Related papers: Integrability of (non-)linear rough differential e…
An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations: hereditary recursion operators, master symmetries,…
We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping…
Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.
We consider a hierarchy of classical Liouville completely integrable models sharing the same (linear) $r$--matrix structure obtained through an $N$--th jet--extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of the…
In this paper, we study the existence and uniqueness of solutions to quadratic Backward Stochastic Differential Equations (QBSDEs for short) with rough driver and square integrable terminal condition. The main idea consists in using both…
The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly…
We consider finite dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the…
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…
The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytical Hamiltonian systems via the differential Galoisian obstruction. In this paper we give a new Morales-Ramis type theorem on the…
Imbibition phenomena have been widely used experimentally and theoretically to study the kinetic roughening of interfaces. We critically discuss the existing experiments and some associated theoretical approaches on the scaling properties…
The long-range spectral density correlations (spectral rigidities $\bar{\Delta}_3(\bar n)$ and related spectral compressibilities) of the $E\otimes (b_1+b_2)$ Jahn-Teller model are found strongly nonuniversal with respect to the Hamiltonian…
Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d. random variables with generic copy $(M,Q)$ such that $M$ is a regular $d\times d$ matrix and $Q$ takes values in $\mathbb{R}^{d}$, we consider the random difference equation (RDE)…
In this note we consider differential equations driven by a signal $x$ which is $\gamma$-H\"older with $\gamma>1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients…
The computation of time dynamics arising in nonlinear time-dependent partial differential equations is an ongoing challenge in numerical analysis, especially once roughness comes into play. Classical numerical schemes in general fail to…
We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf-Cole transformation. More…
Strongly nonlinear flows, which commonly arise in geophysical and engineering turbulence, are characterized by persistent and intermittent energy transfer between various spatial and temporal scales. These systems are difficult to model and…
We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well…
We give a proof of the strong existence and the regularity of stochastic differential equations driven by a Brownian motion and a measurable, Markovian drift without no regularity hypothesis except that the Girsanov exponential associated…