Lipschitz Vector Fields Research Articles

Shape and stability of rotation sets for incompressible flows on the torus

AbstractIn this paper we introduce the concept of rotation cones, determined by rotation sets, and use these to describe the stability of rotation sets of incompressible and fixed-point free continuous flows on the torus $$\mathbb{T}^{d}$$ T d , d ⩾ 2. The previous concept is equivalent and extends the stability of rotation numbers (and rotation vectors) in the special case of fixed-point free continuous flows on $$\mathbb{T}^{2}$$ T 2 . We prove that incompressible Lipschitz vector fields with stable rotation sets have convex rotation cones with non-empty interior, and that all extremal edges are collinear with vectors in ℤd. If, in addition, the rotation cones are proper, then these are polyhedral with finitely many edges. Finally we prove that the set of vector fields with stable rotation sets is C0-open and dense among those having proper rotation cones.

An improved RRT behavioral planning method for robots based on PTM algorithm

For multi-dimensional high-order nonlinear systems with unstable path quality in parameter and extension terms, we developed a new fast search random tree strategy. First, we established a high-order Lipschitz vector field dynamic system to adapt to high-order systems of multi-degree-of-freedom robots, with the complex obstacle function being one of its key components. Secondly, we designed a classification gap filtering network layer (Classification LSTM) to screen training data models and ensure the global stability of data in path design. Additionally, the visual sensors deployed in the unit area effectively implement the path marking backtracking strategy and dead zone path simplification. Finally, three examples are provided to verify the effectiveness of this design method.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures.In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.

On the Transport of Currents

AbstractIn this work, we consider some evolutionary models for k-currents in $$\mathbb {R}^d$$ R d . We study a transport-type equation which can be seen as a generalisation of the transport/continuity equation and can be used to model the movement of singular structures in a medium, such as vortex points/lines/sheets in fluids or dislocation loops in crystals. We provide a detailed overview of recent results on this equation obtained mostly in (Bonicatto et al. Transport of currents and geometric Rademacher-type theorems. arXiv:2207.03922, 2022; Bonicatto et al. Existence and uniqueness for the transport of currents by Lipschitz vector fields. arXiv:2303.03218, 2023). We work within the setting of integral (sometimes merely normal) k-currents, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and a number of Rademacher-type differentiability results. These differentiability results are sharp and can be formulated in terms of a novel condition we called “Negligible Criticality condition” (NC), which turns out to be related also to Sard’s Theorem. We finally provide a new stability result for integral currents satisfying (NC) in a uniform way.

A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

Existence and uniqueness for the transport of currents by Lipschitz vector fields

This work establishes the existence and uniqueness of solutions to the initial-value problem for the geometric transport equationddtTt+LbTt=0 in the class of k-dimensional integral or normal currents Tt (t being the time variable) under the natural assumption of Lipschitz regularity of the driving vector field b. Our argument relies crucially on the notion of decomposability bundle introduced recently by Alberti and Marchese. In the particular case of 0-currents, this also yields a new proof of the uniqueness for the continuity equation in the class of signed measures.

The entropy production of stationary diffusions

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.

Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Let mathfrak {X}^{0,1}_nu (M) be the subset of divergence-free Lipschitz vector fields defined on a closed Riemannian manifold M endowed with the Lipschitz topology Vert ,{cdot },Vert _{0,1} where nu is the volume measure. Let mathfrak {X}^{0,1}_{nu ,ell }(M)subset mathfrak {X}^{0,1}_nu (M) be the subset of vector fields satisfying the ell -property, a property that implies C^1-regularity nu -almost everywhere. We prove that there exists a residual subset with respect to Vert ,{cdot },Vert _{0,1} such that Pesin’s entropy formula holds, i.e. for any the metric entropy equals the integral, with respect to nu , of the sum of the positive Lyapunov exponents.

Formula omitted]-compactness of some classes of integral functionals depending on vector fields

In this paper we achieve some Γ-compactness results for suitable classes of integral functionals depending on a given family of Lipschitz vector fields, with respect to both the strong Lp−topology and the strong WX1,p−topology.

Lipschitz Carnot-Carathéodory Structures and their Limits

In this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not boundedly compact and the convergence is not uniform on compact sets. We discuss several examples in which our convergence result can be applied. Among them, we prove a subFinsler Mitchell’s Theorem with continuously varying norms, and a general convergence result for Carnot-Carathéodory distances associated to subspaces and norms on the Lie algebra of a connected Lie group.

$\Gamma$-Convergence for Functionals Depending on Vector Fields. II. Convergence of Minimizers

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers and momenta of those functionals. Moreover, we apply these results to the periodic homogenization in Carnot groups and to prove a $H$-compactness theorem for linear differential operators of the second order depending on $X$.

Lipschitz stratification of complex hypersurfaces in codimension 2

We show that the Zariski canonical stratification of complex hypersurfaces is locally bi-Lipschitz trivial along the strata of codimension two. More precisely, we study the Zariski equisingular families of surface, not necessarily isolated, singularities in {\mathbb{C}^3} . We show that a natural stratification of such a family, given by the singular set and the generic family of polar curves, provides a Lipschitz stratification in the sense of Mostowski. In particular such families are bi-Lipschitz trivial, with trivializations obtained by integrating Lipschitz vector fields.

Geodesic fields for Pontryagin type C0-Finsler manifolds

Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.

Topological aspects of incompressible flows

In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C 0 incompressible flows generated by Lipschitz vector fields. We prove that a C 0 -generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non-wandering set. In particular, a C 0 -generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C 0 -generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive.

Final dynamics of systems of nonlinear parabolic equations on the circle

<abstract><p>We consider the class of dissipative reaction-diffusion-convection systems on the circle and obtain conditions under which the final (at large times) phase dynamics of a system can be described by an ODE with Lipschitz vector field in $ \mathbb{R}^{N} $. Precisely in this class, the first example of a parabolic problem of mathematical physics without the indicated property was recently constructed.</p></abstract>

Ergodicity effects on transport-diffusion equations with localized damping

The main objective of this paper is to study the time decay of transport-diffusion equation with inhomogeneous localized damping in the multi-dimensional torus. The drift is governed by an autonomous Lipschitz vector field and the diffusion by the standard heat equation with small viscosity parameter $\nu$. In the first part we deal with the inviscid case and show some results on the time decay of the energy using in a crucial way the ergodicity and the unique ergodicity of the flow generated by the drift. In the second part we analyze the same problem with small viscosity and provide quite similar results on the exponential decay uniformly with respect to the viscosity in some logarithmic time scaling of the \mbox{type $t\in [0,C_0\ln(1/\nu)]$}.

Γ-convergence for functionals depending on vector fields. I. Integral representation and compactness

Given a family of locally Lipschitz vector fields X(x)=(X1(x),…,Xm(x)) on Rn, m≤n, we study functionals depending on X. We prove an integral representation for local functionals with respect to X and a result of Γ-compactness for a class of integral functionals depending on X.

Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

The problem of determining a periodic Lipschitz vector field $b=(b_1, \dots, b_d)$ from an observed trajectory of the solution $(X_t: 0 \le t \le T)$ of the multi-dimensional stochastic differential equation \begin{equation*} dX_t = b(X_t)dt + dW_t, \quad t \geq 0, \end{equation*} where $W_t$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^2$-loss in any dimension, and also for supremum norm loss when $d \le 4$. Further, when $d \le 3$, nonparametric Bernstein-von Mises theorems are proved for the posterior distributions of $b$. From this we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu_b$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.

On Lipschitz vector fields and the Cauchy problem in homogeneous groups

We introduce a class of “Lipschitz horizontal” vector fields in homogeneous groups, for which we show equivalent descriptions, e.g., in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory.

On the periodic orbits, shadowing and strong transitivity of continuous flows

We prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in Bomfim and Varandas (2015). In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic orbits hold for a C0-Baire generic subset of Lipschitz vector fields, that generate continuous flows. Similar results also hold for C0-generic homeomorphisms and, in particular, we deduce that chain recurrent classes of C0-generic homeomorphisms have the gluing orbit property.