Related papers: Formalization of Brownian motion in Lean
Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the…
We derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and…
We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable.
G-Brownian motion has a very rich and interesting new structure which nontrivially generalizes the classical one. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a…
We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed.…
The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this…
The relativistic generalization of the Brownian motion is discussed. We show that the transformation property of the noise term is determined by requiring for the equilibrium distribution function to be Lorentz invariant, such as the…
In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha_t:=\sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\}$ for $t \geq 0$. The argmin process $\alpha$ is stationary,with invariant…
The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and…
We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^2$, $\gamma<\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to…
The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first…
We investigate the classical Brownian motion of a particle in a two-dimensional noncommutative (NC) space. Using the standard NC algebra embodied by the sympletic Weyl-Moyal formalism we find that noncommutativity induces a non-vanishing…
In this article, we will first introduce a class of Gaussian processes, and prove the quasi-invariant theorem with respect to the Gaussian Wiener measure, which is the law of the associated Gaussian process. In particular, it includes the…
In this article, we study the family of probability measures (indexed by a positive real number t), obtained by penalization of the Brownian motion by a given functional of its local times at time t. We prove that this family tends to a…
We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two…
A class of Gaussian processes generalizing the usual fractional Brownian motion for Hurst indices in (1/2,1) and multifractal Brownian motion introduced in Ralchenko and Shevchenko (Theory Probab Math Stat 80, 2010) and Boufoussi et al.…
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion. As an application of a recent result of ours on exponential functionals of Brownian motion, we show in this paper that, for every fixed $t>0$, the process given by…
Basic properties of Brownian motion are used to derive two results concerning birth-death chains. First, the probability of extinction is calculated. Second, sufficient conditions on the transition probabilities of a birth-death chain are…
We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in…
This is a pedagogical introduction to Brownian motion on the occasion of the 100th anniversary of Einstein's 1905 paper on the subject. After briefly reviewing Einstein's work in its contemporary context, we pursue some lines of further…