Abstract :
[en] One of the purposes of this thesis is to use Malliavin calculus and
Stochastic Taylor expansions to study the densities of interacting
systems of stochastic differential equations (SDE), seen as
projections of SDE onto a low-dimensional space, and to control the
dependence of the constants on the dimension of the background
space. The setting includes time-dependent SDE and a relatively large
class of path-dependent SDE. Several results also shed light on the
classical theory of SDE, independently of the control on the
constants.
In Part 1, assuming the system satisfies suitably defined projected
equivalents of the classic ellipticity or weak Hörmander conditions,
we prove Gaussian estimates in terms of the Euclidean distance where,
provided natural assumptions, for a fixed target-space
dimension, the constants depend polynomially on the background dimension, and, in the elliptic case, on the number of
driving Brownian motions.
In Part 2, we first define suitable generalisations of
(time-dependent) control distances and prove Kusuoka-Stroock type
results without control on the constants.
In particular, we obtain a time-dependent extension of a result of
Léandre about SDE with non-trivial drifts, i.e., drifts which are not
uniformly contained in the span of the other vector fields.
Then, we introduce a condition which we call the `Progressive
Hörmander condition' and prove similar control-type estimates valid
under this assumption, with polynomial control on the growth
of the constants with background space dimension. The condition is of independent interest in the
study of SDE, and shows the connection between the classic
works of Ben Arous, Kusuoka, Léandre and Stroock, and the more recent
works of Bally, Caramellino, Delarue, Menozzi and Pigato. To main
technique required is the study of density and scaling properties of
some careful choice of linear combinations of terms of the signature
of the driving path.
In Part 3, we introduce a stricter condition called the `separated
progressive Hörmander condition', and prove lower bounds and local
strict positivity under this assumption. (By `local' we mean local
around the solution of the deterministic ODE driven by a null control,
rather than local round the initial point.) The main technical
difficulty is the identification of points contained in the interior
of the support of the log-signature of the path in
the d dimensional Euclidean space composed of d Brownian motions and a deterministic linear
component.
The purpose of Part 4 is to use some results and techniques of the
rest of the thesis to prove extensions of a theorem of Löcherbach
about uniformly elliptic interacting branching diffusions.
