The paper proves global universal approximation theorems for non‑anticipative and general path‑dependent functionals on spaces of piecewise linear paths, showing that linear functionals of the corresponding signatures are dense in $L^p$ and weighted norms. It demonstrates that these results apply to piecewise linear interpolations of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter $H>1/4$, and provides quantitative convergence rates for signatures of such approximations toward their continuous‑time counterparts. Consequently, the authors obtain $L^p$‑approximation results for path‑dependent functionals of Gaussian processes and for random ordinary and stochastic differential equations driven by Brownian motion.
By Mihriban Ceylan, David J. Pr\"omel
arXiv:2608.28446v1 Announce Type: cross
Abstract: For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes...
By Michael Unser
The paper introduces the expected signature kernel as a similarity measure for probability laws on path space, directly applicable to signature maximum mean discrepancies. It characterizes this kernel for group-valued inhomogeneous Lévy processes by solving an extended Goursat-type system, providing exact solutions in the continuous case and well‑posed infinite‑dimensional systems with error‑controlled truncations for jump processes. The authors also show that conditioning on sampled quadratic variations yields an unbiased, lower‑variance statistic for conditionally Gaussian martingales, and present computable finite‑grid implementations that improve variance and test power.
By Peter K. Friz, Paul P. Hager
arXiv:2310. 09149v3 Announce Type: replace-cross Abstract: We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints.
By Keaton Hamm, Varun Khurana
The paper explores the relationship between Gaussian processes and Gaussian random elements within reproducing kernel Banach spaces. It demonstrates that the covariance operator of a weak second‑order Radon probability measure on such a space is uniquely defined by a symmetric positive‑definite function, and in the Gaussian case it identifies which functions correspond to covariance operators via γ‑radonifying operators. Leveraging these findings, the authors extend Driscoll’s theorem to the setting of reproducing kernel Banach spaces.
By Toni Karvonen, Rasmus Kleist H{\o}rlyck S{\o}rensen
arXiv:2609.14922v1 Announce Type: cross
Abstract: For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $\alpha.$...
By Yixuan Zhang, Qiaomin Xie
The paper establishes a link between the adjoint equations of the Itô and rough path formulations of the Pontryagin Maximum Principle (PMP) for stochastic optimal control, showing that the Itô adjoint equals the conditional expectation of the rough adjoint. It introduces a rough stochastic PMP for adapted controls without using forward–backward SDEs, and then derives a unified PMP that connects the two frameworks via Itô–Stratonovich conversion and duality identities. Applications include rederiving the adjoint matching method for fine‑tuning generative models and proposing an indirect shooting method for certain feedback problems.
By Thomas Lew
arXiv:2609. 39144v2 Announce Type: new Abstract: We prove a sharp Gaussian approximation for the invariant law of constant-stepsize SGD with bounded additive noise generated by an exogenous uniformly ergodic Markov chain.
By Junghoon Seo
arXiv:2608. 08040v1 Announce Type: cross Abstract: We introduce conditional cylindrical neural networks for approximating functionals of conditional laws in McKean-Vlasov equations with common noise.
By Nacira Agram, Reda Hmioui, Jan Rems
arXiv:2606. 16610v1 Announce Type: cross Abstract: Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood.
By Marta Gentiloni Silveri, Giovanni Conforti, Alain Durmus
arXiv:2607. 14361v1 Announce Type: cross Abstract: We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$.
By Anastasis Kratsios, Giulia Livieri, Philipp Schmocker
arXiv:2610.19888v1 Announce Type: cross
Abstract: Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent $H \in (0,1)$ and a generalised diffusion coeffi...
By Daniel S. Han, Vaishnavi Ananthanarayanan, Matthew J. Simpson