Theory Of Pseudo-differential Operators Research Articles

On the rôle of singular functions in extending the probabilistic symbol to its most general class

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving Hunt behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.

THE NONLOCAL PROBLEM FOR FRACTAL DIFFUSION EQUATION

Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.

Реалізація математичної моделі в умовах невизначеності і можливих ризиків

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

On recent progress of single-realization recoveries of random Schrödinger systems

<p style='text-indent:20px;'>We consider the recovery of some statistical quantities by using the near-field or far-field data in quantum scattering generated under a single realization of the randomness. We survey the recent main progress in the literature and point out the similarity among the existing results. The methodologies in the reformulation of the forward problems are also investigated. We consider two separate cases of using the near-field and far-field data, and discuss the key ideas of obtaining some crucial asymptotic estimates. We pay special attention on the use of the theory of pseudodifferential operators and microlocal analysis needed in the proofs.</p>

Semilinear degenerate elliptic boundary value problems via the Semenov approximation

This paper is devoted to the study of static bifurcation theory for a class of degenerate boundary value problems for semilinear elliptic differential operators of second-order which includes as particular cases the Dirichlet and Robin problems. The purpose of this paper is to generalize some results of Szulkin (Nonlinear Anal TMA 6:95–116, 1982) to the degenerate case. We study the behavior of nontrivial solution branches for bounded nonlinear terms, and prove that the branches turn back towards the first eigenvalue of the linearized degenerate boundary value problem. Our proof is based on global static bifurcation theory for positive mappings in ordered Banach spaces, with the special emphasis on the Semenov approximation in Chemistry. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the \(L^{p}\) theory of pseudo-differential operators.

Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves

Spectral theory of pseudodifferential operators of degree 0 and an application to forced linear waves

Global pseudodifferential operators of infinite order in classes of ultradifferentiable functions

We develop a theory of pseudodifferential operators of infinite order for the global classes $\mathcal{S}_{\omega}$ of ultradifferentiable functions in the sense of Bj\"orck, following the previous ideas given by Prangoski for ultradifferentiable classes in the sense of Komatsu. We study the composition and the transpose of such operators with symbolic calculus and provide several examples.

Propagation principle for parabolic H-measures

We extend results obtained by Francfort (An introduction to H-measures and their applications. Variational problems in materials science. Birkhauser, Basel, pp 85–110, 2006) to parabolic H-measures developed by Antonic and Lazar (J Funct Anal 265:1190–1239, 2013). The well known theory of pseudodifferential operators is extended to parabolic classes of symbols and operators and used to obtain results applicable to a wide class of partial differential equations. The main result is the propagation principle which is then applied to the Schrodinger equation and the vibrating plate equation.

A $$C^*$$ C ∗ -algebraic approach to the principal symbol II

We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of Sukochev and Zanin (J Oper Theory, 2019) and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes’ trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes’ trace theorem to operators with non-smooth symbols in three examples: the Lie group $$\mathrm {SU}(2)$$ , noncommutative tori and Moyal planes.

Solvability of interior transmission problem for the diffusion equation by constructing its Green function

Abstract Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.

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.

On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra

Since its invention in 1979 the Feichtinger algebra has become a useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several other topics. It is easily defined on locally compact Abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper develops the theory of Feichtinger’s algebra in a linear and comprehensive way. The material gives an entry point into the subject and it will also bring new insight to the expert. A further goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to this space, over operators on it, and to aspects of its dual space. Additional results include a seemingly forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, a new identification of Feichtinger’s algebra as the unique Banach space in $$L^{1}$$ with certain properties, and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications, and a list of related function space constructions is included.

‐SMALL DEVIATIONS FOR WEIGHTED STATIONARY PROCESSES

We find logarithmic asymptotics of $L_2$-small deviation probabilities for weighted stationary Gaussian processes (both for real and complex-valued) having power-type discrete or continuous spectrum. As in the recent work by Hong, Lifshits and Nazarov, our results are based on the spectral theory of pseudo-differential operators developed by Birman and Solomyak.

The initial boundary value problem for free-evolution formulations of general relativity

We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate primarily on boundaries that are geometrically determined by the outermost normal observer to spacelike slices of the foliation. We present high-order-derivative boundary conditions for the gauge, constraint violating and gravitational wave degrees of freedom of the formulation. Second order derivative boundary conditions are presented in terms of the conformal variables used in numerical relativity simulations. Using Kreiss–Agranovich–Métivier theory we demonstrate, in the frozen coefficient approximation, that with sufficiently high order derivative boundary conditions the initial boundary value problem can be rendered boundary stable. The precise number of derivatives required depends on the gauge. For a choice of the gauge condition that renders the system strongly hyperbolic of constant multiplicity, well-posedness of the initial boundary value problem follows in this approximation. Taking into account the theory of pseudo-differential operators, it is expected that the nonlinear problem is also well-posed locally in time.

Exact Fourier inversion formula over manifolds

We show an exact (i.e. no smooth error terms) Fourier inversion type formula for differential operators over Riemannian manifolds. This provides a coordinate free approach for the theory of pseudo-differential operators.

Theory of relativistic heat polynomials and one-sided Lévy distributions

The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These types of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper, we report on new links between fractional diffusion, quantum relativistic equations, and particular families of polynomials, linked to the Bessel polynomials in Carlitz form and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance.

Mathematical quantum Yang–Mills theory revisited

A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang-Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang-Mills spectrum is point and countable. In addition a fine structure of the spectrum is elucidated. KEYS: Millennium Yang-Mills problem; Finite propagation speed; Sobolev inequalities; Nuclear vector spaces; Infinite-dimensional holomorphy; Friedrichs operator extensions; Variational spectral principle; Symbols and spectral theory of pseudo-differential operators.

Fractional-In-Time and Multifractional-In-Space Stochastic Partial Differential Equations

Abstract This paper derives the weak-sense Gaussian solution to a family of fractional-in-time and multifractional-in-space stochastic partial differential equations, driven by fractional-integrated-in-time spatiotemporal white noise. Some fundamental results on the theory of pseudodifferential operators of variable order, and on the Mittag-Leffler function are applied to obtain the temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the derived solution.

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in [26], we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. Three approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped with complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work [1]. The second approach, which complements the first one, is variational and based on the Dirichlet principle. The last approach, which complements the second one, is also variational and uses the multiplier technique. Using these approaches, we are able to obtain new results on the well-posedness of these equations for which the conditions on the coefficients are imposed “partially” or “not strictly” on the interfaces of sign changing coefficients. This allows us to rediscover and extend known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is not in Hloc1(Rd) as usual and possibly not even in Lloc2(Rd). The optimality of our results is also discussed.

In Memoriam

Professor Rémi Vaillancourt passed away on November 29, 2015. He was born in Maniwaki, Quebéc, Canada, in 1934. After completing his undergraduate studies at the University of Ottawa, he defended his PhD thesis at the Courant Institute of Mathematical Sciences in New York under the supervision of P.D. Lax and K.O. Friedrichs in 1969. Later, together with A.P. Calderón, Rémi Vaillancourt obtained important results in the theory of pseudo-differential operators. Since 1970, professor Vaillancourt had worked at the University of Ottawa. He had more than 180 published papers in refereed journals. Professor Rémi Vaillancourt was very active in promoting mathematics and mathematical education. In 1975–1981, he served as a vice-president and president of the Canadian Mathematical Society and later as the president of the Canadian Society of Applied Mathematics. He was also the chairman of the Canadian National Commitee for the International Mathematical Union in 1979–1988. For many years, professor Rémi Vaillancourt was a member of the editorial board of the journal „Scientific proceedings of Riga Technical University”. His contribution to research with the staff of the Department of Engineering Mathematics of the Riga Technical University in the areas of non-destructive testing and fluid mechanics was very significant. We worked with Rémi Vaillancourt for many years. He had an extremely pleasant personality and was always willing to help his colleagues. We had a great pleasure to meet Rémi’s family in Ottawa. During these years, research collaboration with Rémi evolved into closer ties also with his family. We were deeply saddened by the news of Rémi’s passing. He will always be in our hearts.