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View article: Kinetic SDEs with subcritical distributional drifts
Kinetic SDEs with subcritical distributional drifts Open
In this paper we study the well-posedness of the kinetic stochastic differential equation (SDE) in $\mathbb R^{2d}(d\geq2)$ driven by Brownian motion: $$\mathord{\rm d} X_t=V_t\mathord{\rm d} t,\ \mathord{\rm d} V_t=b(t,X_t,V_t)\mathord{\r…
Strong and weak well-posedness of McKean-Vlasov SDEs driven by $α$-stable processes under unified condition Open
In this paper, we consider $α\in (0,2)$ and establish the strong well-posedness of McKean--Vlasov SDEs driven by an $α$-stable process with a Hölder (Besov) kernel $K \in \mathbf{C}^β$, where $β> 1-α$. This condition coincides with the wel…
View article: Quantitative approximation to density dependent SDEs driven by $α$-stable processes
Quantitative approximation to density dependent SDEs driven by $α$-stable processes Open
Based on a class of moderately interacting particle systems, we establish a quantitative approximation for density-dependent McKean-Vlasov SDEs and the corresponding nonlinear, nonlocal PDEs. The SDE is driven by both Brownian motion and p…
Supercritical McKean-Vlasov SDE driven by cylindrical $α$-stable process Open
In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $α$-stable process in $\mathbb{R}^d$ with $α\in (0,1)$: $$ \mathord{\rm d} X_t = (K * μ_{t})(X_t)\mathord{\rm d}t + \ma…
View article: Averaging principle for SDEs with singular drifts driven by $α$-stable processes
Averaging principle for SDEs with singular drifts driven by $α$-stable processes Open
In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for n…
Quantitative approximation of stochastic kinetic equations: from discrete to continuum Open
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.…
Convergence rate of the Euler-Maruyama scheme to density dependent SDEs driven by $α$-stable additive noise Open
In this paper, we establish the weak convergence rate of density-dependent stochastic differential equations with bounded drift driven by $α$-stable processes with $α\in(1,2)$. The well-posedness of these equations has been previously obta…
View article: Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations
Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations Open
Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence an…
View article: Propagation of chaos for moderately interacting particle systems related to singular kinetic Mckean-Vlasov SDEs
Propagation of chaos for moderately interacting particle systems related to singular kinetic Mckean-Vlasov SDEs Open
We study the propagation of chaos in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by alpha-stable processes. Diffusion parts include Brownian (alpha=2) and pure-jump…
View article: SDEs with supercritical distributional drifts
SDEs with supercritical distributional drifts Open
Let $d\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sqrt{2}{\rm d} W_t, $$ where $b$ belongs to the space ${\mathb…
SDE driven by cylindrical $α$-stable process with distributional drift Open
For $α\in (1,2)$, we study the following stochastic differential equation driven by a non-degenerate symmetric $α$-stable process in $\mathbb{R}^d$: \begin{align*} {\rm d} X_t=b(t,X_t){\mathord{\rm d}} t+σ(t,X_{t-}){\mathord{\rm d}} L_t^{(…
View article: Second order fractional mean-field SDEs with singular kernels and measure initial data
Second order fractional mean-field SDEs with singular kernels and measure initial data Open
In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton…
Strong and weak convergence for averaging principle of DDSDE with singular drift Open
In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove…
Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions Open
In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle tra…
Well-posedness of density dependent SDE driven by $α$-stable process with Hölder drifts Open
In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $α$-stable processes with $α\in(1,2)$. The existence part is based on Euler's approximation as \cite{HRZ20}, while, …
Singular kinetic equations and applications Open
In this paper we study singular kinetic equations on $\mathbb{R}^{2d}$ by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the globa…
Euler scheme for density dependent stochastic differential equations Open
In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on th…
Schauder's estimates for nonlocal equations with singular Lévy measures Open
In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(α)}_{κ,σ} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $α\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb R^d\to\mathbb R$ i…
View article: Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $α$-stable processes
Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $α$-stable processes Open
We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is …
View article: Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes
Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes Open
We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: \\[ \\mathrm{d} X_{t}=\\sigma (t, X_{t-})\\mathrm{d} Z_{t}+b (t, X_{t})\\mathrm{d} t,\\ \\ X_{0}=x\\in{\\mathbb {R}} …
Schauder's estimate for nonlocal kinetic equations and its applications Open
In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in $\mathbb R^{2d}$ with Hölder coe…
Hörmander's hypoelliptic theorem for nonlocal operators Open
In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect…