-Adic Markov process and the problem of the first return over balls

UDC 511.225, 519.217, 511.225.1, 303.532 We consider the pseudodifferential operator defined as where and study the Markov process associated to this operator. We also study the first passage time problem associated to for

A groupoid approach to pseudodifferential calculi

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural {\mathbb{R}^{\times}_{+}}-action. Specifically, a properly supported semiregular distribution on {M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the {\mathbb{R}^{\times}_{+}}-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

P-wave extrapolation in anisotropic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme, and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Asymptotics for the Ostrovsky-Hunter Equation in the Critical Case

We consider the Cauchy problem for the Ostrovsky-Hunter equation ∂x∂tu-b/3∂x3u-∂xKu3=au, t,x∈R2, u0,x=u0x, x∈R, where ab>0. Define ξ0=27a/b1/4. Suppose that K is a pseudodifferential operator with a symbol K^ξ such that K^±ξ0=0, Im K^ξ=0, and K^ξ≤C. For example, we can take K^ξ=ξ2-ξ02/ξ2+1. We prove the global in time existence and the large time asymptotic behavior of solutions.

Singular equivariant asymptotics and Weyl’s law. On the distribution of eigenvalues of an invariant elliptic operator

We study the spectrum of an invariant, elliptic, classical pseudodifferential operator on a closed Riemannian manifold