Tauberian and Abelian theorems for rapidly decaying distributions and their applications to stable laws

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f ^(α,ρ)(x) of “extreme” stable laws with parameters (α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ^(α,ρ)(x). This asymptotics had been found in [1–5] by a more complicated method.     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Linear information for approximation of the Itô integrals

We study optimal approximation of stochastic integrals in the Itô sense when linear information, consisting of certain integrals of trajectories of Brownian motion, is available. Upper bounds on the nth minimal error, where n is the fixed cardinality of information, are obtained by the Wagner–Platen algorithm and are O(n ^???3/2) or O(n ^???2), depending on considered class of integrands. We also show that Ω(n ^???2) is a lower bound which holds even for very smooth integrands.     

A Stochastic Taylor-Like Expansion in the Rough Path Theory

In this paper, we establish a Taylor-like expansion in the context of rough path theory for a family of Itô maps indexed by a small parameter. We treat not only the case that the roughness p satisfies [p]=2, but also the case that [p]≥3. As an application, we discuss the Laplace asymptotics for Itô functionals of Brownian rough paths.     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

On a system of piecewise linear functions with vanishing integrals on <Emphasis Type="Italic">R</Emphasis>

For an admissible sequence T we define an orthonormal system consisting of piecewise linear functions with vanishing integrals on R. Necessary and sufficient conditions on T are found for the corresponding system to be a basis in H¹(R).     

On the asymptotics of a Bessel-type integral having applications in wave run-up theory

Rapidly oscillating integrals of the form

$$I(r,h) = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {{e^{\frac{i}{h}F(r\cos \phi )}}G(r\cos \phi )d\phi ,} $$

where F(r) is a real-valued function with nonvanishing derivative, arise when constructing asymptotic solutions of problems with nonstandard characteristics such as the Cauchy problem with spatially localized initial data for the wave equation with velocity degenerating on the boundary of the domain; this problem describes the run-up of tsunami waves on a shallow beach in the linear approximation. The computation of the asymptotics of this integral as h → 0 encounters difficulties owing to the fact that the stationary points of the phase function F(r cos ?) become degenerate for r = 0. For this integral, we construct an asymptotics uniform with respect to r in terms of the Bessel functions J ₀(z) and J ₁(z) of the first kind.

     

Unexpected spectral asymptotics for wave equations on certain compact spacetimes

We study the spectral asymptotics of wave equations on certain compact spacetimes, where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime S¹×S². For the Laplacian on S¹×S², theWeyl asymptotic law gives a growth rate O(s^3/2) for the eigenvalue counting function N(s) = #{λ_j: 0 ≤ λ_j ≤ s}. For the wave operator, there are two corresponding eigenvalue counting functions: N^±(s) = #{λ_j: 0 < ±λ_j ≤ s}, and they both have a growth rate of O(s²). More precisely, there is a leading term π²s²/4 and a correction term of as^3/2, where the constant a is different for N^±. These results are not robust in that if we include a speed of propagation constant to the wave operator, the result depends on number theoretic properties of the constant, and generalizations to S¹ × S^q are valid for q even but not q odd. We also examine some related examples.     

On Functions with Zero Integral Moments over Balls of Fixed Radius

We consider precise conditions ensuring that a function having zero integrals over all balls of fixed radius is equal to zero. The case where, together with zero integrals for f over congruent balls, the functions x_jf for all j have zero integrals over these balls is completely investigated. An inversion formula recovering a function in the class C^∞ from indicated integral moments is derived.     

A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S ¹) be the group of orientation preserving C ^?∞? diffeomorphisms of S ¹. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H ^3/2 metric on the Lie algebra diff(S ¹). The canonical Brownian motion they constructed lives in the group Homeo(S ¹) of Hölderian homeomorphisms of S ¹, which is larger than the group Diff(S ¹). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S ¹), rather than in the larger group Homeo(S ¹).     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

Number of lattice points in a sphere

A new estimate is obtained for the residue R in the asymptotics of the number of integer points in a ball of radius a. The estimate has the form R ? a ^{17/14 + ?} .     

Algebraic Theory of Causal Double Products

Corresponding to each “rectangular” double product in the form of a formal power series R[h] with coefficients in the tensor product ?(?)⊙ ? (?) with itself of the Itô Hopf algebra, we construct “triangular” elements T[h] of ?(?) satisfying ΔT[h] = T[h]⁽¹⁾ R[h]T{h]⁽²⁾. In Fock space representations of ?(?) by iterated quantum stochastic integrals when ? is the algebra of Itô differentials of the calculus, these correspond to “causal” double product integrals in a single Fock space.     

Transmission of Waves Through a Small Aperture in the Cross-Wall in an Acoustic Waveguide

We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter ε > 0. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value Λ _k of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λ ^ε = Λ _k + ε²μ² of the spectral parameter with μ ≤ μ₀, which generates scattering coefficients depending on μ > 0 and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics.     

Boundary Non-crossings of Brownian Pillow

Let B ₀(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]²→? be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability

$\psi(u;h):=\mathbf{P}\big\{B_{0}(s,t)+h(s,t)\leq u(s,t),\forall s,t\in[0,1]\big\}.$

Further we investigate the asymptotic behaviour of ψ(u;γ h) with γ tending to ∞ and solve a related minimisation problem.

     

An Obstacle Problem for Nonlocal Equations in Perforated Domains

In this paper we analyze the behavior of solutions to a nonlocal equation of the form J ? u (x) ? u (x) = f (x) in a perforated domain Ω ? A ^?? with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, u ≥ ψ in Ω ? A ^?? . We show that, assuming that the characteristic function of the domain Ω ? A ^?? verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly ^? in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u ^?? and we find the limit problem that is satisfied in the limit. When \(\mathcal {X} \not \equiv 1\) in this limit problem an extra term appears in the equation as well as a modification of the obstacle constraint inside the domain.     

The number of S-precomplete classes in the functional system <Emphasis Type="Italic">P</Emphasis><Stack><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript><Superscript><Emphasis Type="Italic">τ</Emphasis></Superscript></Stack>

The problem on the number of precomplete classes in the functional system P _κ ^τ is considered, elements of P _κ ^τ are deterministic S-functions defined on words of length τ composed from letters of an alphabet of cardinality κ. An asymptotics for the number of S-precomplete classes in P _κ ^τ is obtained for arbitrary fixed κ and τ tending to infinity.     

One-dimensional Lagrangians generated by a quadratic form

In this paper we apply the classical variational calculus to the Lagrangians of particular type. Namely, we establish formulas for the 1^st integrals and propose a technique for obtaining invariant 1^st integrals. We also deduce the differential homogeneity conditions for Lagrangians with respect to multiplication of a path x(t) by a function c(t) and with respect to the change of the parameter t = t(s).     

Asymptotic behavior on (−∞, 0) of the spectral measures of a family of singular Sturm-Liouville operators

We find the asymptotics as λ/? → ?∞ of the density of the spectral measure of the Sturm-Liouville operator in L ²(0,+∞) generated by the expression ?y″ + ?q(x)y, ? > 0, with the boundary condition y(0) cos α+y′(0) sinα = 0. The potential q(x) tends to ?∞ as x → +∞ and is assumed to satisfy the Sears condition and some additional regularity conditions.