Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1102.5619

Differential Structure and Flow equations on Rough Path Space

Zhongmin Qian·Jan Tudor
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor series. The resulting object has a clear interpretation, even for non-smooth rough paths, which we take to be an element of the tangent space. We can associate it uniquely to an equivalence class of curves, with equivalence defined by our differential structure. Thus, for a functional on rough path space, we can define the derivative in a tangent direction analogous to defining the derivative in a Cameron-Martin direction of a functional on Wiener space. Our tangent space contains many more directions than the Cameron-Martin space and we do not require quasi-invariance of Wiener measure. In addition we also locally (globally) solve the associated flow equation for a class of vector fields satisfying a local (global) Lipshitz type condition.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhongmin Qian, Jan Tudor. 2011-02-28. Differential Structure and Flow equations on Rough Path Space. https://arxiv.org/abs/1102.5619

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generalized Edgeworth expansions for integer-valued additive functionals of uniformly elliptic Markov chains

We obtain asymptotic expansions for probabilities $\bbP(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $\DS S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we relate existence of the standard Edgeworth expansions of order $r$ to the rate of equidistributions of $S_N$ modulo $m$ for small positive integers $m.$

math.PR↗
arXiv2026

Permutations from Random Walk

Xavier and Yushi run a "random race" as follows. An atomless probability distribution $μ$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws $\mathbf{X}_i$ from $μ$ and advances that distance, while Yushi advances by an independent drawing $\mathbf{Y}_i$. After $n$ such moves, what is the probability that Yushi led all the way? That the answer (namely, $4^{-n}\binom{2n}{n}$) is independent of $μ$ follows from a classical theorem of Darling, stating that for symmetric atomless increments, the distribution of each individual rank in the permutation obtained by ranking the partial sums is independent of the step law. We give a self-contained proof and extend the result to the permutations generated by partial sums of uniformly random signed permutations of any fixed, finite, generic set of reals. For atomless increments with mean zero and finite variance, without assuming symmetry, we show that random-walk permutations approach a random object that we call the "Wiener permuton," whose expected pattern densities equal the probabilities of the corresponding permutations generated by finite random walks with centered Laplace increments. Finally, we exhibit an infinite family of constructions whose limiting permutons interpolate between the Wiener permuton and the recursive separable permuton; each has the same intensity permuton, providing a single two-dimensional extension of the classical arcsine law for all of them.

math.PR↗
arXiv2026

On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

math.PR↗