Title:

The Existence of Solutions for Rough Differential Equations

In one of the last Saint Flour lectures in 2004, T. Lyons remarked that a Peano
theorem for rough differential equations had not yet been proved. The first successful
attempt to prove this existence result was carried out by A.M. Davie. Using generalized
Euler schemes, Davie showed that when working in finite dimensions, differential
equations driven by geometric prough paths, for 1 ~ p < 3, have solutions whenever
the vector fields have one degree less of smoothness than what is required for the
existence of a unique solution in Lyons' Universal Limit theorem. P. Friz and N.
Victoir have used geodesic approximations in the free nilpotent group to construct
higher order Euler schemes and thus generalize Davie's results to the general case
p ~ 1. Again this construction works in finite dimensions.
In this thesis we present a new proof for Peano's theorem for rough differential
equations, which is valid in infinite dimensions under an appropriate compactness
assumption on the vector fields. Our approach, which is different from the ones
mentioned above, makes full use of Lyons' Universal Limit theorem and is based
on the construction of a family of rough polynomial approximations, each of which
is a concatenation of rough path solutions of different equations. After giving a
brief overview of the theory of rough paths, we prove several compactness results for
multiplicative functionals and integrals along rough paths. \Ve then present our proof
for Peano's theorem, which in the infinite dimensional case uses these compactness
arguments.
In the last part of the thesis, we focus on the relationship between the solutions
of It6Stratonovich stochastic differential equations and the rough path solutions of
the equations driven by the Brownian rough path. In particular we address the
issue concerning the different conditions imposed on the vector fields in these two
approaches. \Ve prove that if the vector field of the rough differential equation driven
by the Brownian rough path is Cl+c and can be expressed as a limit in V' (pi> p) of
a sequence of C~+c functions which satisfy an appropriate ellipticity condition, then
the rough path solution of the equation can be defined as the almost sure limit in the
pvariation metric of a sequence of solutions of stochastic differential equations.
