Related papers: Global existence for rough differential equations …
In this paper we consider rough differential equations on a smooth manifold $\left( M\right) .$ The main result of this paper gives sufficient conditions on the driving vector-fields so that the rough ODE's have global (in time) solutions.…
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…
We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical derivatives.
We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly.…
We study a class of linear first and second order partial differential equations driven by weak geometric $p$-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving…
We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations.
We introduce strong p-completeness and use them for studying the continuous dependence of solutions of SDE's on non-compact manifolds. We obtain conditions for the existence of global smooth solution flow, and prove their diffeomorphism…
We construct global-in-time solutions for semilinear parabolic rough partial differential equations. We work on a scale of Banach spaces tailored to the controlled rough path approach and derive suitable a-priori estimates of the solution…
We study rough differential equations driven by controlled rough paths in the level-$2$ regime $1/3<\alpha\le 1/2$. Given a reference rough path $\mathbf X=(1,X,\mathbb X)$ and an $\mathbf X$-controlled driver $\mathbf Z=(Z,Z')$, we first…
We devise in this work a simple mechanism for constructing flows on a Banach space from approximate flows, and show how it can be used in a simple way to reprove from scratch and extend the main existence and well-posedness results for…
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…
We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.
Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.
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…
We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of stochastic area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation…
This paper establishes the existence and uniqueness of solutions for rough differential equations driven by reduced rough paths with low regularity, specifically in the roughness regime $\frac{1}{3} < \alpha \leq \frac{1}{2}$. While the…
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 prove the existence of a global solution for the filament equation with inital condition given by a geometric rough path in the sense of Lyons (1998).Our work gives a positive answer to a question left open in recent publications:…
In this paper, we consider a class of evolution equations driven by finite-dimensional $\gamma$-H\"{o}lder rough paths, where $\gamma\in(1/3,1/2]$. We prove the global-in-time solutions of rough evolution equations(REEs) in a sutiable…
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were…