Local dimensions of measures on infinitely generated self-affine sets

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space.     

Local independence of fractional Brownian motion

Let _σ(t,t′)${\sigma }_{(t,t^{\prime })}$ be the sigma-algebra generated by the differences _Xs−_Xs′$X_s-X_{s^{\prime }}$ with s,s^′∈(t,t^′)$s,s^{\prime }\in (t,t^{\prime })$, where _{(Xt)−∞<t<∞}${\left(X_t\right)}_{-\infty <t<\infty }$ is the fractional Brownian motion with Hurst index H∈(0,1)$H\in (0,1)$. We prove that for any two distinct timepoints _t1$t_1$ and _t2$t_2$ the sigma-algebras _{σ(t1−ε,t1+ε)}${\sigma }_{(t_1-\epsilon ,t_1+\epsilon )}$ and _{σ(t2−ε,t2+ε)}${\sigma }_{(t_2-\epsilon ,t_2+\epsilon )}$ are asymptotically independent as ε↘0$\epsilon \searrow 0$. We show the independence in the strong sense that Shannon’s mutual information between the two σ$\sigma$-algebras tends to zero as ε↘0$\epsilon \searrow 0$. Some generalizations and quantitative estimates are also provided.     

On the time inhomogeneous skew Brownian motion

This paper is devoted to the construction of a solution for the “Inhomogeneous skew Brownian motion” equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Étoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.     

Jordan curves in the level sets of additive Brownian motion

This paper studies the topological and connectivity properties of the level sets of additive Brownian motion. More precisely, for each excursion set of this process from a fixed level, we give an explicit construction of a closed Jordan curve contained in the boundary of this excursion set, and in particular, in the level set of this process.

     

Uniform perfectness of self-affine sets

Let be affine maps of Euclidean space with each nonsingular and each contractive. We prove that the self-affine set of is uniformly perfect if it is not a singleton.

     

The local time of iterated Brownian motion

We define and study the local time process {L ^*(x,t);x¹,t0} of the iterated Brownian motion (IBM) {H(t):=W ₁(|W ₂ (t)|); t0}, whereW ₁(·) andW ₂(·) are independent Wiener processes.Research supported by Hungarian National Foundation for Scientific Research, Grant No. T 016384.Research supported by an NSERC Canada Grant at Carleton University, Ottawa.Research supported by a PSC CUNY Grant, No. 6-66364.     

The exact Hausdorff measure of the level sets of Brownian motion

Summary We show that if s(t, x) is the local time of a Brownian motion B, and (t)=(2t¦log|logt)^1/2 then –m({s=x})=s(t,x) for all t>=0 and x real a.s., where –m(E) is the Hausdorff -measure of E. This solves a problem of Taylor and Wendel who proved the above equality, up to a multiplicative constant, for x=0.     

Dimension spectra of self-affine sets

The dimension spectrumH(δ) is a function characterizing the distribution of dimension of sections. Using the multifractal formula for sofic measures, we show that the dimension spectra of irreducible self-affine sets (McMullen’s Carpet) coincide with the modified Legendre transform of the free energy Ψ_d(β). This variational relation leads to the formula of Hausdorff dimension of self-affine sets, max(δ +H(δ)) = Ψ_d(η), whereη is the logarithmic ratio of the contraction rates of the affine maps.     

Local times for Brownian motion on the Sierpinski carpet

Summary Jointly continuous local times are constructed for Brownian motion on the Sierpinski carpet. A consequence is that the Brownian motion hits points. The method used is to analyze a sequence of eigenvalue problems.Research partially supported by NSF grant DMS 87-01073     

The intrinsic local time sheet of Brownian motion

Summary McGill showed that the intrinsic local time process (t, x), t 0, x , of one-dimensional Brownian motion is, for fixedt>0, a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.     

Brownian motion with negative drift and convex level sets in space-time

Suppose a, b, and are reals witha<b and consider the following diffusion equation

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

Sojourn time type functionals of a process obtained by pasting together two Brownian motion processes

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 1, pp. 29–33, January, 1989.     

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

Approximation via regularization of the local time of semimartingales and Brownian motion

Through a regularization procedure, a few schemes for approximation of the local time of a large class of continuous semimartingales and reversible diffusions are given. The convergence holds in the ucp sense. In the case of standard Brownian motion, we have been able to bound the rate of convergence in L²$L^2$, and to establish the a.s. convergence of some of our schemes.