Related papers: On nonlinear rough paths
We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the…
Rough path analysis is developed in the full Besov scale. This extends, and essentially concludes, an investigation started by [Pr\"omel--Trabs, Rough differential equations driven by signals in {B}esov spaces. J. Diff. Equ. 2016], further…
We show how to construct path integrals for quantum mechanical systems where the space of configurations is a general non-compact symmetric space. Associated with this path integral is a perturbation theory which respects the global…
Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic…
Donsker's invariance principle is shown to hold for random walks in rough path topology. As application, we obtain Donsker-type weak limit theorems for stochastic integrals and differential equations.
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration.…
We consider the rough differential equation $dY=f(Y)d\bm \om$ where $\bm \om=(\omega,\bbomega)$ is a rough path defined by a Brownian motion $\omega$ on $\RR^m$. Under the usual regularity assumption on $f$, namely $f\in C^3_b (\RR^d,…
The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets. Its semilinear form was later identified as a non-Markovian backward stochastic…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by…
Classification of datasets into two or more distinct classes is an important machine learning task. Many methods are able to classify binary classification tasks with a very high accuracy on test data, but cannot provide any easily…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…
We give evaluations in closed form of certain non linear differential equations
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
In this article we will analyse how to compute the contribution of each input value to its aggregate output in some nonlinear models. Regression and classification applications, together with related algorithms for deep neural networks are…
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants.…
A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear…