Sample path properties of the stochastic flows (original) (raw)
Sample Path Properties of Brownian Motion
1976
This is a set of lecture notes based on a graduate course given at the Berlin Mathematical School in September 2011. The course is based on a selection of material from my book with Yuval Peres, entitled Brownian motion, which was published by Cambridge University Press in 2010.
Stratonovich’s signatures of Brownian motion determine Brownian sample paths
Probability Theory and Related Fields, 2013
The signature of Brownian motion in R d over a running time interval [0, T ] is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension d ≥ 2, almost all Brownian motion sample paths (running up to time T) are determined by its signature over [0, T ]. 1 2
2021
We study the existence of the stochastic flow associated to a linear stochastic evolution equation dX = AXdt+ ∑ k BkXdWk, on a Hilbert space. Our first result covers the case where A is the generator of a C0semigroup, and (Bk) is a sequence of bounded linear operators such that ∑ k ‖Bk‖ < +∞. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert-Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well.
The Brownian motion and the canonical stochastic flow on a symmetric space
Transactions of the American Mathematical Society, 1994
We study the limiting behavior of Brownian motion xt on a symmetric space V = G/K of noncompact type and the asymptotic stability of the canonical stochastic flow Ft on 0(V). We show that almost surely, xt has a limiting direction as it goes to infinity. The study of the asymptotic stability of Ft is reduced to the study of the limiting behavior of the adjoint action on the Lie algebra & of G by the horizontal diffusion in G. We determine the Lyapunov exponents and the associated filtration of F¡ in terms of root space structure of S .
Sample path properties of bifractional Brownian motion
Bernoulli, 2007
Let B H,K = B H,K (t), t ∈ R + be a bifractional Brownian motion in R d . We prove that B H,K is strongly locally nondeterministic. Applying this property and a stochastic integral representation of B H,K , we establish Chung's law of the iterated logarithm for B H,K , as well as sharp Hölder conditions and tail probability estimates for the local times of B H,K .
A central limit theorem for Gibbs measures relative to Brownian motion
Probability Theory and Related Fields, 2005
We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.
An analysis of stochastic flows
Communications on Stochastic Analysis, 2014
This is a survey article devoted to the stochastic flows with singular interaction. It presents the recent results of the authors which show how the basic for the smooth stochastic differential equations statements can be obtained for the flow with coalescence. The proposed approaches allows to get the large deviations for the Arratia flow, Krylov-Veretennikov expansion for the general stochastic semi-group, discrete time approximation for the Harris flows. Also the transformation of the compacts by the semi-group of the finite-dimensional projections is considered.
Some sample path properties of multifractional Brownian motion
Stochastic Processes and their Applications, 2015
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 smooth cases which lead to sample paths locally similar to a fractional Brownian motion (fBm). The main goal of this paper is therefore to extend these results to a more general frame and consider any type of continuous Hurst function. More specifically, we mainly focus on obtaining a complete characterization of the pointwise Hölder regularity of the sample paths, and the Box and Hausdorff dimensions of the graph. These results, which are somehow unusual for a Gaussian process, are illustrated by several examples, presenting in this way different aspects of the geometry of the mBm with irregular Hurst functions.
Global limit theorems on the convergence of multidimensional random walks to stable processes
Stochastics and Dynamics, 2015
Symmetric heavily tailed random walks on Z d , d ≥ 1, are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in x, t, |x| + t → ∞,) asymptotic behavior of the transition probability p(t, 0, x) is obtained. The examples indicate that the regularity conditions are essential.