Time-dependent Drift Coefficients Research Articles

Stochastic differential equations with critically irregular drift coefficients

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.

Concentration inequalities for additive functionals: A martingale approach

This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic and naturally includes time-inhomogeneous and non-stationary processes as well as initial laws concentrated on a single point. The class of processes studied includes martingales, Markov processes and general square integrable càdlàg processes. The general approach is complemented by a simple and direct method for martingales, diffusions and discrete-time Markov processes. The method is illustrated by deriving concentration inequalities for the Polyak–Ruppert algorithm, SDEs with time-dependent drift coefficients “contractive at infinity” with both Lipschitz and squared Lipschitz observables, some classical martingales and non-elliptic SDEs.

A Prognostic Model for Stochastic Degrading Systems With State Recovery: Application to Li-Ion Batteries

Many industrial systems inevitably suffer performance degradation. Thus, predicting the remaining useful life (RUL) for such degrading systems has attracted significant attention in the prognostics community. For some systems like batteries, one commonly encountered phenomenon is that the system performance degrades with usage and recovers in storage. However, almost all of the current prognostic studies do not consider such a recovery phenomenon in stochastic degradation modeling. In this paper, we present a prognostic model for deteriorating systems experiencing a switching operating process between usage and storage, where the system degradation state recovers randomly after the storage process. The possible recovery from the current time to the predicted future failure time is incorporated in the prognosis. First, the degradation state evolution of the system is modeled through a diffusion process with piecewise but time-dependent drift coefficient functions. Under the concept of first hitting time, we derived the lifetime and RUL distributions for systems with specific constant working mode. Further, we extended the results of RUL distribution in specific constant working mode to the case of stochastic working mode, which is modeled through a flexible two-state semi-Markov model (SMM) with phase-type distributed interval times. The unknown parameters in the present model are estimated based on the observed condition monitoring data of the system, and the SMM model is identified on the basis of the operating data. A numerical study and a case study of Li-ion batteries are carried out to illustrate and demonstrate the proposed prognostic method. Experimental results indicate that the presented method can improve the accuracy of lifetime and RUL estimation for systems with state recovery.

An Adaptive Prognostic Approach via Nonlinear Degradation Modeling: Application to Battery Data

Remaining useful life (RUL) estimation via degradation modeling is considered as one of the most central components in prognostics and health management. Current RUL estimation studies mainly focus on linear stochastic models, and the results under nonlinear models are relatively limited in literature. Even in nonlinear degradation modeling, the estimated RUL is aimed at a population of systems of the same type or depend only on the current degradation observation. In this paper, an adaptive and nonlinear prognostic model is presented to estimate RUL using a system's history of the observed data to date. Specifically, a general nonlinear stochastic process with a time-dependent drift coefficient is first adopted to characterize the dynamics and nonlinearity of the degradation process. In order to render the RUL estimation depending on the degradation history to date, a state-space model is constructed, and Kalman filtering is applied to update one key parameter in the drifting function through treating this parameter as an unobserved state variable. To update the hidden state and other parameters in the state-space model simultaneously and recursively, the expectation maximization algorithm is used in conjunction with Kalman smoother to achieve this aim. The probability density function of the estimated RUL is derived with an explicit form, and some commonly used results under linear models turn out to be its special cases. Finally, the implementation of the presented approach is illustrated by numerical simulations, and an application for estimating the RUL of lithium-ion batteries is used to demonstrate the superiority of the method.

Harnack inequalities for SDEs with multiplicative noise and non-regular drift

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers [7, 21]. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved.

DRIFT COEFFICIENTS OF CHARGED PARTICLES IN TURBULENT MAGNETIC FIELDS

Using test-particle simulations, the off-diagonal elements of the diffusion tensor are evaluated numerically. The comparison of the so-obtained time-dependent drift coefficients with analytical approximations shows that, for weak turbulence strengths or for slab turbulence geometry, the weak scattering result provides an excellent agreement with the numerical results. For two- or three-dimensional turbulence geometry, however, neither the classical scattering result nor alternative analytical approaches provide an accurate description of the numerically obtained values. Furthermore, the influence is discussed of a non-constant energy range in the turbulence spectrum and of non-static turbulence, for which the time dependence is modeled using magnetohydrodynamic plasma waves.

Similarity solutions of a class of perturbative Fokker-Planck equation

In a previous work, a perturbative approach to a class of Fokker-Planck equations, which have constant diffusion coefficients and small time-dependent drift coefficients, was developed by exploiting the close connection between the Fokker-Planck equations and the Schrodinger equations. In this work, we further explore the possibility of similarity solutions of such a class of Fokker-Planck equations. These solutions possess definite scaling behaviors, and are obtained by means of the so-called similarity method.

L p -regularity for parabolic operators with unbounded time-dependent coefficients

We establish the maximal regularity for nonautonomous Ornstein–Uhlenbeck operators in L p-spaces with respect to a family of invariant measures, where \({p \in (1, +\infty)}\) . This result follows from the maximal L p-regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on \({L^{p}(\mathbb{R}^{N} )}\) with Lebesgue measure.

Infinite dimensional Kolmogorov operators with time dependent drift coefficients

Infinite dimensional Kolmogorov operators with time dependent drift coefficients