A characterization of P2(μ) ≠ L2(μ)

It is shown that P²(μ) ≠ L²(μ) if and only if there exists a probability measure ν with ν ⊥ μ and 6pz.dfnc;_1,ν ? Cz.dfnc;pz.dfnc;_2,μ for all polynomials p and a fixed constant C < ∞. The relationship between this method and theorems of Berger and of Carey and Pincus concerning rationally cyclic vectors for powers of nonnormal subnormal Operators is examined.     

Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential

Consider a Markovian standard semigroup P_t, t≥o, with potential kernel U=∫P_tdt on a locally compact space E. Let μ be a finite measure on E with locally finite potential μ^U and X_t, t≥O, the process having (P_t) as transition semigroup and μ as initial law. Then for a measure ν on E the following two statements are equivalent:

1. μ^U≥νU;
2. there exists a “randomized” stopping time T such that X_T is distributed according to ν.

     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

Abstract Riccati equations in an L1 space of finite measure and applications to transport theory

We consider operator-valued Riccati initial-value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R₀. Here A to D and R₀ have values as non-negative bounded linear operators in L¹ (μ), where μ is a finite measure, and T is a closed non-negative operator in L¹ (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.     

Some well-posed functional equations which generate semigroups

Consider the abstract linear functional equation (FE) (Dx)(t) = f(t) (t ? 0), x(t) = ?(t) (t ? 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L^p-space or C₀-space on R = (?t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R;B;η) which vanish on R^? = (?∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R^? and Y(R⁺; B; η). Examples with B = Cⁿ on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.     

Cores for parabolic operators with unbounded coefficients

Let be a family of elliptic differential operators with unbounded coefficients defined in RN+1. In [M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., in press], under suitable assumptions, it has been proved that the operator G:=A−Ds generates a semigroup of positive contractions (Tp(t)) in Lp(RN+1,ν) for every 1?p<+∞, where ν is an infinitesimally invariant measure of (Tp(t)). Here, under some additional conditions on the growth of the coefficients of A, which cover also some growths with an exponential rate at ∞, we provide two different cores for the infinitesimal generator Gp of (Tp(t)) in Lp(RN+1,ν) for p∈[1,+∞), and we also give a partial characterization of D(Gp). Finally, we extend the results so far obtained to the case when the coefficients of the operator A are T-periodic with respect to the variable s for some T>0.     

Continuous approximation of random function

Modifying the methods of Lee [J. Math. Anal. Appl.61 (1977), 1–6], we show that each μ-measurable mapping f on a normal space T into a separable linear metric space E is almost continuous, where μ is a Radon probability measure. It is shown that for every ε > 0 there exists a compact subset K_ε ? T with μ(K_ε) > 1 ? ε and an elementary function g(t) = ∑ⁿ_{i = 1}h_i(t) x_i such that μ(t?K_ε; f(t) ≠ g(t)) < ε, where x_i?E and h_i(t) are real bounded continuous functions with disjoint supports.     

Parameter-dependent operator equations of the Riccati type with application to transport theory

We obtain an existence result for global solutions to initial-value problems for Riccati equations of the form R′(t) + TR(t) + R(t)T = T^ρ A(t)T^1?ρ + T^ρ B(t)T^1?ρ R(t) + R(t)T^ρC(t) T^1?ρ + R(t)T^ρD(t)T^1?ρ R(t), R(0)=R₀, where 0 ? ρ ? 1 and where the functions R and A through D take on values in the cone of non-negative bounded linear operators on L¹ (0, W; μ). T is an unbounded multiplication operator. This problem is of particular interest in case ρ = 1 since it arisess in the theories of particle transport and radiative transfer in a slab. However, in this case there are some serious difficulties associated with this equation, which lead us to define a solution for the case ρ = 1 as the limit of solutions for the cases 0 < ρ < 1.     

On Dynkin's Markov property of random fields associated with symmetric processes

Let p(t, x, y) be a symmetric transition density with respect to a σ-finite measure m on (E, $\text{E}$), g(x,y)=∫p(t,x,y)dt, and $\text{M}\text{={σ-finite measures μ?0:∫g(x,y)μ(}\text{d}\text{x)μ(}\text{d}\text{y)<∞}}$. There exists a Gaussian random field $\text{Φ={?}{}_{\mathrm{\mu }}\text{:μ?}\text{M}\text{}}$ with mean 0 and covariance $\text{E}\text{?}{}_{\mathrm{\mu }}\text{?}{}_{\mathrm{\nu }}\text{=∫g(x,y)μ(}\text{d}\text{x)ν(}\text{d}\text{y)}$. Letting $\text{F}\text{(B)=σ{?}{}_{\mathrm{\mu }}\text{:μ(B}{}^{\mathrm{c}}\text{)=0}}$ we consider necessary and sufficient conditions for the Markov property (MP) on sets B, C: $\text{F}$(B), $\text{F}$(C) c.i. given $\text{F}$(B ∩ C). Of crucial importance is the following, proved by Dynkin: $\text{E}\text{{?}{}_{\mathrm{\mu }}\text{∣}\text{F}\text{(B)}=?}{}_{\mathrm{\mu B}}$, where μ_B is the hitting distribution of the process corresponding to p, m with initial law μ. Another important fact is that ?_μ=?_ν iff μ, ν have the same potential. Putting these together with an additional transience assumption, we present a potential theoretic proof of the following necessary and sufficient condition for (MP) on sets B, C: For every x?E, T_B∩C=T_B+T_C∮ θ_TB=T_C+T_B∮θ_TC a.s. P^x where, for D ? $\text{E}$, T_D is the hitting time of D for the process associated with p, m. This implies a necessary condition proved by Dynkin in a recent preprint for the case where B∪C=E and B, C are finely closed.     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

Suppose that ? ⁿ is the p-dimensional space with Euclidean norm ∥ ? ∥, K (? ^p ) is the set of nonempty compact sets in ? ^p , ?₊ = [0, +∞), D = ?₊ × ? ^m × ? ⁿ × [0, a], D ₀ = ?₊ × ? ^m , F ₀: D ₀ → K (? ^m ), and co F ₀ is the convex cover of the mapping F ₀. We consider the Cauchy problem for the system of differential inclusions $$\dot x \in \mu F(t,x,y,\mu ),\quad \dot y \in G(t,x,y,\mu ),\quad x(0) = x_0 ,\quad y(0) = y_0$$ with slow x and fast y variables; here F: D → K (? ^m ), G: D → K (? ⁿ ), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F ₀, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ₀ > 0 such that for any μ ∈ (0, μ₀] and any solution (x _μ(t), y _μ(t)) of the problem under consideration, there exists a solution u _μ(t) of the problem ${\dot u}$ ∈ μ co F ₀ (t, u), u(0) = x ₀ for which the inequality ∥x _μ(t) ? u _μ(t)∥ < ε holds for each t ∈ [0, 1/μ].     

Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition

Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule u_ε(R_ε(t),t)=εR_ε′(t) at the moving boundary are considered. We prove, when ε approaches zero, R_ε(t) converges to R(t) in C^1+δ/2[0,T] for any finite T>0, 0<δ<1.     

Transportation-information inequalities for Markov processes

In this paper, one investigates the transportation-information T _c I inequalities: α(T _c (ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space ${(\mathcal{X}, d)}$ , where μ is a given probability measure, T _c (ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on ${\mathcal{X}^2}$ , I(ν|μ) is the Fisher–Donsker–Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that T _c I is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(·|μ). The tensorization property of T _c I and comparisons of T _c I with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of T _c I and several examples are worked out.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk

We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk D, into the Bloch-type space B_ν as well as the little Bloch-type space B_ν,0, consisting respectively of all holomorphic functions f on D such that sup_z∈Dν(z)|f^′(z)|<∞, that is, of all holomorphic functions f on D such that lim_|z|→1ν(z)|f^′(z)|=0, for some weight function ν. As a byproduct of our results, norm of the operator is calculated when B_ν is replaced by B_ν/C.     

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.     

Une série génératrice des formes paraboliques de Maass

A certain periodic function F_μ,ν(z), an eigenfunction of the Laplacian on the upper half-plane with respect to z depending on some parameters μ, ν, is not automorphic, but the function μ → F_μ,ν(z)−F_μ,ν(−1/0 extends to a larger domain than the function F_μ,ν(z) itself. Consequently, at the poles of this latter function of μ, the coefficients of the polar parts provide non-analytic modular forms: all Maass cusp forms are finite linear combinations of forms obtained in this way, allowing the second parameter ν to vary     

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.     

Browder-Weyl theorems, tensor products and multiplications

A Banach space operator T∈B(X) satisfies Browder's theorem if the complement of the Weyl spectrum σw(T) of T in σ(T) equals the set of Riesz points of T; T is polaroid if the isolated points of σ(T) are poles (no restriction on rank) of the resolvent of T. Let Φ(T) denote the set of Fredholm points of T. Browder's theorem transfers from A,B∈B(X) to S=LARB (resp., S=A⊗B) if and only if A and B^∗ (resp., A and B) have SVEP at points μ∈Φ(A) and ν∈Φ(B) for which λ=μν∉σw(S). If A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to LARB; again, restricting ourselves to the completion of X⊗Y in the projective topology, if A and B are finitely polaroid, then the polaroid property transfers from A∈B(X) and B∈B(Y) to A⊗B.     

Temps de sortie d’un intervalle borné pour l’intégrale du mouvement brownien

Let (B_t)_t≥0 be a standard Brownian motion starting at y, X_t = x+ ∫₀^tB_s, ds, x ∈ (a, b). Let us set T_ab = inf{t > 0 : X_t ∉ (a,b)}. In this paper, we compute the moments of the random variable B_Ta,b, and deduce the probability law of B_Ta,b. We show how to obtain the expectation E_(x,y)(T_ab^mB_Tabⁿ). We also explicitly determine the probabilities P_(x,y){X_Tab = a} and P_(x,y) { X_Tab = b}.