Related papers: Isolated zeros for Brownian motion with variable d…
Define the incremental fractional Brownian field $B_{H}(s+\tau)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the…
This paper derives the asymptotic behavior of $$\mathbb{P} \{ \int\limits_0^\infty \mathbb{I}\Big(B_H(s)-c_1s>q_1u, B_H(s)-c_2s>q_2u\Big)ds>T_u\},\quad u \to \infty,$$ where $B_H$ is a fractional Brownian motion, $c_1,c_2,q_1,q_2>0,\ H \in…
The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp.…
We introduce a new method to detect the zeros of the Riemann zeta function which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few `half-isolated' zeros. By combining this with classical methods,…
The mixed fractional Brownian motion - the sum of independent fractional and standard Brownian motions - is known to be a semimartingale if the Hurst exponent $H$ of its fractional component satisfies $H > 3/4$. The question posed in the…
We establish a uniform Hausdorff dimension result for the inverse image sets of real-valued strictly $\alpha$-stable L\'evy processes with $1< \alpha\le 2$. This extends a theorem of Kaufman for Brownian motion. Our method is different from…
We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide…
Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point x in the plane such that $H^{\phi_\alpha}({t > 0 :…
In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a H\"older continuous function of order $\beta \in (\frac13,\frac12)$. We also…
We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in the case $H={1\over 2}$, convergence to the averaged solution takes place in…
We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at $-\infty$. We show that these processes do not have to have…
We are interested in existence of solutions to the $d$-dimensional equation \begin{equation*} X_t=x_0+\int_0^t b(X_s)ds + B_t, \end{equation*} where $B$ is a (fractional) Brownian motion with Hurst parameter $H\leqslant 1/2$ and $b$ is an…
In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. Then, we show that, when the delay…
We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of…
We show that Brownian motion of a one-dimensional domain wall in a large but finite system yields a $\omega^{-3/2}$ power spectrum. This is successfully applied to the totally asymmetric simple exclusion process (TASEP) with open…
We study the distribution of first-passage functionals ${\cal A}= \int_0^{t_f} x^n(t)\, dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to…
In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter…
We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Sch\"{u}tz-type formula is derived for the transition probability. We investigate an…
Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations <x(t)x(s)> ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H…
Let $B=(B_t)_{t\in {\mathbb{R}}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_T)_{t\in {\mathbb{R}}}$ is a Brownian motion independent…