arXiv subjects
Publications and source records attributed to Chengcheng Ling.
We study the weak convergence of a generic tamed Euler-Maruyama scheme for kinetic stochastic differential equations (SDEs) with integrable drifts. We show that the marginal density of the considered scheme converges at rate 1/2 to the corresponding marginal density of the SDE. The convergence rate is independent from the criticality gap, which is new compared to previous results.
We study a singular stochastic equation driven by a regular noise of fractional Brownian type with Hurst index $H \in (1,\infty)\setminus\mathbb{Z}$ and drift coefficient $b \in \mathcal{C}^\alpha$, where $\alpha > 1 - \frac{1}{2H}$. The strong well-posedness of this equation was first established in [Ger23], a phenomenon referred to as regularization by regular noise. In this note, we provide a corresponding numerical analysis. Specifically, we show that the Euler-Maruyama approximation $X^n$ converges strongly to the unique solution $X$ with rate $n^{-1}$. Furthermore, under the additional assumption $b \in \mathcal{C}^1$, we show that $n(X - X^n)$ converges to a non-trivial limit as $n \to \infty$, thereby confirming that the rate $n^{-1}$ is in fact optimal upper bound for this scheme.
We establish the existence of weak solutions to a class of distribution-dependent stochastic differential equations (DDSDEs) with possibly degenerate multiplicative noise and singular coefficients. Extending the weak existence techniques introduced by Bechtold & Hofmanova 2023 to a distribution-dependent framework, we utilize pathwise averaging and local-time decomposition methods to show how irregular noise effectively regularizes analytical challenges associated with degeneracies in stochastic systems.
We consider the strong numerical approximation for a fourth-order stochastic nonlinear SPDE driven by space-time white noise on {{$d=1,2,3$}}-dimensional torus. We consider its full discretisation with a spectral Galerkin scheme in space and Euler scheme in time. We show the convergence with almost spatial rate $2-\frac{d}{2}$ and $\frac{6-d}{4}$-temporal rate obtained mainly via \it{Stochastic Sewing} technique.
We study the strong approximation of the solutions to singular stochastic kinetic equations (also referred to as second-order SDEs) driven by $\alpha$-stable processes, using an Euler-type scheme inspired by [11]. For these equations, the stability index $\alpha$ lies in the range $(1,2)$, and the drift term exhibits anisotropic $\beta$-H\"older continuity with $\beta >1 - \frac{\alpha}{2}$. We establish a convergence rate of $(\frac{1}{2} + \frac{\beta}{\alpha(1+\alpha)} \wedge \frac{1}{2})$, which aligns with the results in [4] concerning first-order SDEs.
The aim of the paper is to show the probabilistically strong well-posedness of rough differential equations with distributional drifts driven by the Gaussian rough path lift of fractional Brownian motion with Hurst parameter $H\in(1/3,1/2)$. We assume that the noise is nondegenerate and the drift lies in the Besov-H\"older space $\mathcal{C}^\alpha$ for some $\alpha>1-1/(2H)$. The latter condition matches the one of the additive noise case, thereby providing a multiplicative analogue of Catellier-Gubinelli in the regime $H\in(1/3,1/2)$.
We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses.
We study the $L^p$ rate of convergence of the Milstein scheme for SDEs when the drift coefficients possess only H\"older regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularisation by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.
We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov-R\"ockner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion. Consequentially, such SDE is strongly complete and forms a random dynamical system. Also, a further application to a boundary value problem is discussed.
We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of the SDE can be split into two terms, one of which is singular and the radial component of the other term has a radial component of sufficient strength in the direction of the origin, then the random dynamical system generated by the SDE admits a pullback attractor.
We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $\sigma$, satisfying Krylov--R\"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,\sigma^i)$, both for It\^o and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean--Vlasov SDEs, criteria for strong compactness of solutions and Wong--Zakai type theorems.
We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, H\"older continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and -- under suitable conditions -- be fine-tuned to achieve a strong convergence rate which is arbitrarily close to the benchmark $0.5$ rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition.
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.
We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system. Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) random dynamical system or not.
We investigate the space-time regularity of the local time associated to Volterra-Lévy processes, including Volterra processes driven by $α$-stable processes for $α\in(0,2]$. We show that the spatial regularity of the local time for Volterra-Lévy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturba\Ption of ODEs by a Volterra-Lévy process which has sufficiently regular local time. Following along the lines of [15], we show existence, uniqueness and differentiablility of the flow associated to such equations.
By studying parabolic equations in mixed-norm spaces, we prove the existence and uniqueness of strong solutions to stochastic differential equations driven by Brownian motion with coefficients in spaces with mixed-norm, which extends Krylov and Röckner's result in [11] and Zhang's result in [18].
The well-posedness of nonlocal elliptic equation with singular drift is investigated in Besov-Hölder spaces. As an application, we show the existence and uniqueness for corresponding martingale problem. Moreover, we prove that the one dimensional distribution of the martingale solution has a density in some Besov space.
In this paper, we prove the existence and uniqueness of maximally defined strong solutions to SDEs driven by multiplicative noise on general space-time domains $Q\subset\mathbb{R}_+\times\mathbb{R}^d$, which have continuous paths on the one-point compactification $Q\cup\partial$ of $Q$ where $\partial \notin Q$ and $Q\cup\partial$ is equipped with the Alexandrov topology. If the SDE is of gradient type (see (2.5) below) we prove that under suitable Lyapunov type conditions the life time of the solution is infinite and its distribution has sub-Gaussian tails. This generalizes earlier work \cite{KR} by Krylov and one of the authors to the case where the noise is multiplicative.