Related papers: Brownian motion with variable drift can be space-f…
The canonical Brownian motion that P. Malliavin in 1999 and then S. Fang in 2002 constructed lives in the group of Holderian homeomorphisms of the circle. In this paper, we present another way to construct a Brownian motion that lives…
We consider a long-range version of self-avoiding walk in dimension $d > 2(\alpha \wedge 2)$, where $d$ denotes dimension and $\alpha$ the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to…
For $d \in \{1,2,3\}$, let $(B^d_t;~ t \geq 0)$ be a $d$-dimensional standard Brownian motion. We study the $d$-Brownian span set $Span(d):=\{t-s;~ B^d_s=B^d_t~\mbox{for some}~0 \leq s \leq t\}$. We prove that almost surely the random set…
We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets are…
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c\`adl\`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous…
Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to…
The fractional Brownian motion of index $0 < H < 1$, H-FBM, with d-dimensional time is considered on an expanding set TG, where G is a bounded convex domain that contains 0 at its boundary. The main result: if 0 is a point of smoothness of…
Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$, $f:\left[0,1\right]\longrightarrow\mathbb{R}^{d}$ a Borel function and $A\subset\left[0,1\right]$ a Borel set. We provide sufficient…
We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with…
Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent…
The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the…
Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha>\frac12$ and $1\leq \beta \leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim…
We investigate a random integral which provides a natural example of an imaginary exponential functional of Brownian motion. This functional shows up in the study of the binary annihilation process, within the Doi-Peliti formalism for…
In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some H\"{o}lder regularity…
Brownian motion near soft surfaces is a situation widely encountered in nanoscale and biological physics. However, a complete theoretical description is lacking to date. Here, we theoretically investigate the dynamics of a two-dimensional…
We show that with probability 1, the trace B[0,1] of Brownian motion in space, has positive capacity with respect to exactly the same kernels as the unit square. More precisely, the energy of occupation measure on B[0,1] in the kernel…
We study a notion of generalized H\"older continuity for functions on $\mathbb{R}^d$. We show that for any bounded function $f$ of bounded support and any $r>0$, the $r$-oscillation of $f$ defined as $osc_r f (x):= \sup_{B_r(x)} f -…
In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author…
Multifractional Brownian motion is an extension of the well-known fractional Brownian motion where the Holder regularity is allowed to vary along the paths. In this paper, two kind of multi-parameter extensions of mBm are studied: one is…
A multidimensional Brownian motion with partial reflection on a hyperplane $S$ in the direction $qN+\alpha $, where $N$ is the conormal vector to the hyperplane and $q\in [-1,1], \alpha \in S$ are given parametres, is constructed and this…