On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

Remarks on the equation dXt = a(Xt)dBt

We prove that if a global solution of the equation dX_t = a(X_t) dB_t, X₀ = x exists for some x ? $\text{R}$ and ∫^∞₀a²(X_s)ds = ∞, then one must have a ≠ 0 a.e.     

Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

Diffusions conditionnelles. I. Hypoellipticité partielle

This paper is devoted to the proof of certain results on conditional diffusions. By using the Malliavin calculus of variations, it is shown that, under minimal conditions on the vectors fields which define a two component diffusion (x_t, z_t) and their Lie brackets, a.s., for any T ? 0, the conditional law of x_t (t > 0), given $\text{B}$(z_s,s?T) a C^∞ density.     

Bounded variation evolution equations

A “fundamental theory” is presented for the equation x(t) = ∫₀^tq(x(s), s) ds where the integral is Stieltjes and x is of bounded variation with values in Rⁿ. This includes the ordinary differential equation (o.d.e.) case with impulses. The principal conclusion is that the corresponding conditions for Carathéodory's o.d.e. problem carry over almost unchanged to the more general case. Areas treated include existence (local and global), uniqueness, dependence, integral funnels, stability, and Picard iterates.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

On the stability properties of perturbed linear nonstationary systems

Several results are presented that relate the stability properties of a perturbed linear nonstationary system ?(t) = (A(t) + B(t)) x(t) to those of an unperturbed linear system ?(t) = A(t) x(t). Similarly, the stability properties of the discrete system x_{k + 1} = (A_k + B_k) x_k are related to those of x_{k + 1} = A_kx_k.     

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ? ⁿ⁺¹, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ? ? ⁿ⁺¹ whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τ?(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ?(x, t) to be the function ?(x, t) = s ²(x, x ⁰) ? pt ², where s(x, x ⁰) is the distance between points x and x ⁰ in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x ⁰ can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k ₀ ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k ₀ and sup _x∈Ω s ²(x, x ⁰) satisfies some smallness condition.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Representation and approximation of solutions to semilinear Volterra equations with delay

In this paper we use a theorem of Crandall and Pazy to provide the product integral representation of the nonlinear evolution operator associated with solutions to the semilinear Volterra equation: x(?)(t) = W(t, τ) ?(0) + ∝_τ^tW(t, s)F(s, x_s(?)) ds.Here the kernel W(t, s) is a linear evolution operator on a Banach space X; I is an interval of the form [?r, 0] or (?∞, 0] and F is a nonlinear mapping of R × C(I, X) into X. The abstract theory is applied to examples of partial functional differential equations.     

Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Semi-stable Markov processes in rn

A Markov process in Rⁿ{x_t} with transition function P_t is called semi-stable of order α>0 if for every a>0, P_t(x, E) = P_at(a^ax, a^aE). Let ?_t(ω)=∫^t₀|x_s(ω)|^-1/α ds, T(t) be its inverse and {y_t}={x_T(t)}.Theorem 1: {Y_t} is a multiplicative invariant process; i.e., it has transition function q_t satisfying q_t(x,E)=q_t(ax,aE) for all a > 0.Theorem 2: If {x_t} is Feller, right continuous and uniformly stochastic continuous on a neighborhood of the origin, then {y_t} is Feller.     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

Periodic solutions of semilinear equations of evolution of compact type

The existence of solutions in a weak sense of x′ + (A + B(t, x))x = f(t, x), x(0) = x(T) is established under the conditions that A generates a semigroup of compact type on a Hilbert space H; B(t,x) is a bounded linear operator and f(t, x) a function with values in H; for each square integrable ?(t) the problem with B(t, ?(t)) and f(t, ?(t)) in place of B(t, x) and f(t, x) has a unique solution; and B and f satisfy certain boundedness and continuity conditions.     

Loewner chains and parametric representation in several complex variables

Let B be the unit ball of with respect to an arbitrary norm. We study certain properties of Loewner chains and their transition mappings on the unit ball B. We show that any Loewner chain f(z,t) and the transition mapping v(z,s,t) associated to f(z,t) satisfy locally Lipschitz conditions in t locally uniformly with respect to z∈B. Moreover, we prove that a mapping f∈H(B) has parametric representation if and only if there exists a Loewner chain f(z,t) such that the family {e^−tf(z,t)}_t?0 is a normal family on B and f(z)=f(z,0) for z∈B. Also we show that univalent solutions f(z,t) of the generalized Loewner differential equation in higher dimensions are unique when {e^−tf(z,t)}_t?0 is a normal family on B. Finally we show that the set S⁰(B) of mappings which have parametric representation on B is compact.