Singular Drift Research Articles

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

Well-Posedness for Path-Distribution Dependent Stochastic Differential Equations with Singular Drifts

Well-Posedness for Path-Distribution Dependent Stochastic Differential Equations with Singular Drifts

Strong and weak convergence for the averaging principle of DDSDE with singular drift

Strong and weak convergence for the averaging principle of DDSDE with singular drift

Stochastic Differential Equations with Local Growth Singular Drifts

Stochastic Differential Equations with Local Growth Singular Drifts

Numerical method for singular drift stochastic differential equation driven by fractional Brownian motion

In this paper, we study the stochastic differential equation with singular drift coefficient driven by fractional Brownian motion with Hurst parameter H∈(12,1). We obtain that the optimal convergence rate of the backward Euler method is 1.0. The constant elasticity of variance model and the Aït-Sahalia interest rate model are performed as numerical experiments to validate our theoretical results.

Unveiling dynamics changes: Singular spectrum analysis-based method for detecting concept drift in industrial data streams

Industrial data streams frequently experience concept drifts. Current drift detection methods, focusing on prediction performance or data distribution, often neglect temporal dependencies and require prior distribution assumptions. These limitations have catalyzed the development of dynamic theory-based approaches that identify drifts by discerning variations in data dynamics. However, the majority of these exhibit high complexity in dynamics characterization and still underperform in industrial drift detection. To overcome these challenges, we propose a singular spectrum analysis-based drift detection method for industry data streams, comprising three modules: common dynamics extraction, noise-robust drift detection, and threshold adaptation. Our approach uses singular spectrum analysis (SSA) as the dynamics representation method instead of traditional high-dimensional phase space embedding and Fourier transform. SSA allows for the efficient description of principal dynamics by intelligently mixing the Fourier basis functions according to given data. The first two modules are specifically engineered to address the common dynamics and stochastic noise inherent in industrial data streams through a well-designed online common dynamics extraction algorithm and a noise-robust detection index respectively. Threshold adaptation module provides a more reasonable way of updating the drift threshold based on the extreme value theory, ensuring global consistency for parameter estimation of extreme value distribution. Our proposed method was tested on simulated examples and a real-world sintering process, exhibiting enhanced drift detection performance.

Stochastic transport equation with singular drift

Stochastic transport equation with singular drift

Stochastic model for barrier crossings and fluctuations in local timescale

The problem of computing the rate of diffusion-aided activated barrier crossings between metastable states is one of broad relevance in physical sciences. The transition path formalism aims to compute the rate of these events by analysing the statistical properties of the transition path between the two metastable regions concerned. In this paper, we show that the transition path process is a unique solution to an associated stochastic differential equation, with a discontinuous and singular drift term. The singularity arises from a local time contribution, which accounts for the fluctuations at the boundaries of the metastable regions. The presence of fluctuations at the local time scale calls for an excursion theoretic consideration of barrier crossing events. We show that the rate of such events, as computed from excursion theory, factorizes into a local time term and an excursion measure term, which bears empirical similarity to the transition state theory rate expression. Since excursion theory makes no assumption about the presence of a transition state in the potential energy landscape, the mathematical structure underlying this factorization ought to be general. We hence expect excursion theory (and local times) to provide some physical and mathematical insights in generic barrier crossing problems.

Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem \(-\hbox{div}( M(x) \nabla u ) = -\hbox{div} (E(x) u) + f\) in a bounded domain of \(\mathbb{R}^N\) with \(N \ge 3\). We are particularly interested in singular \(E\) with \(\hbox{div} E \ge 0\). We start by recalling known existence results when \(|E| \in L^N\) that do not rely on the sign of \(\hbox{div} E \). Then, under the assumption that \(\hbox{div} E \ge 0\) distributionally, we extend the existence theory to \(|E| \in L^2\). For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of \(E\) singular at one point as \(Ax /|x|^2\), or towards the boundary as \(\hbox{div} E \sim \hbox{dist}(x, \partial \Omega)^{-2-\alpha}\). In these cases the singularity of \(E\) leads to \(u\) vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. \(\partial u / \partial n < 0\), fails in the presence of such singular drift terms \(E\). For more information see https://ejde.math.txstate.edu/Volumes/2024/13/abstr.html

One-dimensional SDEs with LPS-type singular drift coefficients and Hölder continuous diffusion coefficients

In this paper, under the assumptions of LPS-type singular drift coefficients and Hölder continuous diffusion coefficients, the existence, uniqueness, stability, non-confluent property and strong Feller property for the solution to a one-dimensional stochastic differential equations are obtained. Our proof are based on the tools of local time and Zvonkin's transform.

Elliptic equations with a singular drift from a weak Morrey space

Elliptic equations with a singular drift from a weak Morrey space

On divergence-free (form-bounded type) drifts

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.

Robin problems for elliptic equations with singular drifts on Lipschitz domains

Robin problems for elliptic equations with singular drifts on Lipschitz domains

Stability and moment estimates for the stochastic singular Φ-Laplace equation

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic p-Laplace equations and, more generally, singular stochastic Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir et al. (2012) [22], and improve the results by Liu and Tölle (2011) [41].

Approximation of distribution-independent and distribution-dependent stochastic differential equations with singular drifts

This work is concerned with convergence rate of Euler–Maruyama scheme for distribution-independent and distribution-dependent stochastic differential equations, where the drifts involved are Dini continuous and unbounded. Via introducing a new Zvonkin-type’s transformation, we investigate convergence rate of Euler–Maruyama scheme for stochastic differential equations with singular coefficients which allows the drifts to be unbounded. Moreover, via the analysis of interacting particle systems, we show an approximation issue on a class of distribution-dependent stochastic differential equations with singular drifts. More specifically, propagation of chaos and convergence rate of the Euler–Maruyama scheme associated with the interacting particle systems are investigated.

Strong uniqueness of finite-dimensional Dirichlet operators with singular drifts

We show [Formula: see text]-uniqueness for any [Formula: see text] and the essential self-adjointness of a Dirichlet operator [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text]. In particular, [Formula: see text] is allowed to be in [Formula: see text] or in [Formula: see text] for some [Formula: see text], while [Formula: see text] is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.

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.

Perturbations of singular fractional SDEs

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.

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

Large deviation principle for distribution dependent S(P)DEs with singular drift

In this paper, we first prove that the large deviation principle is preserved under a homeomorphism. As an application, we establish the large deviation principle for distribution dependent stochastic differential equations with Dini continuous drift, semilinear distribution dependent stochastic partial differential equations with Dini continuous drift, respectively.