arXiv subjects

Panpan Ren

Publications and source records attributed to Panpan Ren.

At least 19 recordsLinked to original sources

arXiv2026

Bismut Formula for Intrinsic Derivative of DDSDEs with Singular Interactions

In recent years, remarkable progress has been made for Distribution dependent stochastic equations (DDSDEs) with singular interactions, existing results include wellposedness, propagation of chaos, entropy cost inequality and ergodicity. As a continuation to the existing study, in this paper we establish Bismut type formulas for the intrinsic derivative of DDSDEs with singular interactions, which extends the existing formula established for the case with Lion's differentiable drifts.

math.PR↗

arXiv2026

Extrinsic derivatives for SDEs and SPDEs with distribution dependent noise

The Bismut formula is a crucial tool characterizing regularities of stochastic systems, and has been extensively studied for various models. However it is not yet available for SDEs with distribution dependent noise. In this paper, we first establish a Bismut type formula for the extrinsic derivative of McKean-Vlasov SDEs driven by distribution dependent noise, then make an extension to a class of distribution dependent SPDEs.

math.PR↗

arXiv2026

McKean-Vlasov SDEs with Local Distributional Interactions: Well-Posedness and Entropy-Cost Estimates

We study McKean-Vlasov SDEs with interaction kernels in $\tt W^{-\dd,k},$ the local negative Sobolev space on $\R^d$ with indexes $\dd \in [0,\infty)$ and $k\in [1,\infty].$ We derive the local well-posedness for any singular indexes $(\dd,k)\in [0,\infty)\times [1,\infty],$ and prove the global well-posedness for any initial distributions provided $\dd+\ff d k<1$. Moreover, the relative entropy and the $\|\cdot\|_{\dd,k*}$-distance induced by $ \tt W^{-\dd,k}$ are estimated for the time-marginal distributions of solutions by using the Wasserstein distance of initial distributions, which describe the regularity of the solution in initial distribution. In particular, the main results apply to Nemytskii-type SDEs which depend on higher order derivatives of the density functions, as well as McKean-Vlasov SDEs with interactions more singular than Riesz kernels.

math.PR↗

arXiv2025

Non-asymptotic convergence bounds of modified EM schemes for non-dissipative SDEs

In this paper, we address the issue on non-asymptotic convergence bounds of Euler-type schemes associated with non-dissipative SDEs. On the one hand, for non-degenerate SDEs with super-linear drifts, we propose a novel modified Euler scheme and establish the corresponding non-asymptotic convergence bound under the multiplicative type quasi-Wasserstein distance by the aid of the asymptotic reflection by coupling. As a direct application of the theory derived, we explore the non-asymptotic convergence bound of the modified tamed/truncated Euler scheme and, as a byproduct, furnish the associated non-asymptotic convergence rate under the $L^1$-Wasserstein distance although the dissipativity at infinity is not in force. On the other hand, we tackle the non-asymptotic convergence analysis of the Euler scheme corresponding to a kind of degenerate SDEs, where the underdamped Langevin SDE is a typical candidate. To handle such setting, we also appeal to a carefully tailored coupling approach, where the ingredient in the coupling construction lies in that a proper metric and a suitable substitute in the cut-off function and the reflection matrix need to be chosen appropriately. In addition, as a consequent application, the non-asymptotic convergence bound and the $L^1$-Wasserstein convergence rate are revealed for the kinetic Langevin sampler.

math.PR↗

arXiv2025

Exponential Ergodicity in Relative Entropy and $L^2$-Wasserstein Distance for non-equilibrium partially dissipative Kinetic SDEs

In this paper, we derive exponential ergodicity in relative entropy for general kinetic SDEs under a partially dissipative condition. It covers non-equilibrium situations where the forces are not of gradient type and the invariant measure does not have an explicit density, extending previous results set in the equilibrium case. The key argument is to establish the hypercontractivity of the associated semigroup, which follows from its hyperboundedness and its $L^2$-exponential ergodicity. Moreover, we obtain exponential ergodicity in the $L^2$-Wasserstein distance by combining Talagrand's inequality with a log-Harnack inequality. These results are further extended to the McKean-Vlasov setting and to the associated mean-field interacting particle systems, with convergence rates that are uniform in the number of particles in the latter case, under small nonlinear perturbations.

math.PR↗

arXiv2025

Stochastic intrinsic gradient flows on the Wasserstein space

We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(\rho d x)=\int_{\mathbb R^d}F(x,\rho(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $\Lambda$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $\Lambda$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(\mu)$ is defined for $\Lambda$-a.e. $\mu$ and that the Gaussian-based reference measure $\Lambda$ can be chosen in such way that the distorted process ${(\mu_t)}_{t\geq 0}$ is a martingale solution for the equation $d\mu_t=-DW_F(\mu_t) d t+d R_t$, $t\geq 0$.

math.PR↗

arXiv2025

Entropy-Cost Inequalities for McKean-Vlasov SDEs with Singular Interactions

For a class of McKean-Vlasov stochastic differential equations with singular interactions, which include the Coulomb/Riesz/Biot-Savart kernels as typical examples (Examples 2.1 and 2.2), we derive the well-posedness and regularity estimates by establishing the entropy-cost inequality. To measure the singularity of interactions, we introduce a new probability distance induced by local integrable functions, and estimate this distance for the time-marginal laws of solutions by using the Wasserstein distance of initial distributions. A key point of the study is to characterize the path space of time-marginal distributions for the solutions, by using local hyperbound estimates on diffusion semigroups.

math.PR↗

arXiv2025

Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs

The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the $L^1$-Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in $L^2$-Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion.

math.PR↗

arXiv2024

Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures

Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $\sigma$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper bound conditions given by the uniform $(L^1+L^\infty)$-norm of the extrinsic derivative. As applications, we construct a class of general type Markov processes on $E$ via quasi-regular Dirichlet forms containing the diffusion, jump and killing terms. Moreover, stochastic extrinsic derivative flows on $E$ are studied by using quasi-regular Dirichlet forms, which in particular provide martingale solutions to SDEs on these two spaces, with drifts given by the extrinsic derivative of entropy functionals.

math.PR↗

arXiv2024

Efficient subgroup testing in change-plane models

Considered here is a hypothesis test for the coefficients in the change-plane regression models to detect the existence of a change plane. The test that is considered is from the class of test problems in which some parameters are not identifiable under the null hypothesis. The classic exponential average tests do not work well in practice. To overcome this drawback, a novel test statistic is proposed by taking the weighted average of the squared score test statistic (WAST) over the grouping parameter's space, which has a closed form from the perspective of the conjugate priors when an appropriate weight is chosen. The WAST significantly improves the power in practice, particularly in cases where the number of the grouping variables is large. The asymptotic distributions of the WAST are derived under the null and alternative hypotheses. The approximation of critical value by the bootstrap method is investigated, which is theoretically guaranteed. Furthermore, the proposed test method is naturally extended to the generalized estimating equation (GEE) framework, as well as multiple change planes that can test if there are three or more subgroups. The WAST performs well in simulation studies, and its performance is further validated by applying it to real datasets.

math.ST↗

arXiv2024

Diffusion Processes on $p$-Wasserstein Space over Banach Space

To study diffusion processes on the p-Wasserstein space $\mathscr P_p$ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathscr P_p,\Lambda)$ for a reference probability $\Lambda$ on $\mathscr P_p$. It is formulated in terms of an upper bound condition with the uniform norm of the intrinsic derivative. We find a versatile class of quasi-regular local Dirichlet forms on $\mathscr P_p$ by using images of Dirichlet forms on the tangent space $L^p(X\to X,\mu_0)$ at a reference point $\mu_0\in\mathscr P_p$. The Ornstein--Uhlenbeck type Dirichlet form and process on $\mathscr P_2$ are an important example in this class. We derive an $L^2$-estimate for the corresponding heat kernel and an integration by parts formula for the invariant measure.

math.PR↗

arXiv2024

Extrinsic Derivative Formula for Distribution Dependent SDEs

A Bismut type formula is established for the extrinsic derivative of distribution dependent SDEs. The main result is illustrated by nondegenerate DDSDEs with space time singular drift, as well as degenerate DDSDEs with weakly monotone coefficients.

math.PR↗

arXiv2023

Entropy Estimate for Degenerate SDEs with Applications to Nonlinear Kinetic Fokker-Planck Equations

The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker Planck equations.

math.PR↗

arXiv2023

Singular Degenerate SDEs: Well-Posedness and Exponential Ergodicity

The well-posedness and exponential ergodicity are proved for stochastic Hamiltonian systems containing a singular drift term which is locally integrable in the component with noise. As an application, the well-posedness and uniform exponential ergodicity are derived for a class of singular degenerated McKean-Vlasov SDEs.

math.PR↗

arXiv2023

Probability Distance Estimates Between Diffusion Processes and Applications to Singular McKean-Vlasov SDEs

The $L^k$-Wasserstein distance $\mathbb{W}_k (k\ge 1)$ and the probability distance $\mathbb{W}_\psi$ induced by a concave function $\psi$, are estimated between different diffusion processes with singular coefficients. As applications, the well-posedness, probability distance estimates and the log-Harnack inequality are derived for McKean-Vlasov SDEs with multiplicative distribution dependent noise, where the coefficients are singular in time-space variables and $(\mathbb{W}_k+\mathbb{W}_\psi)$-Lipschitz continuous in the distribution variable. This improves existing results derived in the literature under the $\mathbb{W}_k$-Lipschitz or derivative conditions in the distribution variable.

math.PR↗

arXiv2023

Bi-Coupling Method and Applications

By developing a new technique called the bi-coupling argument, we estimate the relative entropy between different diffusion processes in terms of the distances of initial distributions and drift-diffusion coefficients. As an application, the entropy-cost inequality is established for McKean-Vlasov SDEs with spatial-distribution dependent noise, which is open for a long time and has potential applications in optimal transport, information theory and mean field particle systems.

math.PR↗

arXiv2022

Ornstein-Uhlenbeck Type Processes on Wasserstein Space

Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having finite second moment, and consider the Riemannian structure on $\mathcal P_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck (OU) type Dirichlet form on $\mathcal P_2$ whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is $L^2$-compact. Perturbations of the OU Dirichlet form are also studied.

math.PR↗

arXiv2021

Singular McKean-Vlasov SDEs: Well-Posedness, Regularities and Wangs Harnack Inequality

The well-posedness and regularity estimates in initial distributions are derived for singular McKean-Vlasov SDEs, where the drift contains a locally standard integrable term and a superlinear term in the spatial variable, and is Lipchitz continuous in the distribution variable with respect to a weighted variation distance. When the superlinear term is strengthened to be Lipschitz continuous, Wangs Harnack inequality is established. These results are new also for the classical Ito SDEs where the coefficients are distribution independent.

math.PR↗