Two-dimensional Brownian Motion Research Articles

A singularly perturbed ruin problem for a two-dimensional Brownian motion in the positive quadrant

Abstract We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

Texture-Image-Oriented Coverless Data Hiding Based on Two-Dimensional Fractional Brownian Motion

In an AI-immersing age, scholars look for new possibilities of employing AI technology to their fields, and how to strengthen security and protect privacy is no exception. In a coverless data hiding domain, the embedding capacity of an image generally depends on the size of a chosen database. Therefore, choosing a suitable database is a critical issue in coverless data hiding. A novel coverless data hiding approach is proposed by applying deep learning models to generate texture-like cover images or code images. These code images are then used to construct steganographic images to transmit covert messages. Effective mapping tables between code images in the database and hash sequences are established during the process. The cover images generated by a two-dimensional fractional Brownian motion (2D FBM) are simply called fractional Brownian images (FBIs). The only parameter, the Hurst exponent, of the 2D FBM determines the patterns of these cover images, and the seeds of a random number generator determine the various appearances of a pattern. Through the 2D FBM, we can easily generate as many FBIs of multifarious sizes, patterns, and appearances as possible whenever and wherever. In the paper, a deep learning model is treated as a secret key selecting qualified FBIs as code images to encode corresponding hash sequences. Both different seeds and different deep learning models can pick out diverse qualified FBIs. The proposed coverless data hiding scheme is effective when the amount of secret data is limited. The experimental results show that our proposed approach is more reliable, efficient, and of higher embedding capacity, compared to other coverless data hiding methods.

Existence of global and explosive mild solutions of fractional reaction–diffusion system of semilinear SPDEs with fractional noise

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction–diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by [Formula: see text] for [Formula: see text], along with [Formula: see text] where [Formula: see text] is the fractional power [Formula: see text] of the Laplacian, [Formula: see text] and [Formula: see text] and [Formula: see text] are real constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that [Formula: see text] with Hurst index [Formula: see text] we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solution to our considered system.

Physical layer security enhanced scheme based on joint encryption of DNA and RNA data perturbation in CO-OFDM system

—In the work, we propose a new chaotic encryption algorithm for optical physical layer security enhanced in coherent optical orthogonal frequency division multiplexing (CO-OFDM) systems. The proposed algorithm adopts the six-dimensional (6D) hyperchaotic to achieve the generation of key sequences, which can effectively improve the unpredictability and security of the system. The encryption scheme includes two steps, first, the binary sequence is disturbed by DNA and RNA two-stage encoding and RNA mutation, and then the initial encrypted data is injected into the algorithm module with two-dimensional continuous chaotic mapping (2D-SCL) and Brownian motion to scramble successively. Simulations in a 16QAM CO-OFDM system at a symbol rate of 30 Gbaud over 80 km of standard single-mode fiber (SSMF) show that a legitimate user is able to successfully decrypt the received signals, and an illegal eavesdropper obtains a useless signal with a bit error rate (BER) of about 0.5. The scheme can tolerate very small deviations and has a key space of approximately 10294, which can effectively withstand illegal attacks. And the peak-to-average power ratio (PAPR) can be reduced by about 1.01 dB.

In defense and revival of Bayesian filtering for thermal infrared object tracking

Deep learning-based methods monopolize the latest research in the field of thermal infrared (TIR) object tracking. However, relying solely on deep learning models to obtain better tracking results requires carefully selecting feature information that is beneficial to representing the target object and designing a reasonable template update strategy, which undoubtedly increases the difficulty of model design. Thus, recent TIR tracking methods face many challenges in complex scenarios. This paper introduces a novel Deep Bayesian Filtering (DBF) method to enhance TIR tracking in these challenging situations. DBF is distinctive in its dual-model structure: the system and observation models. The system model leverages motion data to estimate the potential positions of the target object based on two-dimensional Brownian motion, thus generating a prior probability. Following this, the observation model comes into play upon capturing the TIR image. It serves as a classifier and employs infrared information to ascertain the likelihood of these estimated positions, creating a likelihood probability. According to the guidance of the two models, the position of the target object can be determined, and the template can be dynamically updated. Experimental analysis across several benchmark datasets reveals that DBF achieves competitive performance, surpassing most existing TIR tracking methods in complex scenarios.

Hidden Markov modeling of single-particle diffusion with stochastic tethering.

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Deep-Learning Estimators for the Hurst Exponent of Two-Dimensional Fractional Brownian Motion

The fractal dimension (D) is a very useful indicator for recognizing images. The fractal dimension increases as the pattern of an image becomes rougher. Therefore, images are frequently described as certain models of fractal geometry. Among the models, two-dimensional fractional Brownian motion (2D FBM) is commonly used because it has specific physical meaning and only contains the finite-valued parameter (a real value from 0 to 1) of the Hurst exponent (H). More usefully, H and D possess the relation of D = 3 − H. The accuracy of the maximum likelihood estimator (MLE) is the best among estimators, but its efficiency is appreciably low. Lately, an efficient MLE for the Hurst exponent was produced to greatly improve its efficiency, but it still incurs much higher computational costs. Therefore, in the paper, we put forward a deep-learning estimator through classification models. The trained deep-learning models for images of 2D FBM not only incur smaller computational costs but also provide smaller mean-squared errors than the efficient MLE, except for size 32 × 32 × 1. In particular, the computational times of the efficient MLE are up to 129, 3090, and 156248 times those of our proposed simple model for sizes 32 × 32 × 1, 64 × 64 × 1, and 128 × 128 × 1.

Two-dimensional Brownian motion of active particle on superfluid helium surface

We report an experimental study of the 2D dynamics of active particles driven by quantum vortices on the free surface of superfluid helium at T = 1.45 К. The particle motion at short times (< 25 ms) relates to anomalous diffusion mode typical for active particles, while for longer times it corresponds to normal diffusion mode. The values of the rotational and translational kinetic energies of the particle allow to determine for the first time the intensity of the particle-vortex interaction and the dissipation rate of the vortex bundle energy. Strong bonding between a particle and a vortex is explained by coupling of normal and superfluid components.

Analytical valuation of vulnerable chained options

In this paper, we present analytical pricing formulae for vulnerable chained options. To derive the price, we provide the joint probability that one of the two-dimensional Brownian motions hits multiple barriers sequentially before a fixed time and the positions of both Brownian motions. Using the derived formulae, we perform numerical analysis to investigate the impacts of counterparty risk on option prices.

The “logarithmic scale” Minkowski dimension of the most visited sites of two-dimensional Brownian motion

This paper studies the sets of visit times to points x on the plane, where a standard two-dimensional Brownian motion makes a substantial number of visits. For this purpose, we introduce the concept of the “logarithmic scale” Minkowski dimension as a tool for measuring the sets of visit times. We prove that, almost surely, there is a point x such that the “logarithmic scale” Minkowski dimension of the set of visit times to x is 1.

Some global topological properties of a free boundary problem appearing in a two dimensional controlled ruin problem

AbstractIn this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. The volatilities of the Brownian motions are different. We control the drifts of these processes and allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way, s.t. the probability that both companies survive is maximized. It turns out that the state space of the problem is divided into two sets. In one set the first company gets the full drift, and in the other set the second one. We describe some topological properties of these sets and their separating curve.

Divergence of the Variance of the Optical Phase in Gain-Switched Semiconductor Lasers Described by Stochastic Rate Equations

We report a theoretical study of the phase diffusion in a gain-switched single-mode semiconductor laser. We use stochastic rate equations for the electric field to analyze the phase statistics of the gain-switched laser. Their use avoids the instabilities obtained with rate equations for the photon number and optical phase when the photon number is small. However, we show that a new problem appears when the field equations are integrated: the variance of the optical phase becomes divergent. This divergence cannot be observed with the numerical integration of the commonly used equations for the photon number and optical phase because of the previous instabilities. The divergence of the phase variance means that this quantity does not reach a fixed value as the integration time step is decreased. We find that the phase variance increases as the integration time step decreases, with no sign of saturation behavior even for tiny steps. We explain the divergence by making the analogy of our problem with two-dimensional Brownian motion. The fact that the divergence appears is not surprising because in 1940 Paul L\`evy demonstrated that the variance of the polar angle in two-dimensional Brownian motion is a divergent quantity. Our results show that stochastic rate equations for the photon number and phase are not appropriate for describing the phase statistics when the photon number is small. Simulation of the stochastic rate equations for the electric field are consistent with L\`evy's results but gives unphysical results since an infinite value is obtained for a quantity that can be measured.

Classifying Images of Two-Dimensional Fractional Brownian Motion through Deep Learning and Its Applications

Two-dimensional fractional Brownian motion (2D FBM) is an effective model for describing natural scenes and medical images. Essentially, it is characterized by the Hurst exponent (H) or its corresponding fractal dimension (D). For optimal accuracy, we can use the maximum likelihood estimator (MLE) to compute the value. However, its computational cost is much higher than other low-accuracy estimators. Therefore, we propose a feasible deep-learning model and find out some promising pretrained models to classify the Hurst exponent efficiently and effectively. For evaluating the efficacy of deep learning models, two types of 2D FBM images were generated—11 classes and 21 classes of Hurst exponents. For comparison, we first used the efficient MLE to estimate the Hurst exponent of each image and then classified them through machine learning models. On the other hand, we used deep learning models to train and classify all images. Experimental results show that our proposed model and some pretrained models are much higher in accuracy than machine learning models for estimates from the efficient MLE. When applied, deep learning models take much lower computational time than the efficient MLE. Therefore, for accuracy and efficiency, we can use deep learning models to replace the role of the efficient MLE in the future.

Collapse and diffusion in harmonic activation and transport

Abstract For an n-element subset U of $\mathbb {Z}^2$ , select x from U according to harmonic measure from infinity, remove x from U and start a random walk from x. If the walk leaves from y when it first enters the rest of U, add y to it. Iterating this procedure constitutes the process we call harmonic activation and transport (HAT). HAT exhibits a phenomenon we refer to as collapse: Informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion. To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among n-element subsets of $\mathbb {Z}^2$ , what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, d? Concerning the former, examples abound for which the harmonic measure is exponentially small in n. We prove that it can be no smaller than exponential in $n \log n$ . Regarding the latter, the escape probability is at most the reciprocal of $\log d$ , up to a constant factor. We prove it is always at least this much, up to an n-dependent factor.

Quantum kinetic theory of flux-carrying Brownian particles

We develop the kinetic theory of the flux-carrying Brownian motion recently introduced in the context of open quantum systems. This model constitutes an effective description of two-dimensional dissipative particles violating both time-reversal and parity that is consistent with standard thermodynamics. By making use of an appropriate Breit–Wigner approximation, we derive the general form of its quantum kinetic equation for weak system-environment coupling. This encompasses the well-known Kramers equation of conventional Brownian motion as a particular instance. The influence of the underlying chiral symmetry is essentially twofold: the anomalous diffusive tensor picks up antisymmetric components, and the drift term has an additional contribution which plays the role of an environmental torque acting upon the system particles. These yield an unconventional fluid dynamics that is absent in the standard (two-dimensional) Brownian motion subject to an external magnetic field or an active torque. For instance, the quantum single-particle system displays a dissipationless vortex flow in sharp contrast with ordinary diffusive fluids. We also provide preliminary results concerning the relevant hydrodynamics quantities, including the fluid vorticity and the vorticity flux, for the dilute scenario near thermal equilibrium. In particular, the flux-carrying effects manifest as vorticity sources in the Kelvin’s circulation equation. Conversely, the energy kinetic density remains unchanged and the usual Boyle’s law is recovered up to a reformulation of the kinetic temperature.

Strong convergence to two-dimensional alternating Brownian motion processes

Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional aspects of the univariate SBM and MMBM can be studied through the limiting behavior of flip-flop processes. Here, we construct two classes of bivariate flip-flop processes whose marginals converge strongly to SBMs and are dependent on each other, which we refer to as alternating two-dimensional Brownian motion processes. While the limiting bivariate processes are not Gaussian, they possess desirable qualities, such as being tractable and having a time-varying correlation coefficient function.

Limit theorems and ergodicity for general bootstrap random walks

Given the increments of a simple symmetric random walk (Xn)n≥0, we characterize all possible ways of recycling these increments into a simple symmetric random walk (Yn)n≥0 adapted to the filtration of (Xn)n≥0. We study the long term behavior of a suitably normalized two-dimensional process ((Xn,Yn))n≥0. In particular, we provide necessary and sufficient conditions for the process to converge to a two-dimensional Brownian motion (possibly degenerate). We also discuss cases in which the limit is not Gaussian. Finally, we provide a simple necessary and sufficient condition for the ergodicity of the recycling transformation, thus generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and solving the discrete version of the open problem of the ergodicity of the general Lévy transformation (see Mansuy and Yor, 2006).

Rate of escape of conditioned Brownian motion

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima: we find an integral test on the function g so that the future minima process drops below the barrier exp{lnt×g(lnlnt)} at arbitrary large times; we show that the future minima process exceeds K t×lnlnlntat arbitrary large times with probability 0 [resp., 1] if K is larger [resp., smaller] than some positive constant. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.