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Consider the stochastic heat equation \begin{equation*} \partial_t u_t(x)=\frac12 \partial^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in(0,T],\, x\in [0,1], \end{equation*} where $b$ is a generalized function, and $\dot W$ is space-time white noise on ${\mathbb R}_+\times[0,1]$. If the drift $b$ is a sufficiently regular function, then it is well-known that any analytically weak solution to this equation is also analytically mild, and vice versa. We extend this result to drifts that are generalized functions, with an appropriate adaptation of the notions of mild and weak solutions. As a corollary of our results, we show that for $b\in L_p({\mathbb R})$, $p\ge1$, this equation has a unique analytically weak and mild solution, thus extending the classical results of Gy\"ongy and Pardoux (1993).

We consider the stochastic reaction-diffusion equation in $1+1$ dimensions driven by multiplicative space-time white noise, with a distributional drift belonging to a Besov-H\"older space with any regularity index larger than $-1$. We assume that the diffusion coefficient is a regular function which is bounded away from zero. By using a combination of stochastic sewing techniques and Malliavin calculus, we show that the equation admits a unique solution.

We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses., Comment: 51 pages

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.

The goal of this article is to establish a central limit theorem for the Euler-Maruyama scheme approximating multidimensional SDEs with elliptic Brownian diffusion, under very mild regularity requirements on the drift coefficients. When the drift is H\"older continuous, we show that the limiting law of the rescaled fluctuations around the true solution is characterised as the unique solution of a hybrid Young-It\^o differential equation. When the drift has positive Sobolev regularity, this limit is characterised by the solution of a transformed SDE. Our result is an extension of the results of Jacod-Kurtz-Protter (1991, 1998) in which SDEs with differentiable coefficients were considered. To compensate for the lack of regularity of the drifts, we utilize the regularisation effect from the non-degenerate noise.

We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov-R\"ockner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion. Additionally, we show that such SDE is strongly complete, i.e. for almost every trajectory of the Brownian motion, the family of solutions with different initial data forms a continuous semiflow for all nonnegative times.

We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, $d\in\mathbb{N}$. For the case where $b\in L_p(\mathbb{R}^d)$, $p\in[1,\infty]$ we show weak existence of solutions to this equation under the condition $$ \frac{d}p<\frac1H-1, $$ which is an extension of the Krylov-R\"ockner condition (2005) to the fractional case. We construct a counter-example showing optimality of this condition. If $b$ is a Radon measure, particularly the delta measure, we prove weak existence of solutions to this equation under the optimal condition $H<\frac1{d+1}$. We also show strong well-posedness of solutions to this equation under certain conditions. To establish these results, we utilize the stochastic sewing technique and develop a new version of the stochastic sewing lemma.

For a general adapted integrable right-continuous with left limits (RCLL) process $(X_t)_{t\in[0,\tau]}$ taking values in a metric space $(\mathcal E,d)$, we show (among other things) that for every $m\in(1,\infty)$ $$ \frac{m-1}{2m-1}\|\sup_{t\in[0,\tau]}\mathbb{E}(d(X_{t-},X_\tau)|\mathcal F_t)\|_m\le \|\sup_{t\in[0,\tau]}d(X_0,X_t)\|_m\le c\frac{m^2}{m-1} \|\sup_{t\in[0,\tau]}\mathbb{E}(d(X_{t-},X_\tau)|\mathcal F_t)\|_m $$ with a universal constant $c$. This is a probabilistic version of Fefferman--Stein estimate for the sharp maximal functions. While the former inequality is derived easily from Doob's martingale inequality, the later inequality is a consequence of John--Nirenberg inequalities for weighted BMO processes, which are obtained in this note. We explain how John--Nirenberg inequalities can be utilized to obtain inequalities for martingales, both old and new alike in a unified way., Comment: update some details

Stroock and Varadhan in 1997 and Geiss in 2005 independently introduced stochastic processes with bounded mean oscillation (BMO) and established their exponential integrability with some unspecified exponential constant. This result is an analogue of the John--Nirenberg inequality for functions of bounded mean oscillation. In this work, we quantify the size of the exponential constant by the modulus of mean oscillation. Some new applications of BMO processes in rough stochastic differential equations, numerical approximations and regularization by noise are discussed., Comment: update on historical contributions

Polymorphism, the phenomenon that a species can exist in many discrete forms, is common in nature, such as hair colors in an animal species, flower colors in a tree species, and blood types in humans, etc. In materials science, it refers to a solid that can exist in multiple forms with different crystalline structures. In the liquid crystals field, however, polymorphism is hard to find because a discontinuous structural variation is basically impossible because of their fluid or partially fluid nature. Herein we show that the B4 and DC phases that for many years have been classified as distinctive phases are connected, in terms of their nano-architectures, based on the study of a single compound, a flexible bent-shaped dimer. The surrounding solvent is the key to assisting the dimeric molecules in morphing and adopting different supramolecular structures at the mesoscale. Furthermore, we accidentally find a novel nanotube-like structure that has not yet been reported in view of the B4/DC phases. Together with the known sponge (DC) and the helical filament (B4) structures, they are just some of the manifestations of the polymorphism in a class of low-birefringence, chiral phase from achiral liquid crystals., Comment: M: 16p, 5f; SI: 9p, 9f

We consider the Allen-Cahn equation $\partial_t u- \Delta u=u-u^3$ with a rapidly mixing Gaussian field as initial condition. We show that provided that the amplitude of the initial condition is not too large, the equation generates fronts described by nodal sets of the Bargmann-Fock Gaussian field, which then evolve according to mean curvature flow.

Smart services based on the Internet of Everything (IoE) are gaining considerable popularity due to the ever-increasing demands of wireless networks. This demands the appraisal of the wireless networks with enhanced properties as next-generation communication systems. Although 5G networks show great potential to support numerous IoE based services, it is not adequate to meet the complete requirements of the new smart applications. Therefore, there is an increased demand for envisioning the 6G wireless communication systems to overcome the major limitations in the existing 5G networks. Moreover, incorporating artificial intelligence in 6G will provide solutions for very complex problems relevant to network optimization. Furthermore, to add further value to the future 6G networks, researchers are investigating new technologies, such as THz and quantum communications. The requirements of future 6G wireless communications demand to support massive data-driven applications and the increasing number of users. This paper presents recent advances in the 6G wireless networks, including the evolution from 1G to 5G communications, the research trends for 6G, enabling technologies, and state-of-the-art 6G projects., Comment: Submitted to IEEE IoT Journal

We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, H\"older continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition., Comment: 69 pages. A new section is added: numerical solution to stochastic transport equation

We establish a simultaneous generalization of It\^o's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering, - control and the conditional analysis of stochastic systems with common noise., Comment: Substantial rewriting, proofs have been simplified and streamlined, length substantially reduced

A stochastic sewing lemma which is applicable for processes taking values in Banach spaces is introduced. Applications to additive functionals of fractional Brownian motion of distributional type are discussed.

We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order $\alpha \in (0,1)$ lead to rate $(1+\alpha)/2$., Comment: 36 pages, revised version. Section 1.4. added on to some extent degenerate/growing coefficients

We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^\beta_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $\beta-1/q\ge-1$, $\beta>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=\delta_0$ which corresponds to the skewed stochastic heat equation. For $\beta-1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gy\"ongy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in L\^e~(2020).

Forecasting has emerged as an important component of informed, data-driven decision-making in a wide array of fields. We introduce a new data model for probabilistic predictions that encompasses a wide range of forecasting settings. This framework clearly defines the constituent parts of a probabilistic forecast and proposes one approach for representing these data elements. The data model is implemented in Zoltar, a new software application that stores forecasts using the data model and provides standardized API access to the data. In one real-time case study, an instance of the Zoltar web application was used to store, provide access to, and evaluate real-time forecast data on the order of 10$^7$ rows, provided by over 20 international research teams from academia and industry making forecasts of the COVID-19 outbreak in the US. Tools and data infrastructure for probabilistic forecasts, such as those introduced here, will play an increasingly important role in ensuring that future forecasting research adheres to a strict set of rigorous and reproducible standards.

Representing images by compact hash codes is an attractive approach for large-scale content-based image retrieval. In most state-of-the-art hashing-based image retrieval systems, for each image, local descriptors are first aggregated as a global representation vector. This global vector is then subjected to a hashing function to generate a binary hash code. In previous works, the aggregating and the hashing processes are designed independently. Hence these frameworks may generate suboptimal hash codes. In this paper, we first propose a novel unsupervised hashing framework in which feature aggregating and hashing are designed simultaneously and optimized jointly. Specifically, our joint optimization generates aggregated representations that can be better reconstructed by some binary codes. This leads to more discriminative binary hash codes and improved retrieval accuracy. In addition, the proposed method is flexible. It can be extended for supervised hashing. When the data label is available, the framework can be adapted to learn binary codes which minimize the reconstruction loss w.r.t. label vectors. Furthermore, we also propose a fast version of the state-of-the-art hashing method Binary Autoencoder to be used in our proposed frameworks. Extensive experiments on benchmark datasets under various settings show that the proposed methods outperform state-of-the-art unsupervised and supervised hashing methods., Comment: Accepted to IEEE Trans. on Image Processing (TIP), 2019. arXiv admin note: substantial text overlap with arXiv:1704.00860