Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes

We study the long time behaviour of a large class of diffusion processes on $R^N$, generated by second order differential operators of (possibly) degenerate type. The operators that we consider {\em need not} satisfy the H\"ormander condition. Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this paper we study UFG diffusions and demonstrate the importance of such a class of processes in several respects: roughly speaking i) we show that UFG processes constitute a family of SDEs which exhibit multiple invariant measures and for which one is able to describe a systematic procedure to determine the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently "less smooth" than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic H\"ormander condition are UFG processes, our paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic, in the sense that they exhibit multiple invariant measures.Comment: 66 page

On small time asymptotics for rough differential equations driven by fractional Brownian motions

We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of Peter Laurenc

Accidents on Rural Interstate and Parkway Roads and Their Relation to Pavement Friction

Friction measurements were made with a skid trailer at 70 mph on 820 miles of rural, four-lane, controlled-access routes on the interstate and parkway systems in Kentucky. These facilities were subdivided into test sections and half-mile sites. Accident experience, friction measurements and traffic volumes were obtained for each subdivision. The expression of accident occurrence which correlated best with skid resistance was wet-surface accidents per 100 million vehicle miles. There was a definite trend exhibiting a rapid decrease of accidents with increasing Skid Number (70 mph) to 26 ± 1; thereafter, with increasing Skid Numbers, the rate of decrease was considerably lessened. This trend was developed using test-section data and verified using half-mile sites. Analysis of Peak Slip Numbers and accident occurrences indicated similar trends to those developed with Skid Numbers

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature. In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlyind path space, and (3) the metric topology associated to ([γ], [σ]) := ||γ* - σ*||₁ using the (constant-speed) tree-reduced representatives γ* and σ* of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G* of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces. We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24]

Skorohod and rough integration for stochastic differential equations driven by Volterra processes

Given a solution Y to a rough differential equation (RDE), a recent result [7] extends the classical Ito-Stratonovich formula and provides a closed-form expression for ∫ Y ○ dX − ∫ Y dX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X, where X is a Gaussian process with finite p-variation less than 3. In this paper, we extend this result to Gaussian processes with finite p-variation such that 3 ≤ p 1/4. As an application we recover Ito formulas in the case where the vector fields of the RDE governing Y are commutative