Michael Röckner, Guohuan Zhao
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorov's backward equations in some second order Sobolev spaces, which is analytically interesting in itself.
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Strong solutions of SDEs with singular (form-bounded) drift via Roeckner-Zhao approach
本文采用Roeckner-Zhao方法，在最小假设下证明了具有奇异漂移的随机微分方程的强适定性。
Form-boundedness and SDEs with singular drift
该研究梳理并改进了带奇异漂移的随机微分方程在若干极小假设下的弱适定性与强适定性相关研究结果。
We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.
The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a $L^p$ variant of De Giorgi's method.
We prove the unique weak solvability of stochastic differential equations with time-inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e.\,with integrability parameter $q>1$ close to $1$. The constructed weak solutions constitute a Feller evolution family. The proofs are based on a detailed Sobolev regularity theory of the corresponding parabolic equation.
In this letter we prove existence and uniqueness of strong solutions to multi-dimensional SDEs with discontinuous drift and finite activity jumps.
Bounds on convergence rate to the invariant distribution for a class of stochastic differential equations (SDEs) with a gradient-type drift are obtained.
A new proof of pathwise uniqueness for SDEs with Sobolev diffusion and integrable drift term is introduced by extending a method from E. Fedrizzi and F. Flandoli (Pathwise uniqueness and continuous dependence of SDEs with non-regular drift, Stochastics 83 (2011), pp. 241--257) to the case of nonconstant diffusion.
In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributionsof a continuous function.
Non-local SDEs, critical drifts and local blow-ups
研究具有奇异时间非齐次一般漂移的α-稳定随机微分方程，漂移条件接近最小可能的尺度不变性，并可能引发粒子系统中强吸引相互作用导致的局部爆炸。
We obtain $T_2(C)$ for stochastic differential equations with Dini continuous drift and $T_1(C)$ stochastic differential equations with singular coefficients.
By using the ultracontractivity of a reference diffusion semigroup, Krylov's estimate is established for a class of degenerate SDEs with singular drifts, which leads to existence and pathwise uniqueness by means of Zvonkin's transformation. The main result is applied to singular SDEs on generalized Heisenberg groups.
In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To this end we construct a sequence of SDEs with globally Lipschitz drifts. We show that the solutions of these SDEs converge to the solution of the original SDE and the p-moments of their Malliavin derivatives are uniformly bounded.
In this paper, the existence and uniqueness of the distribution dependent SDEs with Hölder continuous drift driven by $α$-stable process is investigated. Moreover, by using Zvonkin type transformation, the convergence rate of Euler-Maruyama method is also obtained. The results cover the ones in the case of distribution independent SDEs.
We establish the well-posedness of stochastic differential equations possessing degenerate diffusions and singular drifts. We prove that SDEs defined on the homogeneous Carnot group, whose hypoelliptic diffusion part is given by the horizontal Brownian motion, admit a unique strong solution for a large class of singular drifts. It considerably generalizes the classical well-posedness results of singular SDEs with non-degenerate diffusions. It also provides an intermediate result between the Cauchy-Lipschitz theorem in ordinary differential equations and the result proved by Krylov and Röckner …
Applying Zvonkin's transform, the exponential convergence in Wasserstein distance for a class of functional SDEs with Hölder continuous drift is obtained. This combining with log-Harnack inequality implies the same convergence in the sense of entropy, which also yields the convergence in total variation norm by Pinsker's inequality.
In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler-Maruyama scheme for such a class of SDEs. We prove convergence of order $1/2$. Finally, we present numerical examples.
In this paper, we investigate stochastic differential equations(SDEs) driven by a class of supercritical $α$-stable process(including the rotational symmetric $α-$stable process) with drift $b$. The weak well-posedness is proved, provided that the $(1-α)$-Hölder semi-norm of $b$ is sufficient small.
Limit theorems for SDEs with irregular drifts
本文研究Hölder连续及分段连续漂移项SDEs加性泛函的极限定理与收敛速率，通过反射耦合与测试函数得到解过程的（拟）Wasserstein距离指数收缩性以支撑结论。
We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + σ(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{p}(\mathbb{R}^d)$, $p>d$ and $p\geq 2$, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem.
In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some regularity such as relative entropy and Sobolev's estimate of invariant probability measure are proved. Finally, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $ν$-almost everywhere stochastic invertible flow, where $ν$ is the Riemannian measure, which is quasi-invariant with respect to $ν$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.
In this paper, the coupling by change of measure is constructed for a class of SDEs with integrable drift and additive noise, from which the Harnack and shift Harnack inequalities are derived. Finally, as applications, the gradient estimate, the regularity of the heat kernel and the distribution properties of the associated transition probability are also obtained. The important tool is Krylov's estimate.
In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.
Small time asymptotics for kinetic SDEs with singular drift
该研究针对动力学Kato类中带奇异漂移的动力学随机微分方程，建立小时间大偏差原理并证明其转移密度的逐点Varadhan型渐近，为相关渐近分析提供理论支撑。
A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
In this paper, using Zvonkin type transform, the large deviation principle is proved for stochastic differential equations with Dini continuous drifts, where the existed methods for large deviation principle are unavailable. The method and result are new in related fields. Moreover, the result is also extended to a class of degenerate stochastic differential equations with Dini continuous drifts.
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 \cite{Flandoli, Zhang11}. 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 s…
Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts
本文基于前人研究，探讨了退化伊藤随机微分方程在局部有界漂移下的点wisewell-posedness问题，通过椭圆正则性结果和额外正则性假设，提供欧几里得空间中每个起始点的点wise分析，主要贡献是弱well-posedness，即弱存在性和唯一性。
Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)μ_{X_t}(dy)d t,\ \ X_0=x, $$ where $μ_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t(\widetilde L_x^p)$ with $\frac dp+\frac2q<2$, where $\widetilde L^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations wi…
Instantaneous blowup for interacting SDEs with superlinear drift
研究了满足有限Osgood条件的相互作用SDEs系统，证明初始轮廓衰减速度慢于特定指数函数时会出现瞬时爆炸。
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
Existence and uniqueness are proved for Mckean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to $W_0$ or $W_0+W_θ$ for some $θ\ge 1$, where $W_0$ is the total variation distance and $W_θ$ is the $L^θ$-Wasserstein distance. This improves some existing results where the drift is either locally bounded in the space variable or continuous in the distribution variable with respect to the Wasserstein distance.
Asymptotic log-Harnack inequality for path-distribution dependent SDEs with infinite memory and Dini drift
针对具有无限记忆和Dini漂移的路径分布依赖随机微分方程，建立了渐近对数哈纳克不等式，该结果在分布无关情形下同样新颖。
In this paper, high-order moment, even exponential moment, estimates are established for the Hölder norm of solutions to stochastic differential equations driven by fractional Brownian motion whose drifts are measurable and have linear growth. As applications, we first study the weak uniqueness of solutions to fractional stochastic differential equations. Moreover, combining our estimates and the Fourier transform, we establish the existence of density of solutions to equations with irregular drifts.
We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito-Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type formula for the first derivatives of the associated diffusion semigroup.
In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path--dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama--Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super--linear growth condition), Gaussian two--sided bound and Hölder conti…
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
$L^p$-sup Convergence of the Euler-Maruyama Scheme for SDEs with Distributional Besov Drift
本文研究了具有分布贝索夫漂移的SDEs的欧拉-马尔萨夫方案的$L^p$-sup收敛性，利用Yamada-Watanabe技术证明了$L^p$范数下的收敛率，并获得了$L^1$-$\sup$范数下的显式收敛率。
Let X be the solution of the multidimensional stochastic differential equationdX(t) = b(t, X(t)) dt + sigma(t, X(t)) dW(t)\, with X(0)=x where W is a standard Brownian motion. We show that when b is measurable and sigma is in an appropriate Sobolev space, the law of X satisfies a uniform quadratic transportation inequality.
We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic Gâteaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.…
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $H\in (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(\cdot,μ) = f\astμ(\cdot) + g(\cdot),\quad f,g\in B^α_{\infty,\infty}, \quad α>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the u…
We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.
The existence and uniqueness of mild solutions are proved for a class of degenerate stochastic differential equations on Hilbert spaces where the drift is Dini continuous in the component with noise and Hölder continuous of order larger than $\ff 2 3$ in the other component. In the finite-dimensional case the Dini continuity is further weakened. The main results are applied to solve second order stochastic systems driven by space-time white noises.
Well-Posedness for SDEs with Logarithmical Critical Distributional Drifts
该研究针对带迭代对数校正的临界分布漂移随机微分方程，通过光滑逼近构造弱解，结合多种估计证明律的唯一性，还得到确定初值解的时间边缘分布密度的双侧高斯估计。
In recent years, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space has begun. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best $L_p$-error rate known so far for approximation of the solution at the final time point is $3/4$ in terms of the number of evaluations of the drivin…
In this paper we study jump-diffusion stochastic differential equations (SDEs) with a discontinuous drift coefficient and a possibly degenerate diffusion coefficient. Such SDEs appear in applications such as optimal control problems in energy markets. We prove existence and uniqueness of strong solutions. In addition we study the strong convergence order of the Euler-Maruyama scheme and recover the optimal rate $1/2$.
We study the zero-noise limit for autonomous, one-dimensional ordinary differential equations with discontinuous right-hand sides. Although the deterministic equation might have infinitely many solutions, we show, under rather general conditions, that the sequence of stochastically perturbed solutions converges to a unique distribution on classical solutions of the deterministic equation. We provide several tools for computing this limit distribution.
Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift
该研究针对满足Serrin型条件的稳定驱动SDE，证明了弱适定性与热核估计，并推导了带截断随机漂移的Euler格式的密度收敛阶。
In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.
We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that process at any given time are derived. The bounds and their optimality is shown by identifying the worst case stochastic differential equation. Then we generalise our findings to a larger class of diffusion coefficients.
A note on weak existence for singular SDEs
本文利用部分 Zvonkin 变换推广 Krylov 关于奇异 SDE 弱存在性的结果，使其适用于无穷远增长或一致局部 Lebesgue 空间中的漂移项。
We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.
Strong rate of convergence for the Euler--Maruyama scheme of SDEs with unbounded Hölder continuous drift coefficient
本文研究了具有无界Hölder连续漂移系数的SDEs中Euler-Maruyama方案的强收敛速率，采用Itô-Tanaka技巧和热核估计进行分析。
We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability conditions and avoid continuity assumptions as far as possible. Our approach is mainly based on Zvonkin's transformation [18], an extended version of the probabilistic approach of Maslowski and Seidler [12] and the convergence concept for random variables in topological spaces in [2].
In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for {which} the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system {possesses} nonlinearity of type $f(|y|)|z|^2,$ where $f$ is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of \cite{ImkDosReis} in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadrati…
Weak Error on the densities for the Euler scheme of stable additive SDEs with Besov drift
针对漂移项为分布意义且属于混合 Lebesgue-Besov 空间的稳定加性 SDE，研究 Euler-Maruyama 离散化的密度弱误差收敛率，得到依赖于参数的收敛速度。
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild H{ö}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $α\in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.
Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.
In this article we study (possibly degenerate) stochastic differential equations (SDE) with irregular (or discontiuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of non-constant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smoot…
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