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Register a new userOn the $frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter $H < 1/2$
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In this paper, we study the $frac{1}{H}$-variation of stochastic divergence integrals $X_t = int_0^t u_s {delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $frac{1}{H}$-variation of $X$ exists in $L^1({Omega})$ and is equal to $e_H int_0^T|u_s|^H ds$, where $e_H = mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $|B_t|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < frac{1}{2}$. Using a multidimensional version of the result on the $frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.
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We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.
In this paper we develop sensitivity analyses w.r.t. the long-range/memory noise parameter for solutions to stochastic differential equations and the probability distributions of their first passage times at given thresholds. Here we consider the case of stochastic differential equations driven by fractional Brownian motions and the sensitivity, when the Hurst parameter~$H$ of the noise tends to the pure Brownian value, of probability distributions of certain functionals of the trajectories of the solutions ${X^H_t}_{tin mathbb{R}_+}$. We first get accurate sensitivity estimates w.r.t. $H$ around the critical Brownian parameter $H=tfrac{1}{2}$ of time marginal probability distributions of $X^H$. We second develop a sensitivity analysis for the Laplace transform of first passage time of $X^H$ at a given threshold. Our technique requires accurate Gaussian estimates on the density of $X^H_t$. The Gaussian estimate we obtain in Section~5 may be of interest by itself.
Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Levy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Holder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Levy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Holder regularity on general indexing collections.
The past few years have witnessed increased attention to the quest for Majorana-like excitations in the condensed matter community. As a promising candidate in this race, the one-dimensional chiral Majorana edge mode (CMEM) in topological insulator-superconductor heterostructures has gathered renewed interests during recent months after an experimental breakthrough. In this paper, we study the quantum transport of topological insulator-superconductor hybrid devices subject to light-matter interaction or general time-periodic modulation. We report half-integer quantized conductance plateaus at $frac{1}{2}frac{e^2}{h}$ and $frac{3}{2}frac{e^2}{h}$ upon applying the so-called sum rule in the theory of quantum transport in Floquet topological matter. In particular, in a photoinduced topological superconductor sandwiched between two Floquet Chern insulators, it is found that for each Floquet sideband, the CMEM admits equal probability for normal transmission and local Andreev reflection over a wide range of parameter regimes, yielding half-integer quantized plateaus that resist static and time-periodic disorder. The $frac{3}{2}frac{e^2}{h}$ plateau has not yet been computationally or experimentally observed in any other superconducting system, and indicates the possibility to simultaneously create and manipulate multiple pairs of CMEMs by light. The robust half-quantized conductance plateaus, due to CMEMs at quasienergies zero or half the driving frequency, are both fascinating and subtle because they only emerge after a summation over contributions from all Floquet sidebands. Such a distinctive transport signature can thus serve as a hallmark of photoinduced CMEMs in topological insulator-superconductor junctions.
Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered.
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