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The origin of fast radio bursts (FRBs), the brightest cosmic explosion in radio bands, remains unknown. We introduce here a novel method for a comprehensive analysis of active FRBs' behaviors in the time-energy domain. Using "Pincus Index" and "Maximum Lyapunov Exponent", we were able to quantify the randomness and chaoticity, respectively, of the bursting events and put FRBs in the context of common transient physical phenomena, such as pulsar, earthquakes, and solar flares. In the bivariate time-energy domain, repeated FRB bursts' behaviors deviate significantly (more random, less chaotic) from pulsars, earthquakes, and solar flares. The waiting times between FRB bursts and the corresponding energy changes exhibit no correlation and remain unpredictable, suggesting that the emission of FRBs does not exhibit the time and energy clustering observed in seismic events. The pronounced stochasticity may arise from a singular source with high entropy or the combination of diverse emission mechanisms/sites. Consequently, our methodology serves as a pragmatic tool for illustrating the congruities and distinctions among diverse physical processes. [ABSTRACT FROM AUTHOR]

Cell signaling occurs in the restricted confines of highly organized intracellular signaling nanodomains. Receptor-associated independent cAMP nanodomains (RAINs) drive local cAMP synthesis and localized signaling. Optical mapping of local cAMP pools has defined local cAMP nanodomains due to buffered diffusion of second messengers. G protein-coupled receptor (GPCR) signaling at subcellular membrane compartments generates isolated intracellular pools of cAMP that drive unique cellular functions. A-kinase anchoring protein (AKAP)-signaling islands regionally distributed throughout the cell potentiate individual protein kinase A (PKA) phosphorylation events at the nanometer scale. Louis Pasteur once famously said 'in the fields of observation chance favors only the prepared mind'. Much of chance is being in the right place at the right time. This is particularly true in the crowded molecular environment of the cell where being in the right place is often more important than timing. Although Brownian motion argues that enzymes will eventually bump into substrates, this probability is greatly enhanced if both molecules reside in the same subcellular compartment. However, activation of cell signaling enzymes often requires the transmission of chemical signals from extracellular stimuli to intracellular sites of action. This review highlights new developments in our understanding of cAMP generation and the 3D utilization of this second messenger inside cells. [ABSTRACT FROM AUTHOR]

Nanofluid is a particle suspension composed of liquid and nanoparticles, and has great potential to enhance mass transfer. Most previous studies have only considered the effect of the Brownian motion of nanoparticles on mass transfer. In this study, the two-dimensional Lattice Boltzmann method (LBM) combined with the cell automation (CA) probabilistic model is proposed to investigate the mass transfer of CO 2 in nanofluids. The model predictions were in good agreement with experimental data, validating the developed mass transfer model. The simulation results indicated that the Brownian motion and grazing effect enhanced the mass transfer in nanofluids, and should be considered simultaneously. Comparing with the pure water, adding 0.1 wt% SiO 2 nanoparticles increased the absorption rate up to 58.3 %, and the effective diffusion coefficients reached 5.41 × 10−9 m2s−1. In addition, changing the physical parameters directly affected the Brownian motion and grazing effect, and changed the effective diffusion coefficient of CO 2 in the nanofluid. The effective diffusion coefficient decreased with an increase in particle size and increased with an increase in fluid temperature. [Display omitted] • The LBM combined with the CA was proposed to investigate the mass transfer of CO 2 in nanofluids. • In our developed model, the nanoparticles can move independently in the solution and absorb CO 2. • The effect of nanoparticles's Brownian motion and grazing effect on mass transfer were both considered. • The effects of different physical parameters on mass transfer in nanofluids were analyzed. [ABSTRACT FROM AUTHOR]

Protein dynamics result from physical, chemical, and biological interactions. Establishment of self-sustained patterning is essential in the Caenorhabditis elegans embryo, where asymmetries of fate determinants are achieved by reaction–diffusion mechanisms. Protein dynamics happen at undetectable spatial/temporal scales. The average molecular behaviour is experimentally observed and described by analytical modelling. Stochastic computations instead allow the prediction of protein dynamics and interactions at the molecular level. Single-molecule approaches help to detect the dynamics of different PAR oligomerisation states and of the switch between slow- and fast-diffusing cytoplasmic proteins in vivo. These measurements allow models with no free parameters and the identification of protein states relevant for asymmetries. Cortical polarity in the C. elegans embryo is important to determine the correct redistribution of cytoplasmic factors, but data show that cytoplasmic polarity signals back to the cortex for the establishment and maintenance of polarity. Molecules inside cells are subject to physical forces and undergo biochemical interactions, continuously changing their physical properties and dynamics. Despite this, cells achieve highly ordered molecular patterns that are crucial to regulate various cellular functions and to specify cell fate. In the Caenorhabditis elegans one-cell embryo, protein asymmetries are established in the narrow time window of a cell division. What are the mechanisms that allow molecules to establish asymmetries, defying the randomness imposed by Brownian motion? Mathematical and computational models have paved the way to the understanding of protein dynamics up to the 'single-molecule level' when resolution represents an issue for precise experimental measurements. Here we review the models that interpret cortical and cytoplasmic asymmetries in the one-cell C. elegans embryo. [ABSTRACT FROM AUTHOR]

This paper studies the ruin probabilities in a homogeneous continuous compound Poisson risk model which is adapted for the perturbed insurance risk model with standard Brownian motion. In such a model, we construct a martingale in terms of a differentiable exponential function based on the discounted and perturbed surplus process. We obtain the exponential upper bounds for the ruin probabilities using Martingale approach and provide a sharper exponential upper bound for the infinite time ruin probability. Moreover, we derive two asymptotic approximation formulas for the finite time ruin probability when claim size distribution belongs to some heavy-tailed families. Finally, several numerical examples are presented to show the effect of constant force of interest on the ruin probabilities and that our results are excellent and reliable. [ABSTRACT FROM AUTHOR]

Diffusion-weighted magnetic resonance imaging (dMRI) is a method of capturing the signal of water molecules diffusing in heterogeneous materials. Gaussian diffusion is interrupted when water mobility is hampered by obstructions in complex structures, and the dMRI signal decay does not match the single exponential decay in Brownian motion. In this study, a concise continuous time random-walk diffusion model is derived with less parameters than the continuous time random walk (CTRW) model and used to characterize the attenuation signal of brain tissue. The fitting results are compared with the CTRW model and the mono-exponential model reflecting the sub-diffusion and the long tail phenomenon of signal decay. Three sample experiments on rat brain and human brain are chosen to evaluate the validity in explaining the anomalous diffusion of water molecules in biological tissues, particularly in brain tissues in diverse directions, which also extends the applications of the concise continuous time random-walk diffusion model. Furthermore, we note that the concise continuous time random-walk diffusion model has practical advantages over the classical exponential model from the perspective of computational accuracy especially in the case of large b values, and has less parameters and is comparable to the CTRW model. [ABSTRACT FROM AUTHOR]

In this paper, we study small-noise asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H ∈ (1 / 2 , 1) and magnitude ϵ H. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establish the large and moderate deviation principles for these types of equations. Besides, we also obtain the central limit theorem, in which the limit process solves a linear equation involving the Lions derivative of the drift coefficient. [ABSTRACT FROM AUTHOR]

We study the tail behavior of max i ≤ N sup s > 0 W i (s) + W A (s) − β s as N → ∞ , with (W i , i ≤ N) i.i.d. Brownian motions and W A an independent Brownian motion. This random variable can be seen as the maximum of N mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around σ 2 2 β log N. Here, we analyze the rare event that this random variable reaches the value ( σ 2 2 β + a) log N , with a > 0. It turns out that its probability behaves roughly as a power law with N , where the exponent depends on a. However, there are three regimes, around a critical point a ⋆ ; namely, 0 < a < a ⋆ , a = a ⋆ , and a > a ⋆ . The latter regime exhibits a form of asymptotic independence, while the first regime reveals highly irregular behavior with a clear dependence structure among the N suprema, with a nontrivial transition at a = a ⋆ . [ABSTRACT FROM AUTHOR]

We investigate how large deviations events cluster in the framework of an infinite moving average process with light-tailed noise and long memory. The long memory makes clusters larger, and the asymptotic behaviour of the size of the cluster turns out to be described by the first hitting time of a randomly shifted fractional Brownian motion with drift. [ABSTRACT FROM AUTHOR]

In this paper a martingale problem for super-Brownian motion with interactive branching is derived. The uniqueness of the solution to the martingale problem is obtained by using the pathwise uniqueness of the solution to a corresponding system of SPDEs with proper boundary conditions. The existence of the solution to the martingale problem and the local Hölder continuity of the density process are also studied. [ABSTRACT FROM AUTHOR]

This paper introduces a new, stochastic mathematical model for the crystallization of emulsion in dispersed media. The mathematical model reads as a stochastic partial differential equation by combining the heat energy equation and the nucleation theory with specified drift and diffusion. We show the existence and uniqueness of the solution of the model by using techniques of stochastic partial differential equations. Numerical experiments are drawn to support the theoretical results. Moreover, comparison of numerical results to experimental ones is provided. [ABSTRACT FROM AUTHOR]

For classical finite time horizon stopping problems driven by a Brownian motion V (t , x) = sup t ≤ τ ≤ 0 E (t , x) [ g (τ , W τ) ] , we derive a new class of Fredholm type integral equations for the stopping set. For a large class of discounted problems, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes. [ABSTRACT FROM AUTHOR]

We consider a multi-dimensional diffusion whose coordinates behave as one-dimensional Brownian motions, evolving independently when apart, but with a sticky interaction when they coincide. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. The diffusion in R n can be viewed as the n -point motions of a stochastic flow of kernels. We use our formulae to study the flow of kernels and show that atoms in the flow are asymptotically exponentially distributed in size at large times. [ABSTRACT FROM AUTHOR]

Nualart and Yoshida (2019) presented a general scheme for asymptotic expansion of Skorohod integrals. However, when applying the general theory to a variation of a process related to a fractional Brownian motion, one repeatedly needs estimates of the order of L p -norms and Sobolev norms of functionals that are complicated as a randomly weighted sum of products of multiple integrals of the fractional Brownian motion. To resolve the difficulties, in this paper, we construct a theory of exponents based on a graphical representation of the structure of the functionals, which enables us to tell an upper bound of the order of functionals by a simple rule. We also show how the exponents change by the actions of the Malliavin derivative and its projection. The exponents are useful in various studies of limit theorems as well as in applications to asymptotic expansion. [ABSTRACT FROM AUTHOR]

We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier–Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in p -variation for maps of Skorokhod type, due to Falkowski and Słominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of p -variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma. [ABSTRACT FROM AUTHOR]

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C ([ − r , T ] ; L p (Ω , W l , q (D))) , C μ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , and the weighted space F μ ɛ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , instead of usual ones C ([ − r , T ] ; L 2 (Ω , H)) , C μ ((− ∞ , T ] ; L 2 (Ω , H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H = L 2 (D) and W l , q (D) , and some well-known fractional tools. In addition, we also study the Hölder continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β → 0 + . [ABSTRACT FROM AUTHOR]

In this work we present how to model an observed time series by a FOU (p) process. We will show that the FOU (p) processes can be used to model a wide range of time series varying from short range dependence to long range dependence, with performance similar to the ARMA or ARFIMA models and in several cases outperforming them. Also, we extend the theoretical results for any FOU (p) processes for the case in which the Hurst parameter is less than 1 / 2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [ 0 , T ]. Lastly, we give a way to obtain explicit formulas for the auto-covariance function for any FOU (p) and we present an application for FOU (2) and FOU (3). [ABSTRACT FROM AUTHOR]

In this work, sulfomethylated lignin (SAL) was firstly synthesized using alkali lignin (AL) as raw materials, and then grafted with pyrrolidone as adsorption groups to prepare sulfomethylated lignin grafted by pyrrolidone (SALN). Results showed that SALN had a lower sulfonic group content than SAL but additionally contained 0.79 mmol/g of pyrrolidone groups. SAL and SALN were separately used as dispersants to prepare 20 wt% of cobalt blue nano pigment paste, and the minimum particle size of 82 nm and 52 nm was achieved when the dispersant dosage was 10 wt% and the grinding time was 4 h. After 7 days of thermal storage at 60 °C, the particle size of nano pigment paste prepared by SALN nearly remained unchanged, but that prepared by SAL increased from 82 nm to 285 nm, and there was obvious precipitation. Therefore, the thermal storage stability of cobalt blue nano pigment paste prepared by SALN was much better than that of SAL. Adsorption isotherm and infrared spectra experimental results indicated a firm multi-point anchoring complexing adsorption was formed between the pyrrolidone group of SALN and Co–O on cobalt blue surface, which made SALN not easy to desorb during collisions caused by Brownian motions. SAL was adsorbed through –C(=O)O−, and thus easy to desorb. Therefore, the stability of the nano pigment paste prepared by SALN was superior to that of SAL. [ABSTRACT FROM AUTHOR]

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [ − 1 , 1 ]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1 / n , 2 / n , ..., [ n t ] / n (for fixed t > 0) is proved under the assumption that n → ∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane. [ABSTRACT FROM AUTHOR]

In this article, we derive a large deviation principle of the strong solution of the 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences. The model consists of the stochastic globally modified Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in Budhiraja et al. (2011) and based on a variational representation on infinite-dimensional Brownian motion. [ABSTRACT FROM AUTHOR]

With { ξ i } i ≥ 0 being a centered stationary Gaussian sequence with non-negative correlation function ρ (i) ≔ E [ ξ 0 ξ i ] and { σ (i) } i ≥ 1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑ i = 1 ℓ σ (i) ξ i , ℓ ≥ 1. For summable correlations ρ , we show that the persistence exponent is universal. On the contrary, for non-summable ρ , even for polynomial weight functions σ (i) ∼ i p the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ (H , p) and study its properties as a function of (p , H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations – e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero – that might be of independent interest. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional Brownian motion, having a random and unobservable drift which can take one of three known values. Assuming that we monitor the position of the process in real time, the problem is to determine as soon as possible and with minimal probabilities of the wrong terminal decisions, which value the drift has taken. We derive the exact solution to the problem in the Bayesian formulation, under any prior probability distribution on the three values that the drift can assume, when the cost of observation is linear. Remarkably, the optimal stopping boundaries of the present problem are non-monotone. [ABSTRACT FROM AUTHOR]

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations. [ABSTRACT FROM AUTHOR]

• S-NZVI/rGO@GG slurry showed excellent stability and mobility performance. • Higher GG concentration, pH, and low ionic strength are beneficial to improve the transportability of S-NZVI/rGO@GG slurry. • Brownian action dominates S-NZVI/rGO@GG slurry deposition in saturated porous media. • CFT, XDLVO, and kinetic fitting were combined to explain attenuated Brownian motion and GG spatial repulsion play a key role in migration. Nanoscale zero-valent iron (NZVI), which has the advantages of small particle size, large specific surface area, and high reactivity, is often injected into contaminated aquifers in the form of slurry. However, the prone to passivation and agglomeration as well as poor stability and mobility of NZVI limit the further application of this technology in fields. Therefore, sulfided NZVI loaded on reduced graphene oxide (S-NZVI/rGO) and guar gum (GG) with shear-thinning properties as stabilizers were used to synthesize S-NZVI/rGO@GG slurries. SEM, TEM, and FT-IR confirmed that the dispersion and anti-passivation of NZVI were optimized in the coupled system. The stability and mobility of the slurry were improved by increasing the GG concentration, enhancing the pH, and decreasing the ionic strength and the presence of Ca2+ ions, respectively. A modified advection-dispersion equation (ADE) was used to simulate the transport experiments considering the strain and physicochemical deposition/release. Meanwhile, colloidal filtration theory (CFT) demonstrated that Brownian motion plays a dominant role in the migration of S-NZVI/rGO@GG slurry, and the maximum migration distance can be increased by appropriately increasing the injection rate. Extended-Derjaguin-Landau-Verwey-Overbeek (XDLVO) theory showed that the excellent stability and migration of S-NZVI/rGO@GG slurry mainly came from the GG spatial forces. This study has important implications for the field injection of S-NZVI/rGO@GG slurry. According to the injection parameters, the injection range of S-NZVI/rGO@GG slurry is effectively controlled, which lays the foundation for the promotion of application in actual fields. [Display omitted] [ABSTRACT FROM AUTHOR]

This article focuses on predefined time synchronization problem for a class of signal switching neural networks with time-varying delays. In the network models, we not only consider the coupling characteristics in the following networks, but also consider the disturbance with standard Brownian motion. In the design of the controller, the control gain is designed as 1 ɛ + T p − t (t ∈ [ T 0 , T p) , ɛ is an optional smaller positive number), which avoids the infinite gain (the control gain is designed as 1 T p − t in other reference). In order to get the predefined time control law, a power function is multiplied to the Lyapunov functional, from which it can get an exponential upper bound function via the derivative and mathematical expectation operation. Utilizing the martingale theory and the method of Laplace matrix, some novel predefined time synchronization criteria are obtained for the leader-following neural networks, meanwhile the following networks can maintain the leader network after achieved synchronization. Based on the special network of the main system, five corollaries separately develop the predefined time synchronization results from different perspectives. An example with some simulation figures and computing results fully exhibits the effectiveness of the achieved synchronization scheme. In this case, although the error signal is disturbed by Brownian motion, the trace signal can still stably converge to zero by this control scheme, meanwhile the predefined-time control effect is achieved. [ABSTRACT FROM AUTHOR]

We study multiplicative SDEs perturbed by an independent additive fractional Brownian motion. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of unique Malliavin differentiable solutions to SDE's driven by a fractional Brownian motion with Hurst parameter H < 1 2 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we derive a Bismut–Elworthy–Li formula with respect to our stock price model for certain classes of vector fields. [ABSTRACT FROM AUTHOR]

We study Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the complexity of the proposed algorithms. • Explicit density formula for the reflected Brownian motion in two dimensional wedges • Monte Carlo simulation algorithms for the reflected or stopped Brownian motion • Estimates on the computation time and adaptations to other reflected processes [ABSTRACT FROM AUTHOR]

In this paper we obtain a Wiener–Hopf type factorization for a real-valued arithmetic Brownian motion with time-dependent drift and volatility. To the best of our knowledge, this paper is the very first step towards realizing the objective of deriving Wiener–Hopf type factorizations for (real-valued) time-inhomogeneous Lévy processes. In order to prove our main theorem, we derive some new results regarding time-inhomogeneous noisy Wiener–Hopf factorization. We demonstrate that in the special case of the arithmetic Brownian motion with constant drift and volatility our main result agrees with classical Wiener–Hopf factorization for this particular time-homogenous Lévy process. [ABSTRACT FROM AUTHOR]

The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in the situation that an underlying Brownian motion is replaced with a stable Lévy process with finite mean and infinite variance. We define a skew stable Lévy process X as a limit of a sequence of stable Lévy processes which are perturbed at zero. We derive a formula for the resolvent of X and show that X is a solution to a stochastic differential equation with a local time. Also, we provide a representation of X in terms of Itô's excursion theory. [ABSTRACT FROM AUTHOR]

In this paper, p th exponential stability and quasi-surely exponential stability of stochastic Hopfield neural networks driven by G -Brownian motion are investigated. Under a sublinear expectation framework, we give several lemmas related to the Halanay inequality and the Burkholder-Davis-Gundy ineqaulity. Our stability results is based on the class of the multidimensional Halanay inequalities, as well as the Burkholder-Davis-Gundy inequalities. The main originality lies in the fact that we consider Knightian uncertainty of the theory model of stochastic Hopfield neural networks. Finally, two numerical examples are presented to illustrate our new theory. [ABSTRACT FROM AUTHOR]

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as X t = B V (t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the L q spectrum of V. We will show that if V is generated by a self-similar measure on R 1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its L q spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1 + 1 2 ⋅ log 2 log 3 . Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely. [ABSTRACT FROM AUTHOR]

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order p ≥ 1 between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index H and the dimension d of the state space, with a "phase-transition" in the rates when d = 2 + 1 / H , akin to the Ajtai–Komlós–Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on R d. [ABSTRACT FROM AUTHOR]

We propose a model of card shuffling where a pack of cards, spread as points on a square table, are repeatedly gathered locally at random spots and then spread towards a random direction. A shuffling of the cards is then obtained by arranging the cards by their increasing x -coordinate values. When there are m cards on the table we show that this random ordering gets mixed in time O log m . Explicit constants are evaluated in a diffusion limit when the position of m cards evolves as an interesting 2 m -dimensional non-reversible reflected jump diffusion in time. Our main technique involves the use of multidimensional Skorokhod maps for double reflections in [ 0 , 1 ] 2 in taking the discrete to continuous limit. The limiting computations are then based on the planar Brownian motion and properties of Bessel processes. [ABSTRACT FROM AUTHOR]

In this article we study multiple S L E κ , for κ ∈ (0 , 4 ] , driven by Dyson Brownian motions. This model was introduced in the unit disk by Cardy, (2003) in connection with the Calogero–Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity κ ∈ (0 , 4 ] for the case of N = 2 driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the driving forces, with assumptions on the modulus of the derivative of the multiple Loewner maps. [ABSTRACT FROM AUTHOR]

In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H ∈ (0 , 1 / 2) ∪ (1 / 2 , 1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications, we provide the Bismut formulas of this kind for both non-degenerate and degenerate cases, and obtain the estimates of the Lions derivative and the total variation distance between the laws of two solutions. [ABSTRACT FROM AUTHOR]