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1.
We study the asymptotic behaviour of the transition density of a Brownian motion in ?, killed at ∂?, where ? c is a compact non polar set. Our main result concern dimension d = 2, where we show that the transition density p ? t (x, y) behaves, for large t, as u(x)u(y)(t(log t)2)−1 for x, y∈?, where u is the unique positive harmonic function vanishing on (∂?) r , such that u(x) ∼ log ∣x∣. Received: 29 January 1999 / Revised version: 11 May 1999  相似文献   

2.
Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H n , n > 2. For ν > 0, the Brownian bridge B (ν) of length ν on H is the process B t , 0 ≤t≤ν, conditioned by B 0 = B ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ3). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999  相似文献   

3.
Let L p (S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥ p . For a system of sets {B t |t ∈ [0, +∞) n } and a given function ψ: [0, +∞) n ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function fL p (S) such that inf {∥fg p p gL p (S), g = 0 almost everywhere on S\B t } = ψ (t), t ∈ [0, +∞) n . As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L 2. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1116–1127, August, 2006.  相似文献   

4.
Let (B t + f(t)) t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B t + f(t)) t∈[0,+∞) are given in this paper.  相似文献   

5.
Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.  相似文献   

6.
Let K⊂ℝ d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(ji) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {xK : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000  相似文献   

7.
We prove the local ergodic theorem inL : Let {T t}t>0 be a strongly continuous semi-group of positive operators onL 1. IfT t is continuous at 0, then ɛ−10 F T 1 * f(x)dtT 0 * f(x) a.e., for everyf∈L . The technique shows how to obtain theL p local ergodic theorems from theL 1-contraction case. It applies also to differentiation ofL p additive processes. Then-dimensional case, which is new, is proved by reduction to then-dimensionalL 1-contraction case, solved by M. Akcoglu and A. del Junco. Research carried out during a sabbatical leave at the Ohio State University.  相似文献   

8.
Given aL 1(ℝ) and A the generator of an L 1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)= −∞ t a(ts)[Au(s)+f(s,u(s))]ds for each f:ℝ×XX almost automorphic in t, uniformly in xX, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that aL 1(ℝ) positive, nonincreasing and log-convex is already sufficient.  相似文献   

9.
Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t  = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.  相似文献   

10.
Let X(t) be an N parameter generalized Lévy sheet taking values in ℝd with a lower index α, ℜ = {(s, t] = ∏ i=1 N (s i, t i], s i < t i}, E(x, Q) = {tQ: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃tQ such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.  相似文献   

11.
Suppose that(T t )t>0 is aC 0 semi-group of contractions on a Banach spaceX, such that there exists a vectorxX, ‖x‖=1 verifyingJ −1(Jx)={x}, whereJ is the duality mapping fromX toP(X *). If |<T t x,f>|→1, whent→+∞ for somefX *, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onxX is the best possible. If the hypothesisJ −1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ −1(Jx) is weakly compact, then if |<T t x, f>|→1, whent→+∞ for somefX *, ‖f‖≤1, there existsyJ −1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL 1.  相似文献   

12.
It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dB t , X(0) = x 0, through b + Y(t), where b > x 0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.  相似文献   

13.
The stochastic equation dX t =dS t +a(t,X t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0=x 0∈ℝ when (ℛeψ(ξ))−1=o(|ξ|−1) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.  相似文献   

14.
Let (ℋ t ) t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ d with covariance matrix I and drift matrix λ(RI), where λ>0 and R is a skew-adjoint matrix, and denote by γ the invariant measure for (ℋ t ) t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L 2(γ ). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ* f(x)=sup  to |ℋ t f(x)| is of weak type 1 with respect to the invariant measure γ . We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L p (γ ) if and only if 1<p≤∞.   相似文献   

15.
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C c (ℝ n )⊂D(A) and A|C c (ℝ n ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|c(1+|ξ|2) and |Imp(x,ξ)|≤c 0Rep(x,ξ). We show that the associated Feller process {X t } t ≥0 on ℝ n is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β x :={λ>0:lim |ξ|→∞ | x y |≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ x :={λ>0:liminf |ξ|→∞ | x y |≤2/|ξ| |ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ x ) that lim t →0 t −1/λ s t |X s x|=0 or ∞ according to λ>β x or λ<δ x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998  相似文献   

16.
Extremes of independent Gaussian processes   总被引:1,自引:0,他引:1  
Zakhar Kabluchko 《Extremes》2011,14(3):285-310
For every n ∈ ℕ, let X 1n ,..., X nn be independent copies of a zero-mean Gaussian process X n  = {X n (t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a n (M n  − b n ), where M n (t) =  max i = 1,...,n X in (t), and a n , b n are normalizing constants. We also provide an analogous characterization for the limits of the process a n L n , where L n (t) =  min i = 1,...,n |X in (t)|.  相似文献   

17.
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   

18.
Let {S n } be a random walk on ℤ d and let R n be the number of different points among 0, S 1,…, S n −1. We prove here that if d≥ 2, then ψ(x) := lim n →∞(−:1/n) logP{R n nx} exists for x≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A⊂ℝ d let Λ t = Λ t (A) be the d-dimensional Lebesgue measure of the `sausage' ∪0≤ s t (B(s) + A). Then φ(x) := lim t→∞: (−1/t) log P{Λ t tx exists for x≥ 0 and has similar properties as ψ. Received: 20 April 2000 / Revised version: 1 September 2000 / Published online: 26 April 2001  相似文献   

19.
Let X i , iN, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping BR. Under a central limit theorem assumption, an asymptotic evaluation of Z n = E (exp (n Φ (∑ i =1 n X i /n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without the central limit theorem assumption. Received: 19 September 1997 / Revised version:22 April 1999  相似文献   

20.
We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX t is equal to the gradient (in space) of L t (x), in a generalized sense, even though xL t (x) is not differentiable. Received: 15 April 1997 / Revised version: 30 January 1998  相似文献   

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