Sojourn Times of Brownian Sheet

Periodica Mathematica Hungarica -     

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian H _n of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process {H _[nt]}_0≤t≤1 behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.     

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

Consider the stochastic heat equation \(\partial_t u = \mathcal{L} u + \dot{W}\), where \(\mathcal{L}\) is the generator of a [Borel right] Markov process in duality. We show that the solution is locally mutually absolutely continuous with respect to a smooth perturbation of the Gaussian process that is associated, via Dynkin’s isomorphism theorem, to the local times of the replica-symmetric process that corresponds to \(\mathcal{L}\). In the case that \(\mathcal{L}\) is the generator of a Lévy process on R ^d , our result gives a probabilistic explanation of the recent findings of Foondun et al. (Trans Am Math Soc, 2007).     

Photoconductivity enhancement in alkali metal doped multiwall carbon nanotubes

Photoconductivity effects in pristine and alkali-metal (K, Li) doped multiwalled carbon nanotubes (CNTs) were studied under xenon (100 mW) and also halogen (10 mW) light continues sources. To perform the measurements, the pristine and alkali doped CNTs were deposited into pores of a silver foam plate with nano-metric porosity by electrophoresis technique. The foam acted as a conducting frame for sweeping the photo-induced electrons to prevent rapid local electron–hole recombination in the CNTs. The radiation spectrum of the xenon source was similar to the Sun light spectrum and under normal ambient condition the photocurrents in the alkali doped samples were enhanced noticeably in comparison with the pristine CNTs. These results present a functional photoconductive performance of a heap of as-prepared alkali-metal doped CNTs that would be applicable as a light sensor without the necessity of separation between metallic and semiconducting CNTs (m- and s-CNTs).     

Images of the Brownian sheet

An -parameter Brownian sheet in maps a non-random compact set in to the random compact set in . We prove two results on the image-set :

(1) It has positive -dimensional Lebesgue measure if and only if has positive -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999).

(2) If , then with probability one, we can find a finite number of points such that for any rotation matrix that leaves in , one of the 's is interior to . In particular, has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989).

This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).' Both ideas may be of independent interest.

We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).

     

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

We study the stochastic heat equation ${\partial_t u = \mathcal{L}u+\sigma(u)\dot W}$ in (1?+?1) dimensions, where ${\dot W}$ is space-time white noise, σ : R → R is Lipschitz continuous, and ${\mathcal{L}}$ is the generator of a symmetric Lévy process that has finite exponential moments, and u ₀ has exponential decay at ±∞. We prove that under natural conditions on σ : (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting (see, however, G?rtner et?al. in Probab Theory Relat Fields 111:17–55, 1998; G?rtner et?al. in Ann Probab 35:439–499, 2007 for the analysis of the location of the peaks in a different model). Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.     

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y),u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process ${\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}$ , in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes ${t \mapsto u(t,x)}$ and ${x \mapsto u(t,x)}$ . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.     

Chung's law for integrated Brownian motion

The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

     

Brownian sheet images and Bessel-Riesz capacity

We show that the image of a 2-dimensional set under -dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ()-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.

     

Harmonic analysis of additive Lévy processes

Let X ₁, . . . ,X _N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets ${F\subseteq{{\bf R}^d}}$ , the Minkowski sum ${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$ , of the range ${\mathfrak{X}({{\bf R}^{N}_{+}})}$ of ${\mathfrak{X}}$ with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.