Related papers: On an Inversion Theorem for Stratonovich's Signatu…

The signature of Brownian motion in $\mathbb{R}^{d}$ 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 $d\geq 2$, almost all Brownian…

Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE \[ X_t=x+\int_0^t\sigma(X_s)\strat db(s)+\int_0^t m(X_s) ds, \] we show that the associated stochastic flow of diffeomorphisms focuses as fast as $…

Score-based diffusion models have emerged as powerful techniques for generating samples from high-dimensional data distributions. These models involve a two-phase process: first, injecting noise to transform the data distribution into a…

This paper proves a Krylov-Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process…

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature…

We consider a diffusion on a bounded domain, assuming that the system is irreducible inside the domain and that the diffusion has varying degree of degeneracy on the domain's boundary. The long-term statistical properties of typical…

A method is proposed for the calculation of diffusion constants for one-dimensional maps exhibiting deterministic diffusion. The procedure is based on harmonic inversion and uses a known relation between the diffusion constant and the…

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every N, we sample a lattice path of N steps with weights given by the coefficient of the corresponding word in the…

This paper provides a new characterization of the stochastic invariance of a closed subset of R^d with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can…

This paper blends two techniques recently developed in [2] and [3] to prove the presence of absolutely continuous spectrum for the multidimensional Schrodinger operator provided that the potential is summable over trajectory with positive…

On the basis of perturbed Kolmogorov backward equations and path integral representation, we unify the derivations of the linear response theory and transient fluctuation theorems for continuous diffusion processes from a backward point of…

Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data,…

Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this…

We study a diffusion process with random space-time dependent coefficients. Moreover the diffusion matrix is allowed to degenerate. An invariance principle is proved provided that the diffusion coefficient is controlled by a time…

We reconsider the problem of diffusion of particles at constant speed and present a generalization of the Telegrapher process to higher dimensional stochastic media ($d>1$), where the particle can move along $2^d$ directions. We derive the…

Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a…

According to a theorem of S. Schumacher and T. Brox, for a diffusion $X$ in a Brownian environment it holds that $(X_t-b_{\log t})/\log^2t\to 0 $ in probability, as $t\to\infty$, where $b_{\cdot}$ is a stochastic process having an explicit…

We introduce a single generative mechanism with which it is able to describe diverse non-stationary diffusions. A non-stationary Markovian replication process for steps is considered, for which we analytically derive time-evolution of the…

We give a proof of Lehoczky's drawdown formula for one-dimensional diffusion processes, using the Poisson structure of the excursions of the diffusion below its running maximum.

Let $\mu$ be a general stochastic measure, where we assume for $\mu$ only $\sigma$-additivity in probability and continuity of paths. We prove that the symmetric integral $\int_{[0,T]}f(\mu_t, t)\circ\,{\rm d}\mu_t$ is well defined. For…