Related papers: Self-intersection local time: Critical exponent, l…

We consider the existence and H\"{o}lder continuity conditions for the $k$-th order derivatives of self-intersection local time for $d$-dimensional fractional Brownian motion, where $k=(k_1,k_2,\cdots, k_d)$. Moreover, we show a limit…

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several…

Let \{B_t^H,t\geq0\} be a d-dimensional fractional Brownian motion. We prove that the approximation of the first-order derivative of self-intersection local time, defined as…

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^t\delta(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^t\delta…

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and…

We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index \beta \in (2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the…

Consider the first exit time of one-dimensional Brownian motion $\{B_s\}_{s\geq 0}$ from a random passageway. We discuss a Brownian motion with two time-dependent random boundaries in quenched sense. Let $\{W_s\}_{s\geq 0}$ be an other…

We consider the local time of the ($1+\beta$)-stable super-Brownian motion with $0<\beta<1$. It is shown by Mytnik and Perkins ({\em Ann. Probab.}, 31(3), 1413 -- 1440, (2003)) that the local time, denoted by $L(t,x)$, is jointly continuous…

Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion and \[ \alpha_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx . \] Let $\eta=N(0,1)$ be independent of $\alpha_{t}$. For each fixed $t$ \[…

In this paper we will examine the derivative of intersection local time of Brownian motion and symmetric stable processes in $R^2$. These processes do not exist when defined in the canonical way. The purpose of this paper is to exhibit the…

In this paper we study the local times of vector-valued Gaussian fields that are `diagonally operator-self-similar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the…

We study the upper tail behaviors of the local times of the additive stable processes. Let $X_1(t),...,X_p(t)$ be independent, d-dimensional symmetric stable processes with stable index $0<\alpha\le 2$ and consider the additive stable…

We prove, for any $\beta >0$, a central limit theorem for the fluctuations of linear statistics in the Sine-$\beta$ process, which is the infinite volume limit of the random microscopic behavior in the bulk of one-dimensional log-gases at…

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain…

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*}…

We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times $t$ and low scatterer densities (Boltzmann-Grad limit). The normalization factor is $\sqrt{t\log…

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of strictly stationary AR(1) and INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient $\alpha \in (0, 1)$ and idiosyncratic…

We consider a system of diffusing particles on the real line in a quadratic external potential and with repulsive electrostatic interaction. The empirical measure process is known to converge weakly to a deterministic measure-valued process…

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description…

We investigate boundary estimates for elliptic operators with stationary random coefficients exhibiting integrable correlations, arising from stochastic homogenization theory. As practical applications, we establish decay estimates for…