the martingale approach. As consequence of the construction of Markov projections, comparison results for path-independent functions (European options) of semimartingales can be reduced to the well investigated problem of comparison of Markovian semimartingales. The Markov projection approach to comparison results does not require one of the semimartingales to be Markovian, which is a common assumption in literature. An idea of Brunick and Shreve (2013) to mimick updated processes leads to a related reduction result to the Markovian case and thus to the comparison of related generators. As consequence, a general comparison result is also obtained for path-dependent functions of semimartingales.

Линейный финитный функционал в весовом пространстве Соболева

In this paper, a representation of a linear functional in the weighted Sobolev space is obtained. The space is normed without use of pseudodifferential operators. The norm contains partial derivatives of all intermediate orders of the test function. The Sobolev space is considered to be of non-Hilbert type. First, we deduce the representation of linear functional via a boundary element of the test space. The boundary element corresponds to the given functional. This way, referring to Clarkson’s inequalities, we prove the uniqueness of the boundary element. Then, to obtain a condition for the boundary element, we differentiate the function built based on the norm. The result leads to a representation of an arbitrary linear functional via the boundary element. When considering the boundary element as unknown, the representation performs as a nonlinear differential equation. Second, we consider a finite linear functional. The extreme function of such a functional was built in our earlier papers. The extreme function is expressed via convolution of the fundamental solution of a linear partial differential equation with a given functional. The functional performs as a distribution in the convolution. Convolution exists if the linear functional is finite. Using this fact, we prove that the representation of a finite linear functional via the boundary element is identical to the representation via the extreme function.

The theory of pseudo-differential operators on the noncommutative n-torus

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on finite-dimensional real vector spaces, one may derive a similar pseudo-differential calculus on noncommutative n-tori, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for n=2,4, as shown in the groundbreaking paper in which the Gauss–Bonnet theorem on the noncommutative two-torus is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we extend these results to finitely generated projective right modules over the noncommutative n-torus. Then we define the corresponding analog of Sobolev spaces and prove equivalents of the Sobolev and Rellich lemmas.

Clarkson’s inequalities for periodic Sobolev space

The validity of Clarkson’s inequalities for periodic functions in the Sobolev space normed without the use of pseudodifferential operators is proved. The norm of a function is defined by using integrals over a fundamental domain of the function and its generalized partial derivatives of all intermediate orders. It is preliminarily shown that Clarkson’s inequalities hold for periodic functions integrable to some power p over a cube of unit measure with identified opposite faces. The work is motivated by the necessity of developing foundations for the functional-analytic approach to evaluating approximation methods.

Wavefront sets and polarizations on supermanifolds

In this paper, we develop the foundation for microlocal analysis on supermanifolds. Making use of pseudodifferential operators on supermanifolds as introduced by Rempel and Schmitt, we define a suitable notion of super-wavefront set for superdistributions which generalizes Dencker’s polarization sets for vector-valued distributions to supergeometry. In particular, our super-wavefront sets detect polarization information of the singularities of superdistributions. We prove a refined pullback theorem for superdistributions along supermanifold morphisms, which as a special case establishes criteria when two superdistributions may be multiplied. As an application of our framework, we study the singularities of distributional solutions of a supersymmetric field theory.

Higher-order conservation laws for the nonlinear Poisson equation via characteristic cohomology

We study higher-order conservation laws of the nonlinearizable elliptic Poisson equation as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. We show that the \({\mathbb{S}^1}\) -symmetry of the PDE leads to a normal form for the undifferentiated conservation laws. Zhiber and Shabat (in Sov Phys Dokl Akad 24(8):607–609, 1979) determine which potentials of nonlinearizable Poisson equations admit nontrivial Lie–Backlund transformations. In the case that such transformations exist, they introduce a pseudo-differential operator that can be used to generate infinitely many such transformations. We obtain similar results using the theory of characteristic cohomology: we show that for higher-order conservation laws to exist, it is necessary that the potential satisfies a linear second-order ODE. In this case, at most two new conservation laws in normal form appear at each even prolongation. By using a recursion motivated by Killing fields, we show that, for the simplest class of potentials, this upper bound is attained. The recursion circumvents the use of pseudo-differential operators. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.

Use of decomposition of the wave equations and pseudo-differential operators for the description of nonparaxial beams and broadband wave packets

We discuss the common approach for deriving the equations for the description of wave beams and wave packets in a wide range of angles and frequencies by decomposition of the initial equations into coupled equations of normal waves with the use of pseudo-differential operators. Estimates of accuracy of the one-wave approximation as a function of accuracy of approximation for the propagation-constant operator are given. We propose to use the method of moments for the description of pulses with a wide spectrum. The transition from the obtained to known equations is described. As an example of using the obtained equations, we analyze the self-focusing and modulation instabilities of plane waves in quartz in the range of transition from the normal to anomalous dispersion of the group velocity.

ω-hypoelliptic differential operators of constant strength

We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in E ω(Ω) which is homogeneous ω-hypoelliptic is also σ-hypoelliptic for any weight function σ= O( ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights ( ω, σ) to conclude that, for any operator P( x, D) with constant strength, the σ-hypoellipticity of the frozen operator P( x 0, D) implies the ω-hypoellipticity of P( x, D). This requires the use of pseudodifferential operators.

On the essential spectrum of the linearized Navier-Stokes operator

We determine the essential spectrum of the linearized Navier-Stokes operator with physical boundary conditions. In contrast to other approaches we do not make use of pseudo-differential operators. We establish a direct proof using only some fundamental results for matrix operators.

The Moyal bracket and the dispersionless limit of the KP hierarchy

A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sato approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, underwhich the Moyal bracket collapses to the Poisson bracket, is particularly simple.

On classical and quantal Kolmogorov entropies

The construction of a quantal Kolmogorov entropy that tends to the classical Kolmogorov entropy in the limit h(cross) to 0 is discussed. The approach is to use sets of functions rather than disjoint partitions and involves the use of pseudo-differential operators.