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Global existence for rough differential equations under linear growth conditions

Gubinelli, Massimiliano; Lejay, Antoine (2009-05), Global existence for rough differential equations under linear growth conditions. https://basepub.dauphine.fr/handle/123456789/3051

Type
Document de travail / Working paper
External document link
http://hal.archives-ouvertes.fr/hal-00384327/en/
Date
2009-05
Pages
20 pages
Metadata
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Author(s)
Gubinelli, Massimiliano
Lejay, Antoine cc
Abstract (EN)
We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential equation with bounded coefficients.
Subjects / Keywords
Rough differential equation; Global existence; Change of variable formula; Explosion in a finite time; Rough path; Geometric rough paths