We study some finite time transport properties of isotropic Brownian flows. Under a certain nondegeneracy condition on the potential spectral measure, we prove that uniform shrinking or expansion of balls under the flow over some bounded time interval can happen with positive probability. We also provide a control theorem for isotropic Brownian flows with drift. Finally, we apply the above results to show that under the nondegeneracy condition the length of a rectifiable curve evolving in an isotropic Brownian flow with strictly negative top Lyapunov exponent converges to zero as $t\to \infty$ with positive probability.
The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engländer in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated exponentially shrinking branching Brownian sausage (BBM-sausage). Using a previous result on the density of the support of BBM, and some well-known resu…
In this paper we consider the asymptotic behaviour of all moments of the interparticle distance and of all mixed moments of an isotropic Brownian stochastic flow which serves as a smooth approximation of the Arratia flow.
In this paper we study the small time asymptotic behavior of the spectral heat content $\widetilde{Q}_D^{(α)}(t)$ of an arbitrary bounded $C^{1,1}$ domain $D$ with respect to the \textit{subordinate killed Brownian motion} in $D$ via an $(α/2)$-stable subordinator. For all $α\in (0,2)$, we establish a two-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$ in all dimensions. When $α\in (1,2)$ and $d\geq 2$, we establish a three-term small time expansion for $\widetilde{Q}_D^{(α)}(t)$.
Limit theorems for squared increment sums of the maximum of two isotropic fractional Brownian fields under fixed-domain asymptotics
Université du Littoral Côte d’Opale（滨海大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，国家经济统计学院）
We re-examine the celebrated Doob--McKean identity that identifies a conditioned one-dimensional Brownian motion as the radial part of a 3-dimensional Brownian motion or, equivalently, a Bessel-3 process, albeit now in the analogous setting of isotropic $α$-stable processes. We find a natural analogue that matches the Brownian setting, with the role of the Brownian motion replaced by that of the isotropic $α$-stable process, providing one interprets the components of the original identity in the right way.
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 the boundary, then the log-asymptotics of probability that H-FBM does not exceed a fixed positive level in TG is $(H - d + o(1)) \log T$, $T\to\infty$. Some generalizations of this result to isotropic but not self-similar Gaussian processes with stationary increments are also considered.
This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [20]. Building upon their work, it is shown by analyzing the associated elliptic boundary-value problem that, almost surely, the smoothed (in the sense that the boundary data is continuous) exit law of the diffusion from large domains converges, as the domain's scale approaches infinity, to that of a Brownia…
Stochastic homogenization is achieved for a class of elliptic and parabolic equations describing the lifetime, in large domains, of stationary diffusion processes in random environment which are small, statistically isotropic perturbations of Brownian motion in dimension at least three. Furthermore, the homogenization is shown to occur with an algebraic rate. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [21], upon whose results the present work relies strongly, and more recently their smoothed exit distributions from large domains were shown to conver…
Central limit theorems for squared increment sums of fractional Brownian fields based on a Delaunay triangulation in $2D$
Université du Littoral Côte d’Opale（法国北部海岸大学）；Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances（里昂大学，里昂第一大学，金融与保险科学学院）；Laboratory in Finance and Insurance - LFA CREST - Center for Research in Economics and Statistics, ENSAE（金融与保险实验室-LFA CREST-经济与统计研究中心，巴黎国立高等经济学院）
In this paper we identify the asymptotic tail of the distribution of the exit time $τ_C$ from a cone $C$ of an isotropic $α$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $Θ_t$. Under some additional regularity assumptions, the angular process $Θ_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asympto…
We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of Brownian noise and periodic boundary conditions, we prove that the pathwise entropy solutions converge to their spatial average and provide an estimate on the rate of convergence. The third main result of the paper is a new regularization result in the spirit of averaging lemmata. This work extends both the framework of pathwise entropy solutions for stochastic scalar c…
