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Samy Tindel

Publications and source records attributed to Samy Tindel.

At least 19 recordsLinked to original sources
arXiv2026

On It\^o-Stratonovich formula for rough sheets

In this paper, we explore a new strategy towards an It\^o-Stratonovich type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously cumbersome. The emergence of mixed differential terms in 2D leads to overlapping iterated integrals, which previously required the construction of exhaustive combinatorial structures. As an example of this kind of structure, let us mention the massive 36-element planar signature introduced by K. Chouk and M. Gubinelli in their influential paper 'Rough sheets'. In this work we propose a simplified setting for rough calculus in the plane, which relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick allows us to significantly streamline the complexity of planar algebraic integration. We illustrate this methodology for a specialized Rough-Young framework: we extend the classical planar change-of-variable formula to paths possessing an asymmetric H\"older regularity: $\gamma_1 > 1/3$ in the first direction and $\gamma_2 > 1/2$ in the second direction. Relying on a structured controlled path expansions, we rigorously minimize the set of iterated integrals needed in the signature, and express the resulting planar change-of-variable formula as the explicit limit of Riemann sums.

math.PR↗
arXiv2026

Computational aspects of the Volterra Signature

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive Picard iterates of linear controlled Volterra equations, making their exact computation of additional mathematical interest. However, the kernel introduces substantial algorithmic challenges. We provide a resolution by first decomposing the Chen-type convolution relation established in [arXiv:2603.04525] into analytic and arithmetic parts, and then introducing several efficient algorithms: a general approximative scheme with quadratic complexity $O(J^2)$ in the number of time steps $J$, an FFT-based acceleration with complexity $O(J\log J)$ for convolution kernels on uniform grids, and an exact recursion with complexity $O(JR^2)$ for kernels admitting a state-space representation of dimension $R$; retaining standard signature complexity in the path dimension and truncation level $N$. We further show that the number of factors in matrix-valued kernels of the form $K(t,s)=\sum_p k_p(t-s)A_p$ do not increase the asymptotic complexity in $J$ and $N$. Finally, we derive a finite-difference predictor--corrector scheme for the associated Volterra signature kernel. All algorithms are implemented in the publicly available JAX-based package "tensordev".

math.NA↗
arXiv2026

A class of d-dimensional directed polymers in a Gaussian environment

We introduce and analyze a broad class of continuous directed polymers in $\mathbb{R}^d$ driven by Gaussian environments that are white in time and spatially correlated, under Dalang's condition. Using an It\^o-renormalized stochastic-heat-equation representation, we establish structural properties of the partition function, including positivity, stationarity, scaling, homogeneity, and a Chapman--Kolmogorov relation. On finite time intervals, we prove Brownian-type pathwise behavior, namely H\"older continuity and identification of the quadratic variation. We then obtain a sharp measure-theoretic dichotomy: the quenched polymer measure is singular with respect to Wiener measure if and only if $\widehat f(\mathbb{R}^d)=\infty$ (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension $d\ge 3$, we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the $1+1$ white-noise setting to higher-dimensional Gaussian environments with general spatial covariance.

math.PR↗
arXiv2026

The Volterra signature

Modern approaches for learning from non-Markovian time series, such as recurrent neural networks, neural controlled differential equations or transformers, typically rely on implicit memory mechanisms that can be difficult to interpret or to train over long horizons. We propose the \emph{Volterra signature} $\mathrm{VSig}(x;K)$ as a principled, explicit feature representation for history-dependent systems. By developing the input path $x$ weighted by a temporal kernel $K$ into the tensor algebra, we leverage the associated Volterra--Chen identity to derive rigorous learning-theoretic guarantees. Specifically, we prove an \emph{injectivity} statement (identifiability under augmentation) that leads to a \emph{universal approximation} theorem on the infinite dimensional path space, which in certain cases is achieved by \emph{linear functionals} of $\mathrm{VSig}(x;K)$. Moreover, we demonstrate applicability of the \emph{kernel trick} by showing that the inner product associated with Volterra signatures admits a closed characterization via a two-parameter integral equation, enabling numerical methods from PDEs for computation. For a large class of exponential-type kernels, $\mathrm{VSig}(x;K)$ solves a linear state-space ODE in the tensor algebra. Combined with inherent invariance to time reparameterization, these results position the Volterra signature as a robust, computationally tractable feature map for data science. We demonstrate its efficacy in dynamic learning tasks on real and synthetic data, where it consistently improves classical path signature baselines.

stat.ML↗
arXiv2026

The Pontryagin maximum principle and $Q$-functions in rough environments

We derive the Pontryagin maximum principle and $Q$-functions for the relaxed control of noisy rough differential equations. Our main tool is the development of a novel differentiation procedure along `spike variation' perturbations of the optimal state-control pair. We then exploit our development of the infinitesimal $Q$-function (also known as the $q$-function) to derive a policy improvement algorithm for settings with entropic cost constraints.

math.OC↗
arXiv2025

On the Fourier dimension of fractional Brownian graphs

In this note we prove that the Fourier dimension of the graph $G(B)$ of a fractional Brownian motion $B$ with Hurst parameter $H\in(0,1/2)$ is equal to 1. This finishes to solve a conjecture by Fraser and Sahlsten. It also yields an exact formula for the gap $\dim_{\rm H}(G(B)) - \dim_{\rm F}(G(B))$ between the Hausdorff dimension and the Fourier dimension of $G(B)$. The proof is based on an intricate combinatorics procedure for multiple integrals related to the covariance function of the fractional Brownian motion.

math.PR↗
arXiv2025

Weighted Besov spaces on Heisenberg groups and applications to the Parabolic Anderson model

This article aims at a proper definition and resolution of the parabolic Anderson model on Heisenberg groups $\mathbf{H}_{n}$. This stochastic PDE is understood in a pathwise (Stratonovich) sense. We consider a noise which is smoother than white noise in time, with a spatial covariance function generated by negative powers $(-Δ)^{-α}$ of the sub-Laplacian on $\mathbf{H}_{n}$. We give optimal conditions on the covariance function so that the stochastic PDE is solvable. A large portion of the article is dedicated to a detailed definition of weighted Besov spaces on $\mathbf{H}_{n}$. This definition, related paraproducts and heat flow smoothing properties, forms a necessary step in the resolution of our main equation. It also appears to be new and of independent interest. It relies on a recent approach, called projective, to Fourier transforms on $\mathbf{H}_{n}$.

math.PR↗
arXiv2024

On ergodic properties of stochastic PDEs

In this note we review several situations in which stochastic PDEs exhibit ergodic properties. We begin with the basic dissipative conditions, as stated by Da Prato and Zabczyk in their classical monograph. Then we describe the singular case of SPDEs with reflection. Next we move to some degenerate (and thus more demanding) settings. Namely we recall some results obtained around 2006, concerning stochastic Navier-Stokes equations with a very degenerate noise. We finish the article by handling some cases with degenerate coefficients. This includes a new result about the parabolic Anderson model in dimension $d\ge 3$, driven by a general class of noises and fairly general initial conditions. In this context, a phase transition is observed, expressed in terms of the noise intensity.

math.PR↗
arXiv2024

A coupling between random walks in random environments and Brox's diffusion

It is known that a properly rescaled version of Sinai's random walk converges in distribution to Brox's diffusion. In this article we quantify this convergence by considering a specific coupling between Sinai's walk and Brox's diffusion. Our method relies on convergence results for martingale problems considered in the rough path setting.

math.PR↗
arXiv2024

On the signature of an image

Over the past decade, the importance of the 1D signature which can be seen as a functional defined along a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a $d$-dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen's relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.

math.CA↗
arXiv2024

Pathwise Relaxed Optimal Control of Rough Differential Equations

This note lays part of the theoretical ground for a definition of differential systems modeling reinforcement learning in continuous time non-Markovian rough environments. Specifically we focus on optimal relaxed control of rough equations (the term relaxed referring to the fact that controls have to be considered as measure valued objects). With reinforcement learning in view, our reward functions encompass forms that involve an entropy-type term favoring exploration. In this context, our contribution focuses on a careful definition of the corresponding relaxed Hamilton-Jacobi-Bellman (HJB)-type equation. A substantial part of our endeavor consists in a precise definition of the notion of test function and viscosity solution for the rough relaxed PDE obtained in this framework. Note that this task is often merely sketched in the rough viscosity literature, in spite of the fact that it gives a proper meaning to the differential system at stake. In the last part of the paper we prove that the natural value function in our context solves a relaxed rough HJB equation in the viscosity sense.

math.OC↗
arXiv2024

Parabolic Anderson model in bounded domains of recurrent metric measure spaces

A metric measure space equipped with a Dirichlet form is called recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models \[ \partial_{t} u(t,x) = Δu(t,x) + βu(t,x) \, \dot{W}_α(t,x) \] where the noise $W_α$ is white in time and colored in space when $α>0$ while for $α=0$ it is also white in space. Both Dirichlet and Neumann boundary conditions are considered. Besides proving existence and uniqueness in the Itô sense we also get precise $L^p$ estimates for the moments and intermittency properties of the solution as a consequence. Our study reveals new exponents which are intrinsically associated to the geometry of the underlying space and the results for instance apply in metric graphs or fractals like the Sierpiński gasket for which we prove scaling invariance properties of the models.

math.PR↗
arXiv2023

Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law

In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.

math.PR↗
arXiv2023

Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates

The Malliavin differentiability of a SDE plays a crucial role in the study of density smoothness and ergodicity among others. For Gaussian driven SDEs the differentiability property is now well established. In this paper, we consider the Malliavin differentiability for the Euler scheme of such SDEs. We will focus on SDEs driven by fractional Brownian motions (fBm), which is a very natural class of Gaussian processes. We derive a uniform (in the step size $n$) path-wise upper-bound estimate for the Euler scheme for stochastic differential equations driven by fBm with Hurst parameter $H>1/3$ and its Malliavin derivatives.

math.PR↗
arXiv2022

Parabolic Anderson model on Heisenberg groups: the Itô setting

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}Δu+u\dot{W}_α$, where $Δ$ is the hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_α; α>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-Δ)^{-α}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_α$; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as $α>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.

math.PR↗
arXiv2022

Hyperbolic Anderson model 2: Strichartz estimates and Stratonovich setting

We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz type estimates for the wave kernel in weighted Besov spaces, by which we can prove the wellposedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

math.PR↗
arXiv2022

2-d signature of images and texture classification

We introduce a proper notion of 2-dimensional signature for images. This object is inspired by the so-called rough paths theory, and it captures many essential features of a 2-dimensional object such as an image. It thus serves as a low-dimensional feature for pattern classification. Here we implement a simple procedure for texture classification. In this context, we show that a low dimensional set of features based on signatures produces an excellent accuracy.

cs.CV↗