Drift Term Research Articles

Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

Cα regularity of weak solutions of non-homogeneous ultraparabolic equations with drift terms

Forecasting Sugar Price Fluctuation In Malaysia Using Geometric Brownian Motion Modelling

A mathematical model known as Geometric Brownian motion has proven to be an effective tool that can be deployed to forecast the price of goods in the future due to the presence of random terms, which represent the stochastic or random fluctuation of prices over a given period of time. The success of this model revolves around the estimation of its governing parameters. To efficiently predict the price of goods, using the Geometric Brownian Motion model (GBM), one needs to determine the value of returns and use the calculated returns to estimate the value of drift as well as volatility. This research used the value of volatility and drift terms obtained from real data of sugar in the months under review. The method has shown to be very reliable in capturing the intelligent trend in the price of sugar in Malaysia. The model was able to stimulate the pattern of the predicted price that shares a great resemblance to the actual price of sugar. The result obtained is very encouraging and places this study with a good note now that the country is just returning to normalcy from the pandemic that has crippled the economies of most developing nations in the world. We used the model to predict the price of sugar for a period of 20 months and the result of our prediction as well as the graph of the predicted prices confirmed the practicability of the model. The obtained values of MAPE and MSE, two of the popular performance metrics, also justified the effectiveness of the GBM in capturing the trend in the data. This model can be classified as a good model that can be deployed to forecast the price of goods that exhibit high volatility.

Uniqueness in Weighted Lebesgue Spaces for an Elliptic Equation with Drift on Manifolds

We investigate the uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of elliptic equations with a drift posed on a complete, noncompact, Riemannian manifold M of infinite volume and dimension Nge 2. Furthermore, in the special case of a model manifold with polynomial volume growth, we show that the conditions on the drift term are sharp.

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

Symmetric Fuzzy Stochastic Differential Equations Driven by Fractional Brownian Motion

We consider symmetric fuzzy stochastic differential equations where diffusion and drift terms arise in a symmetric way at both sides of the equations and diffusion parts are driven by fractional Brownian motions. Such equations can be used in real-life hybrid systems, which include properties of being both random and fuzzy and reflecting long-range dependence. By imposing on the mappings occurring in the equation the conditions of Lipschitzian continuity and additional constraints by an integrable stochastic process, we construct an approximation sequence of fuzzy stochastic processes and apply this to prove the existence of a unique solution to the studied equation. Finally, a model from population dynamics is considered to illustrate the potential application of our equations.

Adaptive Output Feedback Control for Nonholonomic Chained Systems with Integral Input State Stability Inverse Dynamics.

This paper investigates a class of nonholonomic chained systems with integral input-to-state stable (iISS) inverse dynamics subject to unknown virtual control directions and parameter uncertainty included in drift terms. First, the system is divided into two interconnected subsystems according to the system's structure. Second, one controller is designed using a switch strategy for state finite escape. Then, another controller and adaptive law are designed by combining a reduced-order state observer and backstepping method after input-state scaling. Finally, simulation results validate the feasibility of the proposed control algorithm.

Composite Patankar-Euler methods for positive simulations of stochastic differential equation models for biological regulatory systems.

Stochastic differential equations (SDE) are a powerful tool to model biological regulatory processes with intrinsic and extrinsic noise. However, numerical simulations of SDE models may be problematic if the values of noise terms are negative and large, which is not realistic for biological systems since the molecular copy numbers or protein concentrations should be non-negative. To address this issue, we propose the composite Patankar-Euler methods to obtain positive simulations of SDE models. A SDE model is separated into three parts, namely, the positive-valued drift terms, negative-valued drift terms, and diffusion terms. We first propose the deterministic Patankar-Euler method to avoid negative solutions generated from the negative-valued drift terms. The stochastic Patankar-Euler method is designed to avoid negative solutions generated from both the negative-valued drift terms and diffusion terms. These Patankar-Euler methods have the strong convergence order of a half. The composite Patankar-Euler methods are the combinations of the explicit Euler method, deterministic Patankar-Euler method, and stochastic Patankar-Euler method. Three SDE system models are used to examine the effectiveness, accuracy, and convergence properties of the composite Patankar-Euler methods. Numerical results suggest that the composite Patankar-Euler methods are effective methods to ensure positive simulations when any appropriate stepsize is used.

Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H1 stability, higher moment H1 stability, L2 stability, and higher moment L2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

The Smoluchowski–Kramer approximation of a generalized Langevin equation with state-dependent damping

The small-mass limit is derived for a generalized Langevin equation (GLE) with state-dependent damping and a kernel expressed as an infinite sum of exponentials. The state dependence includes both the current and the past. By some bounded estimates and tightness, as the mass tends to 0, the GLE is shown to converge in distribution to a limit equation with additional drift terms that come from the current and the past state dependence, respectively.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns

Neuronal networks are complex systems of interconnected neurons responsible for transmitting and processing information throughout the nervous system. The building blocks of neuronal networks consist of individual neurons, specialized cells that receive, process, and transmit electrical and chemical signals throughout the body. The formation of neuronal networks in the developing nervous system is a process of fundamental importance for understanding brain activity, including perception, memory, and cognition. To form networks, neuronal cells extend long processes called axons, which navigate toward other target neurons guided by both intrinsic and extrinsic factors, including genetic programming, chemical signaling, intercellular interactions, and mechanical and geometrical cues. Despite important recent advances, the basic mechanisms underlying collective neuron behavior and the formation of functional neuronal networks are not entirely understood. In this paper, we present a combined experimental and theoretical analysis of neuronal growth on surfaces with micropatterned periodic geometrical features. We demonstrate that the extension of axons on these surfaces is described by a biased random walk model, in which the surface geometry imparts a constant drift term to the axon, and the stochastic cues produce a random walk around the average growth direction. We show that the model predicts key parameters that describe axonal dynamics: diffusion (cell motility) coefficient, average growth velocity, and axonal mean squared length, and we compare these parameters with the results of experimental measurements. Our findings indicate that neuronal growth is governed by a contact-guidance mechanism, in which the axons respond to external geometrical cues by aligning their motion along the surface micropatterns. These results have a significant impact on developing novel neural network models, as well as biomimetic substrates, to stimulate nerve regeneration and repair after injury.

Polynomial Propagation of Moments in Stochastic Differential Equations

We address the problem of approximating the moments of the solution, , of an Itô stochastic differential equation (SDE) with drift and diffusion terms over a time grid . In particular, we assume an explicit numerical scheme for the generation of sample paths and then obtain recursive equations that yield any desired noncentral moment of as a function of the initial condition . The core of the methodology is the decomposition of the numerical solution into a “central part” and an “effective noise” term. The central term is computed deterministically from the ordinary differential equation (ODE) that results from eliminating the diffusion term in the SDE, while the effective noise accounts for the stochastic deviation from the numerical solution of the ODE. For simplicity, we describe the proposed methodology based on an Euler–Maruyama integrator, but other explicit numerical schemes can be exploited in the same way. We also apply the moment approximations to construct estimates of the 1-dimensional marginal probability density functions of based on a Gram–Charlier expansion. Both for the approximation of moments and 1-dimensional densities, we describe how to handle the cases in which the initial condition is fixed (i.e., for some deterministic and known ) or random. In the latter case, we resort to polynomial chaos expansion (PCE) schemes in order to approximate the target moments. The methodology has been inspired by the PCE and differential algebra methods used for uncertainty propagation in astrodynamics problems. Hence, we illustrate its application for the quantification of uncertainty in a 2-dimensional Keplerian orbit perturbed by a Wiener noise process.

Stochastic differential equation modelling of cancer cell migration and tissue invasion

Invasion of the surrounding tissue is a key aspect of cancer growth and spread involving a coordinated effort between cell migration and matrix degradation, and has been the subject of mathematical modelling for almost 30 years. In this current paper we address a long-standing question in the field of cancer cell migration modelling. Namely, identify the migratory pattern and spread of individual cancer cells, or small clusters of cancer cells, when the macroscopic evolution of the cancer cell colony is dictated by a specific partial differential equation (PDE). We show that the usual heuristic understanding of the diffusion and advection terms of the PDE being one-to-one responsible for the random and biased motion of the solitary cancer cells, respectively, is not precise. On the contrary, we show that the drift term of the correct stochastic differential equation scheme that dictates the individual cancer cell migration, should account also for the divergence of the diffusion of the PDE. We support our claims with a number of numerical experiments and computational simulations.

Particle approximation of the doubly parabolic Keller-Segel equation in the plane

In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any N≥2, we show tightness in N of its empirical measure, and that any weak limit point of this empirical measure, as N→∞, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.

Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Padé and log-Padé approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the Padé approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus.We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative (more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The conformally symplectic maps depend on a dissipative parameter and a drift term, which is needed to get the existence of invariant attractors. The 4D maps are obtained coupling (i) two symplectic standard maps, or (ii) two conformally symplectic standard maps, or (iii) a symplectic and a conformally symplectic standard map.Concerning the results, Padé and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.

Periodic measures for the stochastic delay modified Swift–Hohenberg lattice systems

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift–Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T-periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.

Existence of Optimal Control for Nonlinear Fokker–Planck Equations in \(\boldsymbol{L^1(\mathbb{R}^d)}\).

This work is concerned with the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker–Planck equation in with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. One obtains in particular the existence for the stochastic optimal control problem governed by McKean–Vlasov SDEs. For this problem, one proves the existence of a stochastic Markov optimal controller in feedback form.

Stable stadial and interstadial states of the last glacial's climate identified in a combined stable water isotope and dust record from Greenland

Abstract. During the last glacial interval, the Northern Hemisphere climate was punctuated by a series of abrupt changes between two characteristic climate regimes. The existence of stadial (cold) and interstadial (milder) periods is typically attributed to a hypothesised bistability in the glacial North Atlantic climate system, allowing for rapid transitions from the stadial to the interstadial state – the so-called Dansgaard–Oeschger (DO) events – and more gradual yet still fairly abrupt reverse shifts. The physical mechanisms driving these regime transitions remain debated. DO events are characterised by substantial warming over Greenland and a reorganisation of the Northern Hemisphere atmospheric circulation, which are evident from concomitant shifts in the δ18O ratios and dust concentration records from Greenland ice cores. Treating the combined δ18O and dust record obtained by the North Greenland Ice Core Project (NGRIP) as a realisation of a two-dimensional, time-homogeneous, and Markovian stochastic process, we present a reconstruction of its underlying deterministic drift based on the leading-order terms of the Kramers–Moyal equation. The analysis reveals two basins of attraction in the two-dimensional state space that can be identified with the stadial and interstadial regimes. The drift term of the dust exhibits a double-fold bifurcation structure, while – in contrast to prevailing assumptions – the δ18O component of the drift is clearly mono-stable. This suggests that the last glacial's Greenland temperatures should not be regarded as an intrinsically bistable climate variable. Instead, the two-regime nature of the δ18O record is apparently inherited from a coupling to another bistable climate process. In contrast, the bistability evidenced in the dust drift points to the presence of two stable circulation regimes of the last glacial's Northern Hemisphere atmosphere.

Well-posedness of a system of SDEs driven by jump random measures

We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the correlation between the driving Brownian motions and jump random measures. Pathwise uniqueness is derived by employing some standard techniques. Then, we use a comparison theorem along with our uniqueness result to construct non-negative, [Formula: see text]-integrable càdlàg solutions as monotone limits of solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to be included. Our approach allows also for a comparison property to be established for the solutions to the systems we investigate. The applicability of certain systems in financial modeling is also discussed.

Mean first exit times of Ornstein–Uhlenbeck processes in high-dimensional spaces

The d-dimensional Ornstein–Uhlenbeck process (OUP) describes the trajectory of a particle in a d-dimensional, spherically symmetric, quadratic potential. The OUP is composed of a drift term weighted by a constant and a diffusion coefficient weighted by σ > 0. In the absence of drift (i.e. θ = 0), the OUP simply becomes a standard Brownian motion (BM). This paper is concerned with estimating the mean first-exit time (MFET) of the OUP from a ball of finite radius L for large . We prove that, asymptotically for , the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by ) has asymptotically no effect on its MFET. This finding might be surprising because, for small , the OUP exit time is significantly larger than BM by a margin that depends on θ. As it allows for the drift to be ignored, it might simplify the analysis of high-dimensional exit-time problems in numerous areas.Finally, our short proof for the non-asymptotic MFET of OUP, using the Andronov–Vitt–Pontryagin formula, might be of independent interest.