On the boundedness of the total variation of the logarithm of a Blaschke product

We establish that, for a Blaschke product B(z) convergent in the unit disk, the condition - ∞ < \smallint ₀¹ log(1 - t)n(t,B)dt\smallint _0^1 \log (1 - t)n(t,B)dt is sufficient for the total variation of logB to be bounded on a circle of radiusr, 0 <r < 1. For products B(z) with zeros concentrated on a single ray, this condition is also necessary. Here, n(t, B) denotes the number of zeros of the functionB (z) in a disk of radiust.     

Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach

Summary This paper deals with the existence of solutions for the implicit Cauchy problem F(t, x, x)=_B, x(t₀)=x₀ in a Banach space B. By using the Kuratowski and the Hausdorff measure of non compactness, we prove an existence theorem for the previous problem (Teorema 1.1) and its extension to non compact intervals (Teorema 2.1). These results generalize the previous ones by R.Conti [1] (in the case B=R), G.Pulvirenti [2] and T. Dominguez Benavides [3], [4] (in the general case). In particular, we relax a Lipschitz condition assumed by all of the abovementioned authors. Some applications of Teorema 2.1 are presented.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E: $({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.$ Here, ${(A(t))_{t\in [0,T]}}In this paper we study the following non-autonomous stochastic evolution equation on a Banach space E:

(SE)    {ll dU(t) = (A(t)U(t) +F(t,U(t))) dt + B(t,U(t)) dWH(t),     t ? [0,T], U(0) = u0.({\rm SE})\quad \left\{\begin{array}{ll} {\rm d}U(t) = (A(t)U(t) +F(t,U(t)))\,{\rm d}t + B(t,U(t))\,{\rm d}W_H(t), \quad t\in [0,T], \\ U(0) = u_0.\end{array}\right.      4. TOWARDS A CHARACTERIZATION OF EVENTUALLY NORM CONTINUOUS SEMIGROUPS ON BANACH SPACES      《Quaestiones Mathematicae》2013,36(1-2):183-190 Abstract We characterize the condition (E) limμ→±∞ ∥R(iμ,A∥ = 0 for a generator A of an exponentially stable semigroup (T(t))t≥0 on an arbitrary Banach space in terms of (T(t))t≥0. As shown earlier on Hilbert spaces this condition is equivalent to the norm continuity of the semigroup for t > 0.      5. Global stabilization of nonlinear systems by quadratic Lyapunov functions      I. E. Zuber  A. Kh. Gelig 《Vestnik St. Petersburg University: Mathematics》2010,43(1):49-53 The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t 0 and x ∈ ℝ n , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t 0 and x ∈ ℝ n . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.      6. Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks   总被引：1，自引：0，他引：1   Liming Wu 《Journal of Theoretical Probability》1999,12(3):661-673 Given a Brownian motion (B t) t0 in R d and a measurable real function f on R d belonging to the Kato class, we show that 1/t 0 t f(B s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.      7. On normal control processes      R. Conti 《Journal of Optimization Theory and Applications》1974,14(5):497-503 Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB p′, r′ of the control spaceL p′([τ, T]),l m r ′ are characterized in terms ofH(t)=X(T)X ?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.      8. Self-packing of centrally symmetric convex bodies in ℝ2      P. G. Doyle  J. C. Lagarias  D. Randall 《Discrete and Computational Geometry》1992,8(1):171-189 LetB be a compact convex body symmetric around0 in ℝ2 which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thatt B can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have       9. Removable singularities of weak solutions to the navier-stokes equations      Hideo Kozono 《偏微分方程通讯》2013,38(5-6):949-966 Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R3. We show that there is an absolute constant ε0 such that ever, y weak solution u with the property that Suptε(a,b)|u(t)|L(D)≤ε0 is necessarily of class C∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (xo, to) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|-1) as x→x 0 uniforlnly in t in some neighbourliood of to, then (xo,to) is actually a removable singularity of u.      10. On unbounded perturbations of semigroups: Compactness and norm continuity   总被引：2，自引：0，他引：2   Miao?LiEmail author  Xiaohui?Gu  Falun?Huang 《Semigroup Forum》2002,65(1):58-70 Let A and B be the generators of strongly continuous semigroups (S(t)) t \geq 0 and (T(t)) t \geq 0 , respectively. Denote by Δ (t) = T(t) -S(t) . We show that if Δ(t) is norm continuous for t>0 and R(λ,B)-R(λ,A) is compact for λ ∈ ρ(A)\cap ρ(B) , then Δ(t) is compact. The converse is true if the perturbing operator is of Miyadera-Voigt-type. A characterization of norm continuity of Δ(t) in terms of the resolvents of the generators is given in Hilbert spaces.      11. Normalized trigonometric splines of Lagrange type      A. A. Makarov 《Vestnik St. Petersburg University: Mathematics》2008,41(3):266-272 B ?-splines, which are a nonpolynomial generalization of the well-known B-splines, are investigated. B ?-splines arise from approximation relations regarded as a system of linear algebraical equations, from which both polynomial and nonpolynomial splines are derived. Third-order normalized trigonometric splines of Lagrange type (zero height) determined by the generating vector function ?(t) = (1, sin t, cos t, sin2 t) T are constructed. These splines are twice continuously differentiable and have minimal compact support. A system of functionals biorthogonal to B ?-splines is defined. The solution of the interpolation problem generated by the resulting biorthogonal system in the space of B ?-splines is found.      12. New oscillation criteria for linear matrix Hamiltonian systems      Yuan Gong Sun 《Journal of Mathematical Analysis and Applications》2003,279(2):651-658 Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t0,∞), (−∞<t0). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.      13. On the spectral mapping theorem for perturbed strongly continuous semigroups      S. Brendle  R. Nagel  J. Poland 《Archiv der Mathematik》2000,74(5):365-378 We consider a strongly continuous semigroup (T(t))t \geqq 0(T(t))_{t \geqq 0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))t \geqq 0(S(t))_{t \geqq 0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))t \geqq 0(S(t))_{t \geqq 0} .      14. The occupation measure of super-Brownian motion conditioned to nonextinction      Laurent Serlet 《Journal of Theoretical Probability》1996,9(3):561-578 We study super-Brownian motion inR d starting from a nontrivial finite measure and conditioned to nonextinction as defined by Evans. If (Y t ) t0 denotes this process, we provide a new approach to the immortal particle representation of (Y t ) t0 . We then show that the measureZ onR d defined byZ(B)= o 1 Y t (B) dt is almost surely finite on compact sets whend5 and almost surely infinite on every ball whend4.      15. Randomly fractionally integrated processes      P. Doukhan  G. Lang  D. Surgailis 《Lithuanian Mathematical Journal》2007,47(1):1-23 Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d t ) t∈ℤ of real numbers; if the parameter sequence is constant d t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε t and B(d)ε t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X t A = A(d)ε t and X t B = B(d)ε t by assuming that d = (d t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε t ). In the case where the mean , we show that large sample properties of X A and X B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X t A ) of a randomly fractionally integrated process X t A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.      16. Extremes of Gaussian random fields with regularly varying dependence structure      Krzysztof Dȩbicki  Enkelejd Hashorva  Peng Liu 《Extremes》2017,20(2):333-392 Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ 2(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t 0. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t 0. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.      17. Extreme points, support points and the Loewner variation in several complex variables      GRAHAM Ian  HAMADA Hidetaka  KOHR Gabriela  KOHR Mirela 《中国科学 数学(英文版)》2012,55(7):1353-1366 In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.      18. The modulator of the local time      Simeon M. Berman 《纯数学与应用数学通讯》1988,41(1):121-132 Let x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, αm(x, 1) is integrable over B. The modulator is a function Mm(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the Lm(B)-valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.      19. On sets of linear differential systems to which perturbed linear systems cannot be reduced      N. A. Izobov  S. A. Mazanik 《Differential Equations》2011,47(11):1563-1568 The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition || B(t) - A(t) || \leqslant const ×e - st ,t \geqslant 0,\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,      20. Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges      Mátyás Barczy  Endre Iglói 《Central European Journal of Mathematics》2011,9(1):65-84 We study Karhunen-Loève expansions of the process(X t (α)) t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

TOWARDS A CHARACTERIZATION OF EVENTUALLY NORM CONTINUOUS SEMIGROUPS ON BANACH SPACES

Abstract

We characterize the condition

(E) lim_μ→±∞ ∥R(iμ,A∥ = 0

for a generator A of an exponentially stable semigroup (T(t))_t≥0 on an arbitrary Banach space in terms of (T(t))_t≥0. As shown earlier on Hilbert spaces this condition is equivalent to the norm continuity of the semigroup for t > 0.     

Global stabilization of nonlinear systems by quadratic Lyapunov functions

The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t ₀ and x ∈ ℝ ⁿ , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t ₀ and x ∈ ℝ ⁿ . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

Self-packing of centrally symmetric convex bodies in ℝ2

LetB be a compact convex body symmetric around0 in ℝ² which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thatt B can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

On unbounded perturbations of semigroups: Compactness and norm continuity

Let A and B be the generators of strongly continuous semigroups (S(t)) _{t \geq 0} and (T(t)) _{t \geq 0} , respectively. Denote by Δ (t) = T(t) -S(t) . We show that if Δ(t) is norm continuous for t>0 and R(λ,B)-R(λ,A) is compact for λ ∈ ρ(A)\cap ρ(B) , then Δ(t) is compact. The converse is true if the perturbing operator is of Miyadera-Voigt-type. A characterization of norm continuity of Δ(t) in terms of the resolvents of the generators is given in Hilbert spaces.     

Normalized trigonometric splines of Lagrange type

B _?-splines, which are a nonpolynomial generalization of the well-known B-splines, are investigated. B _?-splines arise from approximation relations regarded as a system of linear algebraical equations, from which both polynomial and nonpolynomial splines are derived. Third-order normalized trigonometric splines of Lagrange type (zero height) determined by the generating vector function ?(t) = (1, sin t, cos t, sin2 t) ^T are constructed. These splines are twice continuously differentiable and have minimal compact support. A system of functionals biorthogonal to B _?-splines is defined. The solution of the interpolation problem generated by the resulting biorthogonal system in the space of B _?-splines is found.     

New oscillation criteria for linear matrix Hamiltonian systems

Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t₀,∞), (−∞<t₀). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.     

On the spectral mapping theorem for perturbed strongly continuous semigroups

We consider a strongly continuous semigroup (T(t))_{t \geqq 0}(T(t))_{t \geqq 0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} .     

The occupation measure of super-Brownian motion conditioned to nonextinction

We study super-Brownian motion inR ^d starting from a nontrivial finite measure and conditioned to nonextinction as defined by Evans. If (Y _t ) _t0 denotes this process, we provide a new approach to the immortal particle representation of (Y _t ) _t0 . We then show that the measureZ onR ^d defined byZ(B)= _o ¹ Y _t (B) dt is almost surely finite on compact sets whend5 and almost surely infinite on every ball whend4.     

Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d _t ) _t∈ℤ of real numbers; if the parameter sequence is constant d _t ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)^−d . They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε _t and B(d)ε _t for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X _t ^A = A(d)ε _t and X _t ^B = B(d)ε _t by assuming that d = (d _t , t ∈ ℤ) is a random iid sequence, independent of the noise (ε _t ). In the case where the mean , we show that large sample properties of X ^A and X ^B are similar to FARIMA(0, , 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter . The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(X _t ^A ) of a randomly fractionally integrated process X _t ^A with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d _t , this reduces to the standard Hermite rank used in Dobrushin and Major [2]. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007.     

Extremes of Gaussian random fields with regularly varying dependence structure

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ ²(?) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ?, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 ? σ(t) has a polynomial decrease to 0 as t → t ₀. In this contribution we consider more general case that 1 ? σ(t) is regularly varying at t ₀. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t ₀. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

The modulator of the local time

Let x(t), 0 ≦ t ≦ 1, be a real measurable function having a local time α(x, t) which is a continuous function of t for almost all x. It is also assumed that, for some m ≧ 2 and some real interval B, α^m(x, 1) is integrable over B. The modulator is a function M_m(t, B), t > 0, denned in terms of α. It is shown that the modulator serves as a measure of the smoothness of the L_m(B)-valued function α(., t) with respect to t. Then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x(t). The results are applied to the case where x(t) is the sample function of a real stochastic process. In this way new results are obtained for large classes of Gaussian and Markov processes.     

On sets of linear differential systems to which perturbed linear systems cannot be reduced

The paper [2] defines the noncoinciding irreducibility sets N ₂(a, σ) and N ₃(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N ₂(a, σ)] the condition

Karhunen-Loève expansions of <Emphasis Type="Italic">α</Emphasis>-Wiener bridges

We study Karhunen-Loève expansions of the process(X _t ^(α)) _t∈[0,T) given by the stochastic differential equation $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) , with the initial condition X ₀^(α) = 0, where α > 0, T ∈ (0, ∞), and (B _t )t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X ^(α). As applications, we calculate the Laplace transform and the distribution function of the L ²[0, T]-norm square of X ^(α) studying also its asymptotic behavior (large and small deviation).