In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are absolutely continuous with respect to Wiener measure. This remarkable sigma-finite measure can be generalized in three other cases: one can start from a two-dimensional Brownian motion, from a recurrent diffusion with values in R_+, and from a discrete, recurrent Markov chain.
We give a general framework for the universality classes of $ σ$-finite measures in penalisation problems with multiplicative weights. We discuss penalisation problems for Brownian motions, Lévy processes and Langevin processes in our framework.
We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubul…
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
本文从粗路理论的角度研究了G-布朗运动的样本路和由G-布朗运动驱动的随机微分方程。作为出发点，我们证明了G-布朗运动的准必然样本路径可以正则地增强到第二水平，从而成为粗糙度为2 < p < 3的几何粗糙路径.这一结果使我们能够在粗糙路径的一般框架下引入路径意义下的G-布朗运动驱动的粗糙微分方程的概念。其次，我们建立了由G-布朗运动驱动的随机微分方程与随机微分方程之间的基本关系。作为应用，我们引入了由G布朗运动驱动的微分流形上的随机微分方程的概念，并利用路径局部化技术从RDE的角度构造了解。这是基于Eells-Elworthy-Malliavin的思想在黎曼流形上引入G-布朗运动的出发点。本文的最后一部分讨论了一类广泛而有趣的G-函数的构造问题，这类G-函数的不变群是正交群。我们还发展了Euler-Maruyama近似的G-布朗运动的独立利益驱动的随机微分方程。
Modelling of multiple simultaneous failures in insurance, finance and other areas of applied probability is important especially from the point of view of pandemic-type events. A benchmark limiting model for the analysis of multiple failures is the classical $d$-dimensional Brownian risk model (Brm), see [1]. From both theoretical and practical point of view, of interest is the calculation of the probability of multiple simultaneous failures in a given time horizon. The main findings of this contribution concern the approximation of the probability that at least $k$ out of $d$ components of Br…
We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge (2018). In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.
Pitman's theorem states that if {Bt, t $\ge$ 0} is a one-dimensional Brownian motion, then {Bt -- 2 inf s$\le$t Bs, t $\ge$ 0} is a three dimensional Bessel process, i.e. a Brownian motion conditioned in Doob sense to remain forever positive. In this paper one gives a similar representation for the Brownian motion in an interval. Due to the double barrier, it is much more involved and only asymptotic. This uses the fact that the interval is an alcove of the Affine Lie algebra A 1 1 .
We consider a diffusion in $\mathbb{R}^n$ whose coordinates each behave as one-dimensional Brownian motions, that behave independently when apart, but have a sticky interaction when they meet. The diffusion in $\mathbb{R}^n$ can be viewed as the $n$-point motion of a stochastic flow of kernels. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. We then use our formulae to study the behaviour of the flow of kernels for the exactly solvable choice of interaction.
We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements [Comets, Popov, Vachkovskaia, 2016]. At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of [Comets, Popov, Vachkovskaia, 2016; Comets, Popov, …
In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-Löf random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed.
We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a multiplicative functional associated with fractional boundary conditions of tempered type. By exploiting such connection we write some functionals in terms of a simple (positive and non-decreasing) process…
Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.
We design numerical schemes for a class of slow-fast systems of stochastic differential equations, where the fast component is an Ornstein-Uhlenbeck process and the slow component is driven by a fractional Brownian motion with Hurst index $H>1/2$. We establish the asymptotic preserving property of the proposed scheme: when the time-scale parameter goes to $0$, a limiting scheme which is consistent with the averaged equation is obtained. With this numerical analysis point of view, we thus illustrate the recently proved averaging result for the considered SDE systems and the main differences wit…
Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Saïd--Kobayashi--Ørsted $(k,a)$-generalized Laguerre semigroup theory
