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1.
For ν(dθ), a σ-finite Borel measure on R d , we consider L 2(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ t 0 e −λ(θ)( t s ) dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ t 0 g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process t (τ)≗Ytt). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001  相似文献   

2.
Letf t be aC 2 Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB x (ε,t)=[y: dist (f s x,f s y)≤ε for all 0≤st], then volB x (ε,t) has the order exp(∫ 0 t φ (f s x)ds) in the continuous time case and exp (Σ s t−1 φ (f s x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in the unstable direction. An application to the theory of large deviations is given. Partially supported by US-Israel BSF. Partially supported by a Darpa grant.  相似文献   

3.
New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)0 K(t, s) F(s, y(s))ds fort ∈ [0,∞).  相似文献   

4.
An Application of a Mountain Pass Theorem   总被引:3,自引:0,他引:3  
We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, uH 1 0(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L -function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ s 0 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.  相似文献   

5.
We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(xy)/θ(xy) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(xy)+τ(x)τ(y), τ(0)=0. In both cases we find θ2=aθ4+bθ2+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x r x s )) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex r are equidistant.
Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux  相似文献   

6.
Let (zj) be a sequence of complex numbers satisfying |zj| ∞ asj → ∞ and denote by n(r) the number of zj satisfying |zj|≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫ (ϕ(t)t logt)−1 dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.  相似文献   

7.
We consider Dirichlet series zg,a(s)=?n=1 g(na) e-ln s{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?n=1 g(na) zn{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series zg,a(s) = ?n=1 g(n a)/ns{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ 0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ 0 satisfies σ 0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ g,α (s) has an analytic continuation to the entire complex plane.  相似文献   

8.
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim r → ∞ e2r s(r) = 0, then (M, g) has to be isometric to ℍ n . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim r → ∞ r 2 s(r) = 0, then (M, g) is isometric to ℝ n , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if Ka on a geodesic ball B p (R) in M and K = a on ∂B p (R), then K = a on B p (R).  相似文献   

9.
In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T max)/D(T min) associated withM[.] in terms of the behaviour of the determinants det [[f rgs](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.  相似文献   

10.
Let F p,t (n) denote the number of the coefficients of (x 1+1x 2+...+x t ) j , 0 ≤jn− 1, which are not divisible by the prime p. Define G p,t (n) = F p,t /n θ and β(p,t) = lim infF p,t )(n)/n θ, where θ = (log)/(log p). In this paper, we mainly prove that G p,t can be extended to a continuous function on ℝ+, and the function G p,t is nowhere monotonic. Both the set of differential points of the function G p,t and the set of non-differential points of the function G p,t are dense in ℝ+. Received February 18, 2000, Accepted December 7, 2000  相似文献   

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