Rough Paths Research Articles

Integration of nonsmooth 2-forms: From Young to Itô and Stratonovich

We show that geometric integrals of the type ∫Ωfdg1∧dg2 can be defined over a two-dimensional domain Ω when the functions f, g1, g2:R2→R are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map g=(g1,g2) when f(x)=F(x,g(x)) with F sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.

L 2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

The aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C 1 drift and C 2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although L q convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L 2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.

Experimental and numerical investigation on the mode I and mode II interlaminar fracture toughness of nitrogen-doped reduced graphene oxide reinforced composites

An experimentaland computational study is carried out to investigate the toughening effect of nitrogen-doped reduced graphene-oxide particles (ND-RGOP) in mode I and mode II delamination of carbon fiber/epoxy laminates. For mode I, both delamination initiation and propagation toughness are enhanced by 126.3% and 119.3% with 0.8 ND-RGOP addition. In addition to mode I toughness enhancement, ND-RGOP reinforcement also increases mode II toughness. The highest mode II delamination toughness value was obtained in 0.4ND-RGOP laminates with an increase of 45.6% compared to neat laminates. However, as seen in the bending tests, the maximum force of the end notch bending (ENF) specimens with 0.8 ND-RGOP added is not lower than that of neat laminates. The interlaminar fracture toughness is high due to the rough path of delamination with ND-RGOP addition, which is also observed by FESEM images of the delamination surfaces. In addition to the experiments, a cohesive zone model with parametric traction stress value was prepared in ABAQUS/CAE to obtain the critical traction stresses in the delamination of laminates. The critical normal traction stress of double cantilever beam (DCB) specimens is increased by over 20% with the addition of ND-RGOP. Besides, the critical interlaminar shear stress in mode II delamination for ENF samples is significantly improved.

Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equations

We study a general class of focusing L 2 L^2 -critical nonlinear Schrödinger equations with lower order perturbations, in the possible absence of the pseudo-conformal symmetry and the conservation law of energy. In dimensions one and two, we construct multi-bubble Bourgain-Wang type blow-up solutions, which behave like a sum of pseudo-conformal blow-up solutions that concentrate at K K distinct singularities, 1 ≤ K > ∞ 1\leq K>{\infty } , and a regular profile. Moreover, we obtain the uniqueness in the energy class where the convergence rate is within the order ( T − t ) 4 + (T-t)^{4+} , for t t close to the blow-up time T T . These results in particular apply to the canonical nonlinear Schrödinger equations and, through the pseudo-conformal transform, yield the existence and conditional uniqueness of non-pure multi-solitons, which behave asymptotically as a sum of multi-solitons and a dispersive part. Thus, the results provide new examples of the mass quantization conjecture and the soliton resolution conjecture. Furthermore, through a Doss-Sussman type transform, we obtain multi-bubble Bourgain-Wang solutions for stochastic nonlinear Schrödinger equations, where the driving noise is taken in the sense of controlled rough path.

Effect of ultrasonic rolling on crack propagation behavior of Ti6Al4V titanium alloy laser welded joints

This study investigated the effect of ultrasonic rolling on the fatigue resistance of Ti6Al4V titanium alloy laser welded joints, with an emphasis on crack growth rate. Multiple analyses were conducted on samples subjected to different numbers of rolling passes, including analysis of surface roughness, morphology, cross-section structure, hardness, fatigue fracture, and residual stress. Ultrasonic rolling was found to considerably enhance the life cycle of the sample. One, three, and five rolling passes resulted in fatigue life increases of 17.14%, 46.38%, and 66.61%, respectively, when compared with untreated samples. Treated samples exhibited rough and intricate crack propagation paths on fracture surfaces, along with numerous secondary cracks. Fatigue striation width decreased by 48.6%, 62.5%, and 69.4% for samples receiving one, three, and five rolling passes, respectively. These fatigue striations, which were narrower and more densely distributed, indicate that ultrasonic rolling effectively increases fatigue life. The ultrasonic rolling treatment enhanced fatigue performance and reduced the rate of crack propagation due to changes in surface compressive residual stress, microstructure, microhardness, and improved surface quality. Crack retardation primarily occurred in initial stage of crack propagation and stable propagation stages, whereas inhibitory effects weakened during the final fracture stage due to superposition and offset of the higher driving force with compressive residual stress. In summary, the study demonstrates that treatment via ultrasonic rolling is an effective method for improving fatigue life and fracture toughness in Ti6Al4V titanium alloy laser-welded joints.

SUPPORT THEOREM FOR PINNED DIFFUSION PROCESSES

AbstractIn this paper, we prove a support theorem of Stroock–Varadhan type for pinned diffusion processes. To this end, we use two powerful results from stochastic analysis. One is quasi-sure analysis for Brownian rough path. The other is Aida–Kusuoka–Stroock’s positivity theorem for the densities of weighted laws of non-degenerate Wiener functionals.

Improved Rapid-Expanding-Random-Tree-Based Trajectory Planning on Drill ARM of Anchor Drilling Robots

Permanent highway support in deep coal mines now depends on the anchor drilling robot’s drill arm. The drilling arm’s trajectory planning using the conventional RRT (rapid-expanding random tree) algorithm is inefficient and has crooked, rough paths. To improve the accuracy of path planning, we propose an improved RRT algorithm. Firstly, the kinematic model of the drill arm of the drill and anchor robot was established, and the improved DH solution parameters and the positive solution of the drill arm kinematics were solved. The end effector’s attainable working space was calculated using the Monte Carlo approach. Additionally, to address the problem of the slow running speed of the RRT algorithm, an artificial potential field factor was introduced to construct virtual force fields at obstacle and target points and calculate the potential field map for the entire reachable workspace to improve the speed of the sampling points close to the target point. At the same time, the greedy approach and the three-time B-sample curve-fitting method were used simultaneously to remove unnecessary points and carry out smooth path processing in order to improve the quality of the drill arm trajectory. This was carried out in order to solve the issue of rough pathways generated by the RRT algorithm. Finally, 50 time-sampling comparison experiments were conducted on 2D and 3D maps. The experimental results showed that the improved RRT algorithm improved the average sampling speed by 20% and reduced the average path length by 14% compared with the RRT algorithm, which verified the feasibility and effectiveness of this improved RRT algorithm. The improved RRT algorithm generates more efficient and smoother paths, which can improve the intelligence of the support process by integrating and automating drilling and anchoring and providing reliable support for coal mine intelligence.

How Rough Path Lifts Affect Expected Return and Volatility: A Rough Model under Transaction Cost

How Rough Path Lifts Affect Expected Return and Volatility: A Rough Model under Transaction Cost

Quasi-geometric rough paths and rough change of variable formula

Quasi-geometric rough paths and rough change of variable formula

A Fourier Analysis Based New Look at Integration

Abstract We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.

Hierarchical collision-free trajectory planning for autonomous vehicles based on improved artificial potential field method

To enable autonomous vehicles to generate smooth and collision-free trajectories and improve their driving performance on structured roads, this paper proposes a hierarchical trajectory planning algorithm based on an improved artificial potential field method. To improve the applicability of the algorithm to complex scenarios, the Frenet coordinate system was established to address these limitations. First, the safety distance model is applied to the risk assessment of the improved artificial potential field method. Then, the hierarchical solution is carried out, and the road solvable convex space and the rough path solution are solved by combining the artificial potential field method. On this basis, the potential field term and the smoothing term cost function are established, and the sequential quadratic programming (SQP) algorithm is used to solve the exact path that meets the requirements of safety and smoothness. Hierarchical planning shortens the solution time by quickly determining the bounds of the convex space. Finally, in the speed planning, in order to take into account the comfort and safety of the occupants, the speed curve is solved by considering the dynamic constraints of the vehicle. The obstacle avoidance effects of the algorithm on static and dynamic obstacles are tested in different simulation scenarios. The results of the simulation experiment show that the proposed algorithm can successfully achieve obstacle avoidance on complex structured roads.

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called Itô-Lyons map in the energy spaces L∞(0,T;Hk)∩L2(0,T;Hk+1) for any k≥1. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the second step where the continuity in the optimal norm is established through an application of the rough Gronwall lemma. Our approach is direct and does not rely on any transformation formula, which permits to treat multidimensional noise. As an easy consequence we then deduce a Wong-Zakai type result, a large deviation principle for the solution and a support theorem.

The Accessibility Problem for Geometric Rough Differential Equations

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields [1] with the rough calculus on manifolds developed by Cass, Litterer and Lyons in [2].

Random attractors for rough stochastic partial differential equations

This paper is devoted to the existence of random attractors for rough partial differential equations driven by nonlinear multiplicative Hölder rough paths with exponents in (1/3,1/2]. Our approach relies upon rough paths theory and stopping times analysis in a suitable scale of interpolation spaces. The core step is to derive the adequate algebraic and analytical properties of a sequence of stopping times, which allows us to establish the required compact tempered absorbing set. The existence of a pullback attractor for the generated random dynamical system is straightforward. An illustrative example is presented by reaction-diffusion equations subjected to fractional Brownian rough paths.

Numerical Attractors via Discrete Rough Paths

Numerical Attractors via Discrete Rough Paths

Functional data analysis with rough sample paths?

Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is imposed to allow for the use of smoothing techniques that annihilate the noise. At the same time, imposing the smoothness assumption excludes a considerable range of stochastic processes, most notably diffusion processes. Under perfect observation of the sample paths, such processes would not need to be excluded from the realm of functional data analysis. In this paper, we introduce a careful modification of existing methods, dubbed the ‘reflected triangle estimator’, and show that this allows for the functional data analysis of processes with nowhere differentiable sample paths, even when these are discretely and noisily observed, including under irregular and sparse designs. Our estimator matches the established rates of convergence for processes with smooth paths, and furthermore attains the same optimal rates as one would get under perfect observation. Thus, with reflected triangle estimation, the scope of applicability of much of the methodology developed for discretely/irregularly/noisily/sparsely sampled functional data is considerably extended. By way of simulation it is shown that the advantages furnished are reflected in practice, hinting at potential closer links with the field of diffusion inference.

Variational Principles on Geometric Rough Paths and the Lévy Area Correction

In this paper, we describe two effects of the L\'evy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the L\'evy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the dynamics is driven by coloured noise, and for rigid body dynamics with double-bracket dissipation, theoretical and numerical results show that this additional deterministic torque shifts the centre of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.

New Directions in the Applications of Rough Path Theory

This article provides a concise overview of some of the recent advances in the application of rough path theory to machine learning. Controlled differential equations (CDEs) are discussed as the key mathematical model to describe the interaction of a stream with a physical control system. A collection of iterated integrals known as the signature naturally arises in the description of the response produced by such interactions. The signature comes equipped with a variety of powerful properties rendering it an ideal feature map for streamed data. We summarise recent advances in the symbiosis between deep learning and CDEs, studying the link with RNNs and culminating with the Neural CDE model. We concluded with a discussion on signature kernel methods.

Volterra Equations Driven by Rough Signals 3: Probabilistic Construction of the Volterra Rough Path for Fractional Brownian Motions

Based on the recent development of the framework of Volterra rough paths (Harang and Tindel in Stoch Process Appl 142:34–78, 2021), we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} and for the standard Brownian motion. The Volterra kernel k(t, s) is allowed to be singular, and behaving similar to |t-s|-γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|t-s|^{-\\gamma }$$\\end{document} for some γ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\ge 0$$\\end{document}. The construction is done in both the Stratonovich and Itô senses. It is based on a modified Garsia–Rodemich–Romsey lemma which is of interest in its own right, as well as tools from Malliavin calculus. A discussion of challenges and potential extensions is provided.

Image Inpainting with Parallel Decoding Structure for Future Internet

Image inpainting benefits much from the future Internet, but the memory and computational cost in encoding image features in deep learning methods poses great challenges to this field. In this paper, we propose a parallel decoding structure based on GANs for image inpainting, which comprises a single encoding network and a parallel decoding network. By adding a diet parallel extended-decoder path for semantic inpainting (Diet-PEPSI) unit to the encoder network, we can employ a new rate-adaptive dilated convolutional layer to share the weights to dynamically generate feature maps by the given dilation rate, which can effectively decrease the number of convolutional layer parameters. For the decoding network composed of rough paths and inpainting paths, we propose the use of an improved CAM for reconstruction in the decoder that results in a smooth transition at the border of defective areas. For the discriminator, we substitute the local discriminator with a region ensemble discriminator, which can attack the restraint of only the recovering square, like areas for traditional methods with the robust training of a new loss function. The experiments on CelebA and CelebA-HQ verify the significance of the proposed method regarding both resource overhead and recovery performance.