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.     

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

Hausdorff graph topology, proximal graph topology and the uniform topology for densely continuous forms and minimal USCO maps

For metric spaces (X, d _x) and (Y, d _y) we consider the Hausdorff metric topology on the set (CL(X × Y), ρ) of closed subsets of the product metrized by the product (box) metric ρ and consider the proximal topology defined on CL(X × Y). These topologies are inherited by the set G(X, Y) of closed-graph multifunctions from X to Y, if we identify each multifunction with its graph. Finally, we consider the topology of uniform convergence τ _uc on the set F(X, 2^Y) of all closed-valued multifunctions, i.e. functions from X to the set (CL(Y),) of closed subsets of Y metrized by the Hausdorff metric . We show the relationship between these topologies on the space G(X, Y) and also on the subspaces of minimal USCO maps and locally bounded densely continuous forms. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank.ubica Holá for suggestions and comments.     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Measure of the multiple self-intersection set of a markov process

Let X(t), t ≧ 0, be a Markov process in R^m with homogeneous transition density p(t; x, y). For a closed bounded set B ? R^m, X is said to have a self-intersection of order r ≧ 2 in B if there are distinct points t₁ < … < t_r such that X(t₁) ∈ B and X(t_j) = X(t¹), for j = 2,…, r. The focus of this work is the Hausdorff measure, suitably defined, of the set of such r-tuples. The main result is that under general conditions on p(t; x, y) as well as the specific condition there is a measure function M(t), defined in terms of the integral above, such that the corresponding Hausdorff measure of self-intersection set is positive, with positive probability. The results are applied to Lévy and diffusion processes, and are shown to extend recent results in this area.     

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P ₀(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P ₀(X) if and only if d ^–1 is hereditarily precompact.     

Evolution equations governed by the sweeping process

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX(t) -N _C(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1. In Sections 5 and 6, a second-order sweeping process is considered:X (t) -N _C(X(t)) (X(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.     

A Nonparametric Estimation Problem From Indirect Observations

This paper deals with a nonparametric estimation problem of an integral-type functional from indirect observations where the observation Y (t) is a sum of a known function of an unobservable process X (t) and a Gaussian white noise, and X (t) is a sum of an unknown function a(t) and a Gaussian process. The minimax lower bound on the quality of nonparametric estimation is derived and an asymptotically efficient estimator is proposed. The paper concludes with some examples including one about reduction to parameter estimation.     

Iterative Pexider equation modulo a subset

Summary LetX, Y, Z be arbitrary nonempty sets,E be a nonempty subset ofZ ^z andK be a groupoid. Assume that {F _t} _{t K} Z ^X, {G _t} _{t K} Y ^X, {H _t} _{t K} Z ^Y are families of functions satisfying the functional equationF _st = k_(s,t) H_s G_t for (s, t) D(K), whereD(K) stands for the domain of the binary operation on the groupoidK andk _(s,t) E for (s, t) D(K). Conditions are established under which the equation can be reduced to the corresponding Cauchy equation. This paper generalizes some results from [4] and [1].     

On the Vector Process Obtained by Iterated Integration of the Telegraph Signal

We analyse the vector process (X ₀(t), X ₁(t),...,X _n(t), t > 0) where , and X ₀(t) is the o two-valued telegraph process.In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed.Special care is given to the case of the process (X ₀(t), X ₁(t), X ₂(t), t > 0) representing a randomly accelerated motion where some explicit results on the probability distribution are derived.     

Iterations of quasi-uniform hyperspace constructions

Summary Given a quasi-uniform space (X,U), we study its Hausdorff quasi-uniformity U_H on the set P₀(X) of nonempty subsets of the set X. In particular we are concerned with the question whether at a certain finite stage iterations of the described Hausdorff hyperspace construction applied to two distinct quasi-uniformities on X will necessarily lead to hyperspaces carrying distinct induced topologies.     

Exponential-cosine operator-valued functional equation in the strong operator topology

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR ⁺ = [0, ). A one parameter family of operators {S(t);t R ⁺},S:R ⁺ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S ²(s))S(t –s), for alls, t R ⁺,s t. Let ,fD(A), and ,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A ²) implies ₀ ^t (t –u(S(u)fduD(B) and B ₀ ^t (t –u)S(u)fdu =S(t)f –f +tAf – 2A ₀ ^t S(u)fdu + 2A ² ₀ ^t (t –u)S(u)fdu.D(B) is seen to be dense inD(A ²). Some regularity properties ofS(t) have also been obtained.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

Sample Path Properties of a Class of Operator Stable Processes

Abstract

Let X = {X(t), t ? ?₊} be an operator stable Lévy process on ? ^d with the exponent B, where B is a diagonal matrix. In the present paper, we consider the asymptotic behavior of the first passage time out of a sphere, and of the sojourn time in a sphere. We shall also determine the exact Hausdorff measure function for the range of X over unit time interval [0, 1].     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

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(j−i) (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 {x∈K : 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     

Hausdorff Measures and Capacities for Increase Times of Stable Processes

Let X be a real-valued stable process. It is known that X possesses increase times iff (X ₁ > 0) > ([1]). In that case, we specify the exact Hausdorff function of the set of increase times, and characterize the subsets of [0,+) which contain increase times in terms of their capacity.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.