An abstract averaging theorem

For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of u_t = ?A(t)u is given by an evolution operator V_?(t, s). Conditions are given under which $\text{V}{}_{\mathrm{?}}\text{(}\text{(t+s)}\text{?}\text{,}\text{s}\text{?}\text{)}$ converges strongly as ? → 0 to a semigroup T(t) generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝}{}_0{}^{\mathrm{T}}\text{A(t)f dt}$.This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S_?(t). Conditions are given under which $\text{S(?}\text{t}\text{?}\text{) S}{}_{\mathrm{?}}\text{(}\text{t}\text{?}\text{)}$ converges strongly to a semigroup generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝ S(?t) AS(t)f dt}$. This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation.     

Explicit error terms for asymptotic expansions of Mellin convolutions

Explicit expressions are derived for the error terms associated with the asymptotic expansions of the convolution integral $\text{I(λ) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) h(λt) dt}$, where h(t) and $\text{?(t)}$ are algebraically dominated at both 0⁺ and + ∞. Examples included are Fourier, Bessel, generalized Stieltjes, Hilbert and “potential” transforms.     

Asymptotic evaluation of two families of integrals

In this paper the integrals $\text{f}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[?(t + τ)]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t + τ)}{}^{\mathrm{?1}}\text{dt}\text{and}\text{g}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[? ¦ ? τ ¦]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t ? τ)}{}^{\mathrm{?1}}\text{dt}$ are investigated for positive real values of the variable τ. Here, m is a nonnegative integer, v is a complex variable with Re(v) > ?1. Both integrals are related to the complex integral Φ_mγ(z) = ∝₀^∞exp[?(t ? z)]t^?γ(ln t)^m(t ? z)^?1dt with 0 ? Re(γ) < 1, the behavior of which is analyzed in detail. The results are applied to obtain asymptotic representations for f_mn(τ) and g_mn(τ), m and n both nonnegative integers, near τ = 0. The latter integrals play a role in the study of the equations of neutron transport and radiative transfer.     

On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

Temporally inhomogeneous scattering theory

A theory of scattering for the time dependent evolution equations $\text{du}\text{dt}\text{= iH}{}_{\mathrm{j}}\text{(t)u, j = 0, 1}$ (¹) is developed. The wave operators are defined in terms of the evolution operators U_j(t, s), which govern (¹). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H₀(t) = ?Δ, Δ denoting the Laplacian, and H₁(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable ($\text{H}{}_{\mathrm{j}}\text{(t) = ∑}{}_{\mathrm{m,\; k\; =\; 1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{mk}}{}^{\mathrm{(j)}}\text{(t)(}\text{?}{}^2\text{?x}{}_{\mathrm{m}}\text{?x}{}_{\mathrm{k}}\text{) + b}{}_{\mathrm{j}}\text{(t)}\text{for}\text{j = 0, 1}$).     

On quick oscillations in functional equations with deviating arguments

For the functional equation L_ny(t) + H(t,y(g(t)) = f(t), n ? 2, where $\text{L}{}_{\mathrm{n}}\text{=}\text{1}\text{p}{}_{\mathrm{n}}\text{(t)}\text{d}\text{dt}\text{1}\text{P}{}_{\mathrm{n\; ?\; 1}}\text{(t)}\text{d}\text{dt}\text{…}\text{d}\text{dt}\text{1}\text{P}{}_1\text{(t)}\text{d}\text{dt}\text{·}\text{P}{}_0\text{(t)}$ sufficient conditions have been found to ensure that a solution is either quickly oscillating or else it is nonoscillatory. Both canonical and noncanonical forms of L_n have been studied.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.     

Existence and boundedness of solutions of abstract nonlinear integrodifferential equations of nonconvolution type

Existence and boundedness theorems are given for solutions of nonlinear integrodifferential equations of type $\text{d}\text{dt}\text{u(t) + Bu(t) + ∝}{}_0{}^{\mathrm{t}}\text{a(t, s) Au(s) ds ? f(t) (t > 0)}$, (1.1) u(0) = u₀, Here A and B are nonlinear, possibly multivalued, operators on a Banach space W and a Hilbert space H, where W ? H. The function f (0, ∞) → H and the kernel a(t, s): $\text{R}$ × $\text{R}$ → $\text{R}$ are known functions. The results of this paper extend the results of Crandall, Londen, and Nohel [4] for equation (1.1). They assumed the kernel to be of the type a(t, s) = a(t ? s). We relax this assumption and obtain similar results. Examples of kernels satisfying the conditions we require are given in section 4.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Convergence of convex integrals in Lp spaces

We consider functionals of the form: I_f(u) = ∝_Tf[t, u(t)]μ(dt), which are defined on spaces L^p(T, R^k), and we study for these functionals the properties of a convergence for which the conjugacy $\text{I}{}_{\mathrm{f}}\text{→ I}{}_{\mathrm{f1}}$ is a continuous operator.     

Asymptotic expansions of Fourier transforms of functions with logarithmic singularities

Asymptotic expansion as x → +∞ is obtained for the infinite Fourier integral $\text{F(x) = ∝}{}_0{}^{\mathrm{\infty }}\text{?(t) e}{}^{\mathrm{ixt}}\text{dt}$, in which $\text{?(t)}$ has a logarithmic singularity of the type t^α?1(?ln t)^β at the origin. Here, Re α > 0 and β is an arbitrary complex number.     

Boundary layer methods for certain nonlinear singularly perturbed optimal control problems

We shall examine the control problem consisting of the system $\text{dx}\text{dt}\text{= f}{}_1\text{(x, z, u, t, ?)}$$\text{?(}\text{dz}\text{dt}\text{) = f}{}_2\text{(x, z, u, t, ?)}$ on the interval 0 ? t ? 1 with the initial values x(0, ?) and z(0, ?) prescribed, where the cost functional J(?) = π(x(1, ?), z(1, ?), ?) + ∝₀¹V(x(t, ?), z(t, ?), u(t, ?), t, ?) dt is to be minimized. We shall restrict attention to the special problem where the f_i's are linear in z and u, V is quadratic in z and independent of z when ? = 0, π and V are positive semidefinite functions of x and z, and V is a positive definite function of u. Under appropriate conditions, we shall obtain an asymptotic solution of the problem valid as the small parameter ? tends to zero. The techniques of constructing such asymptotic expansions will be stressed.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

An ∞-dimensional inhomogeneous Langevin's equation

On a modified space Φ′ from the space $\text{J}$′ of tempered distributions, it is proven that a stochastic equation, $\text{X(t) = γ + W(t) + ∝}{}_0{}^{\mathrm{t}}\text{L}{}^1\text{(s) X(s) ds}$, has a unique solution, where W(t) is a Φ′-valued Brownian motion independent of a Φ′-valued Gaussian random variable γ and $\text{L}{}^1\text{(s)}$ is an integro-differential operator. As an application, a fluctuaton result (or central limit theorem) is shown for interacting diffusions.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).     

On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.