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See detailKusuoka-Stroock type bounds for densities related to low-dimensional projections of high-dimensional SDE
Ledent, Antoine Patrick Isabelle Eric UL

Doctoral thesis (2017)

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 ... [more ▼]

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. [less ▲]

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