Laws of the iterated logarithm for high-dimensional wiener sausage

Let {β(s), s ≥ 0} be the standard Brownian motion in ℝ ^d with d ≥ 4 and let |W _r (t)| be the volume of the Wiener sausage associated with {β(s), s ≥ 0} observed until time t. From the central limit theorem of Wiener sausage, we know that when d ≥ 4 the limit distribution is normal. In this paper, we study the laws of the iterated logarithm for | W_r (t) | - \mathbbE| W_r (t) |\left| {W_r (t)} \right| - \mathbb{E}\left| {W_r (t)} \right| in this case.     

A large-deviation result for the range of random walk and for the Wiener sausage

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     

Brownian bridge on hyperbolic spaces and on homogeneous trees

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     

On the First Hitting Time of a One-dimensional Diffusion and a Compound Poisson Process

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.     

Spectral invariance for algebras of pseudodifferential operators on besov-triebel-lizorkin spaces

The algebra of pseudodifferential operators with symbols inS _1,δ ⁰ , δ<1, is shown to be a spectrally invariant subalgebra of ℒ(b _p,q ^s ) and ℒ(F _p,q ^s ). The spectrum of each of these pseudodifferential operators acting onB _p,q ^s orF _p,q ^s is independent of the choice ofs, p, andq.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H 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.     

Hölder continuity for spatial and path processes via spectral analysis

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)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

Numerical Computation of First-Passage Times of Increasing Lévy Processes

Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$E(t) = \inf \{s: D(s) > t \} is a process which has arisen in many applications. Of particular interest is the mean first-hitting time U(t)=\mathbbEE(t)U(t)=\mathbb{E}E(t). This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function [(U)\tilde](l) = (lf(l))^-1\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}, where ϕ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.     

Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V ³ such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let B_s(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V ³ such that y is an internal vertex on an induced path in G between x and z. The ternary relations B_s(P){\cal B}_s(P) and B_s(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation B_s(P)=B_s(G){\cal B}_s(P)={\cal B}_s(G).     

Increment sizes of the principal value of Brownian local time

Let W be a standard Brownian motion, and define Y(t)= ∫₀ ^t ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y. Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

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.     

Stochastic differential equations for Dirichlet processes

We consider the stochastic differential equation dX _t = a(X _t )dW _t + b(X _t )dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness results when a is in C ^1/2 and b is only a generalized function, for example,the distributional derivative of a H?lder function or of a function of bounded variation. When b = aa′, that is, when the generator of the SDE is the divergence form operator ℒ = , a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral. Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

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).     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

Annihilating power values of co-commutators with generalized derivations

Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u)) ⁿ  = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s ₄, the standard identity in four variables.     

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001