Singular Drift Research Articles

Stochastic differential equations with Sobolev diffusion and singular drift and applications

In this paper, we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points and the Malliavin differentiability with respect to sample paths. We also establish Bismut–Elworthy–Li’s formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier–Stokes equations given by Constantin–Iyer [Comm. Pure Appl. Math. 61 (2008) 330–345] to prove the local well-posedness of NSEs in $\mathbb{R}^{d}$ with initial values in the first-order Sobolev space $\mathbb{W}^{1}_{p}(\mathbb{R}^{d};\mathbb{R}^{d})$ provided $p>d$.

Fundamental Solution of the Fractional Diffusion Equation with a Singular Drift*

We prove pointwise upper estimates of a fundamental solution of a linear evolution equation with fractional Laplacian and with a singular drift term. Such a result has already been obtained in [Maekawa and Miura, Journal of Functional Analysis 264 (2013) 2245–2268] by generalizing the classical ideas of Carlen, Kusuoka, and Stroock. Here, we propose a different and direct method to show such type of estimates, which appears also to be more elementary.

On the stochastic regularity of distorted Brownian motions

We systematically develop general tools to apply Fukushima's absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density w.r.t. the reference measure and methods to estimate drift potentials comfortably. We then apply our results to distorted Brownian motions and construct weak solutions to singular stochastic differential equations, i.e. equations with possibly unbounded and discontinuous drift and reflection terms which may be the sum of countably many local times. The solutions can start from any point of the explicitly specified state space. We consider different kind of weights, like Muckenhoupt $A_2$ weights and weights with moderate growth at singularities as well as different kind of (multiple) boundary conditions. Our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected Brownian motions with singular drift in bounded and unbounded multi-dimensional domains

SPDEs with two reflecting walls and two singular drifts

We study SPDEs with two reflecting walls Λ1<Λ2 and two singular drifts c1(X−Λ1)ϑ, c2(Λ2−X)ϑ, driven by space–time white noise. First, we establish the existence and uniqueness of the solutions X for ϑ≥0, which satisfy Λ1≤X≤Λ2. Second, we obtain the following pathwise properties of the solutions X. If ϑ>3, then a.s. Λ1<X<Λ2 for all t≥0; If 0<ϑ<3, then X hits Λ1 or Λ2 with positive probability in finite time. Thus ϑ=3 is the critical parameter for X to hit reflecting walls.

Fractional operators with singular drift: smoothing properties and Morrey–Campanato spaces

We investigate some smoothness properties for a linear transport-diffusion equation involving a class of non-degenerate Lévy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some Hölder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey–Campanato space for which the regularizing properties of the Lévy operator suffice to obtain global Hölder continuity.

Weak-type Estimates and Potential Estimates for Elliptic Equations with Drift Terms

We consider divergence form elliptic equations with a strongly singular drift term −div(A∇u)+b⋅∇u=μ in a domain \({\Omega } \subset \mathbb {R}^{n}\) (n≥3). We give a weak-type \(L^{1} - L^{n / (n - 2), \infty }\) estimate for a solution to the Dirichlet problem with homogeneous boundary condition. Moreover, we give a two-sided pointwise potential estimate for a weak solution and its applications.

Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities

Abstract In the present paper we prove upper and lower bounds of the heat kernel for the operator Δ - ∇ ⁡ ( | x | - α ) ⋅ ∇ {\Delta-\nabla({|x|^{-\alpha}})\cdot\nabla} in ℝ n ∖ { 0 } {\mathbb{R}^{n}\setminus\{0\}} , where α &gt; 0 {\alpha&gt;0} . We obtain these bounds from an isoperimetric inequality for a measure e - | x | - α ⁢ d ⁢ x {\mathrm{e}^{-{|x|^{-\alpha}}}dx} on ℝ n ∖ { 0 } {\mathbb{R}^{n}\setminus\{0\}} . The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.

Stochastic particle approximation of the Keller–Segel equation and two-dimensional generalization of Bessel processes

We are interested in the two-dimensional Keller–Segel partial differential equation. This equation is a model for chemotaxis (and for Newtonian gravitational interaction). When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$ of bacteria to the chemo-attractant is larger than $8\pi$. We investigate its approximation by a system of $N$ two-dimensional Brownian particles interacting through a singular attractive kernel in the drift term. In the very subcritical case $\chi<2\pi$, the diffusion strongly dominates this singular drift: we obtain existence for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a weak solution of the Keller–Segel equation. We also show that for any $N\ge2$ and any value of $\chi>0$, pairs of particles do collide with positive probability: the singularity of the drift is indeed visited. Nevertheless, when $\chi<2\pi N$, it is possible to control the drift and obtain existence of the particle system until the first time when at least three particles collide. We check that this time is a.s. infinite, so that global existence holds for the particle system, if and only if $\chi\leq8\pi(N-2)/(N-1)$. Finally, we remark that in the system with $N=2$ particles, the difference between the two positions provides a natural two-dimensional generalization of Bessel processes, which we study in details.

New insights into the problem with a singular drift term in the complex Langevin method

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables. One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD. We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process. In fact, this problem can occur in general when the stochastic process involves a singular drift term. We show, however, in a simple example that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.

Heat kernel estimates for [formula omitted] under gradient perturbation

For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a→0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.

On Weak Solutions of Elliptic Equations with Singular Drifts

We consider the Dirichlet problem for second-order linear elliptic equations with singular drift terms given by a vector field $u$. $W^{1,p}$-estimates for weak solutions are quite well known, prov...

On some functional inequalities for skew Brownian motion

We study the Poincare and logarithmic Sobolev inequalities. We provide several constructions of skew Brownian motion; this is an example of diffusion with singular drift interesting from different points of view. We obtain inequalities for skew Brownian motion that naturally generalize the Gaussian case. It turns out that for skew Brownian motion the estimates depend on the local time of the process.

Some remarks on Lefschetz thimbles and complex Langevin dynamics

Lefschetz thimbles and complex Langevin dynamics both provide a means to tackle the numerical sign problem prevalent in theories with a complex weight in the partition function, e.g. due to nonzero chemical potential. Here we collect some findings for the quartic model, and for U(1) and SU(2) models in the presence of a determinant, which have some features not discussed before, due to a singular drift. We find evidence for a relation between classical runaways and stable thimbles, and give an example of a degenerate fixed point. We typically find that the distributions sampled in complex Langevin dynamics are related to the thimble(s), but with some important caveats, for instance due to the presence of unstable fixed points in the Langevin dynamics.

Quasi-invariance of the Stochastic Flow Associated to Itô’s SDE with Singular Time-Dependent Drift

In this paper, we consider the Itô SDE $$\begin{aligned} d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in \mathbb {R}^d, \end{aligned}$$ where $$W_t$$ is a $$d$$ -dimensional standard Wiener process and the drift coefficient $$b:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}^d$$ belongs to $$L^q(0,T;L^p(\mathbb {R}^d))$$ with $$p\ge 2, q>2$$ and $$\frac{d}{p} +\frac{2}{q}<1$$ . In 2005, Krylov and Röckner (Probab Theory Relat Fields 131(2):154–196, 2005) proved that the above equation has a unique strong solution $$X_t$$ . Recently, it was shown by Fedrizzi and Flandoli (Stoch Anal Appl 31:708–736, 2013) that the solution $$X_t$$ is indeed a stochastic flow of homeomorphisms on $$\mathbb {R}^d$$ . We prove in the present work that the Lebesgue measure is quasi-invariant under the flow $$X_t$$ .

Stable process with singular drift

Suppose that d≥2 and α∈(1,2). Let μ=(μ1,…,μd) be such that each μi is a signed measure on Rd belonging to the Kato class Kd,α−1. In this paper, we consider the stochastic differential equation dXt=dSt+dAt, where St is a symmetric α-stable process on Rd and for each j=1,…,d, the jth component Atj of At is a continuous additive functional of finite variation with respect to X whose Revuz measure is μj. The unique solution for the above stochastic differential equation is called an α-stable process with drift μ. We prove the existence and uniqueness, in the weak sense, of such an α-stable process with drift μ and establish sharp two-sided heat kernel estimates for such a process.

Reactive trajectories and the transition path process

We study the trajectories of a solution \(X_t\) to an Ito stochastic differential equation in \({\mathbb { R}}^d\), as the process passes between two disjoint open sets, \(A\) and \(B\). These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets \(A\) and \(B\) represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process \(Y_t\), which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process \(X_t\). As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by Weinan and Vanden-Eijnden.

The Small Time Asymptotics of SPDEs with Reflection

We study stochastic partial differential equations with singular drifts and with reflection, driven by space-time white noise with nonconstant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. As a byproduct, the Hölder continuity of the solution is obtained. The strong Feller property is also obtained. Moreover, we show large deviation principle.

Hölder continuity property of the densities of SDEs with singular drift coefficients

We prove that the solution of stochastic differential equations with deterministic diffusion coefficient admits a Holder continuous density via a condition on the integrability of the Fourier transform of the drift coefficient. In our result, the integrability is an important factor to determine the order of Holder continuity of the density. Explicit examples and some applications are given.

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [Oper. Res. 56 (2008), 607–617] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler–Maruyama Monte Carlo method from 𝒪 ( TOL - 3 ) ${{{\mathcal {O}}({\mathrm {TOL}}^{-3})}}$ to 𝒪 ( TOL - 2 log ( TOL - 1 ) 2 ) ${{{\mathcal {O}}({\mathrm {TOL}}^{-2}\log ({\mathrm {TOL}}^{-1})^{2})}}$ for a mean square error of 𝒪 ( TOL 2 ) ${{{\mathcal {O}}({\mathrm {TOL}}^2)}}$ . Later, the work [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] presented an MLMC method using a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretization MLMC method. This work improves the adaptive MLMC algorithms presented in [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is 𝒪 ( TOL - 4 ) ${{{\mathcal {O}}({\mathrm {TOL}}^{-4})}}$ . For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy 𝒪 ( TOL ) ${{{\mathcal {O}}({\mathrm {TOL}})}}$ from 𝒪 ( TOL - 3 ) ${{{\mathcal {O}}({\mathrm {TOL}}^{-3})}}$ for the adaptive single level algorithm to essentially 𝒪 ( TOL - 2 log ( TOL - 1 ) 2 ) ${{{\mathcal {O}}({\mathrm {TOL}}^{-2}\log ({\mathrm {TOL}}^{-1})^2)}}$ for the adaptive MLMC algorithm.

The Patlak–Keller–Segel model of chemotaxis on with singular drift and mortality rate

We extend the classical Patlak–Keller–Segel model of chemotaxis on in the following two directions: the presence of a logarithmically singular prescribed drift term, which aggregates mass at a singular point, and the effect of a linear logistic term (mortality rate) which induces an overall exponential decay of the mass.The results characterize conditional global existence (existence of solutions at all time for selected initial data) as well as unconditional blow-up at a finite time, with respect to the total mass and second moment of the initial data.