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In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and show that Hausdorff…

The geometry of the multifractional Brownian motion (mBm) is known to present a complex and surprising form when the Hurst function is greatly irregular. Nevertheless, most of the literature devoted to the subject considers sufficiently…

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 study the long-time behaviour of solutions to a class of $d$-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H \in (0,1)$. The drift consists of a dissipative Lipschitz term and a…

Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_{t}=\int _{0}^{t}e^{2B_{s}}ds,\,t\ge 0$. Starting from a simple observation of generalized inverse…

We construct and study branching fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. The construction relies on a generalization of the discrete approximation of fractional Brownian motion (Hammond and Sheffield, Probability…

It is well known that for standard Brownian motion $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$ its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability 1 contains 0 as an interior point for each $t > 0$ (see…

We prove that the Hausdorff dimension of the set of points where a function in the Zygmund class in the euclidean space has bounded divided differences, is bigger or equal to 1. A similar result for functions in the Small Zygmund class is…

Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It\^{o}'s formula for…

In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension…

A convex function $f:[a,b]\to\mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^{b}f(t)dt\leq \frac{f(a)+f(b)}{2}. $$ Motivated by the above estimates, in this paper we…

We show that the longitudinal position $x(t)$ of a particle in a $(d+1)$-dimensional layered random velocity field (the Matheron-de Marsily model) can be identified as a fractional Brownian motion (fBm) characterized by a variable Hurst…

In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to…

This paper concerns a variational representation formula for Wiener functionals. Let $B=\{ B_{t}\} _{t\ge 0}$ be a standard $d$-dimensional Brownian motion. Bou\'e and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$…

We consider the individual points on a Martin-L\"of random path of Brownian motion. We show (1) that Khintchine's law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the…

Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is that the increment process generated by the bifractional Brownian motion…

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*}…

Let $B^{\alpha_i}$ be an $(N_i,d)$-fractional Brownian motion with Hurst index ${\alpha_i}$ ($i=1,2$), and let $B^{\alpha_1}$ and $B^{\alpha_2}$ be independent. We prove that, if $\frac{N_1}{\alpha_1}+\frac{N_2}{\alpha_2}>d$, then the…

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show…

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive additive processes…