Related papers: On nonlinear rough paths
In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of…
We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent SDEs containing running…
Nonuniform ellipticity is a classical topic in the theory of partial differential equations. While several results in regularity theory have been adding up over decades, many basic issues, as for instance the validity of Schauder theory and…
The present paper refers to the theory and the practice of the systems regarding non-linear systems and their applications. We aimed the integration of these systems to elaborate their response as well as to highlight some outstanding…
Here we study the abstract nonlinear differential equation of second order that in special case is the equation of the type of equation of traffic flow. We prove the solvability theorem for the posed problem under the appropriate conditions…
This paper introduces the path derivatives, in the spirit of Dupire's functional It\^o calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the…
In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].
A classic problem in analysis is to solve nonlinear equations of the form \begin{equation*} F(x)=0, \end{equation*} where $F:D^n\to \mathbb{R}^m$ is a continuous map of the closed unit disk $D^n\subset\mathbb{R}^n$ in $\mathbb{R}^m$. A…
Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one…
We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian…
We argue that there should exist a "noncommutative Fourier transform" which should identify functions of noncommutative variables (say, of matrices of indeterminate size) and ordinary functions or measures on the space of paths. Some…
This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric \Pi-rough paths in our terminology) sketched by Lyons ("Differential equations driven by rough signals", Revista Mathematica Iber. Vol 14, Nr.…
We review a possible framework for (non)linear quantum theories, into which linear quantum mechanics fits as well, and discuss the notion of ``equivalence'' in this setting. Finally, we draw the attention to persisting severe problems of…
We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial…
We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we…
A diagram approach to classical nonlinear stochastic field theory is introduced. This approach is intended to serve as a link between quantum and classical field theories, resulting in an independent constructive characterisation of the…