Special Session 88: Recent developments in stochastic analysis and related topics

Random Attractors for McKean-Vlasov S(P)DEs

Mengyu Cheng Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: Random attractor is an important concept of understanding the long-tine behavior of random systems induced by stochastic evolution equations. In this talk, we will consider the existence of random attractors for McKean-Vlasov S(P)DEs on the product space $\mathbb{R}^n \times \mathcal{P}(\mathbb{R}^n)$.

Symmetry and functional inequalities for stable Levy-type operators

Lu-Jing Huang Fujian Normal University Peoples Rep of China Co-Author(s):    Tao Wang

Abstract: In this talk, we establish the sufficient and necessary conditions for the symmetry of the following stable L\`evy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$ \mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\mathrm{d}}{\mathrm{d} x}, $$ where $a,b$ are the continuous positive and differentiable functions, respectively. We then study the criteria for functional inequalities, such as logarithmic Sobolev inequalities, Nash inequalities and super-Poincar\`e inequalities under the assumption of symmetry. Our approach involves the Orlicz space theory and the estimates of the Green functions. This is based on a joint work with Tao Wang.

Moments and tails of the Gaussian multiplicative chaos

Yichao Huang Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: The Seiberg bounds form a set of necessary and sufficient conditions under which correlations functions in Liouville conformal field theory are well-defined. Since the probabilistic construction of Liouville correlations functions by David, Kupiainen, Rhodes and Vargas, a probabilistic version of the Seiberg bounds can be obtained via the theory of Gaussian Multiplicative Chaos. We will give a brief review on this construction, and then explain its boundary version, where a new class of Gaussian Multiplicative Chaos emerges naturally. We will discuss finer estimates on the right tail of the Gaussian multiplicative chaos measures if time permits.

Long time behaviors of mean field interacting particle systems and McKean-Vlasov equations

Wei Liu Wuhan University Peoples Rep of China Co-Author(s):    Arnaud Guillin, Liming Wu and Chaoen Zhang.

Abstract: In this talk, we will present our recent studies about the long time behaviors of mean-field interacting particle systems and the McKean-Vlasov equation, by using two different methods: coupling method and functional inequalities.

Kernel Variable Importance Measure with Applications

Yanyan Liu Wuhan University Peoples Rep of China Co-Author(s):    Huang bingyao, Peng Liuhua, Liuyanyan

Abstract: This paper proposes a kernel variable importance measure (KvIM) based on the maximum mean discrepancy (MMD). KvIM can effectively measure the importance of an individual dimension in contributing to the distributional difference by constructing weighted MMD and applying perturbations to evaluate MMD changes through assigned weights. It has advantages such as non-parametric, model-free, comprehensive consideration of the dependencies among dimensions, and suitability for high-dimensional data. Furthermore, the consistency of empirical KvIM under general conditions and its theoretical properties in high-dimensional settings were studied. In addition, we also apply KvIM to classification problems and streaming datasets and propose a KvIM-enhanced classification approach and renewable empirical KvIM accordingly. Numerous numerical studies illustrate that the proposed method is feasible and effective.

Existence and uniqueness results for strongly degenerate McKean-Vlasov equations with rough coefficients

Andrea Pascucci University of Bologna Italy Co-Author(s):    Alessio Rondelli, Alexander Yu Veretennikov

Abstract: We present existence results for weak solutions to a broad class of degenerate McKean-Vlasov equations with rough coefficients, expanding upon and refining the techniques recently introduced by the third author. Under certain structural conditions, we also establish results concerning both weak and strong well-posedness.

Asymptotic behavior of multi-scale stochastic systems

Xiaobin Sun Jiangsu Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we explore the asymptotic behavior of various kinds of multi-scale stochastic (partial) differential equations. Specifically, we prove the strong and weak averaging principles, the central limit theorem, as well as diffusion approximation within the context of these multi-scale stochastic systems. Additionally, we present the key ideas in the proofs of the optimal convergence rates.

Wright-Fisher stochastic heat equations with irregular drifts

Zhenyao Sun Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Clayton Barnes and Leonid Mytnik

Abstract: Consider $[0,1]$-valued random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W \] where $b(1)\leq 0\leq b(0)$ and $\dot W$ is a space-time white noise. In this talk, we present the weak existence and uniqueness of the above equation for a class of drifts $b(u)$ that may be irregular at the points where the noise is degenerate, that is, at $u=0$ or $u=1$. This class of drifts includes non-Lipschitz drifts like $b(u) = u^q(1-u)$ for every $q\in (0,1)$, and some discontinuous drifts like $b(u) = \mathbf 1_{(0,1]}(u)-u$. This demonstrates a regularization effect of the multiplicative space-time white noise without assuming the standard assumption that the noise coefficient is Lipschitz and non-degenerate. The method we apply is a further development of a moment duality technique that uses branching-coalescing Brownian motions as the dual particle system. To handle an irregular drift in the above equation, particles in the dual system are allowed to have a number of offspring with infinite expectation, even an infinite number of offspring with positive probability. We show that, even though the branching mechanism with infinite number of offspring causes explosions in finite time, immediately after each explosion the total population comes down from infinity due to the coalescing mechanism. Our results on this dual particle system are of independent interest.

Exponential contractivity of modified Euler schemes for SDEs with super-linearity

Jian Wang Fujian Normal University Peoples Rep of China Co-Author(s):

Abstract: As a well-known fact, the classical Euler scheme works merely for SDEs with coefficients of linear growth. In this paper, we propose a novel variant of Euler schemes, which is applicable to SDEs with super-linear drifts and encompasses the classical Euler scheme, the tamed Euler scheme and the truncated Euler scheme, as three typical candidates. On the one hand, by exploiting an approach based on the refined basic coupling, we show that the proposed Euler recursion is exponentially contractive under a mixed probability distance (i.e., the total variation distance plus the $L^1$-Wasserstein distance). On the other hand, by invoking a trick on the coupling by reflection, we, in particular, demonstrate that the tamed Euler scheme is exponentially contractive under the $L^1$-Wasserstein distance. In addition, as an important application, a quantitative $L^1$-Wasserstein error bound between the exact invariant probability measure and the numerical counterpart concerning the SDEs under consideration and the associated tamed Euler scheme, respectively. Last but not least, we want to stress that the theory derived in the present work is available for SDEs, where, most importantly, the drifts involved are allowed to be highly nonlinear.

The first eigenvalue of one-dimensional elliptic operators with killing

Yingchao Xie Jiangsu Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we investigate the first eigenvalue for one-dimensional elliptic operators with killing. Two-sided approximation procedures and basic estimates of the first eigenvalue are given in both the half line and the whole line. The proofs are based on the h-transform, Chen`s dual variational formulas and the split technique. In particular, a few examples are presented to illustrate the power of our results.

$W_d$-convergence rate of EMs for invariant measures of supercritical stable SDEs

Xiaolong Zhang Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Peng Chen, Lihu Xu, Xiaolong Zhang and Xicheng Zhang

Abstract: Through establishing the regularity estimates for nonlocal Poisson/Stein equations under certain H\older and dissipativity conditions on the coefficients, we derive the $W_d$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by $\alpha$-stable noises with $\alpha \in (\frac{1}{2}, 2)$, where $W_d$ denotes the Wasserstein metric corresponding to the distance $d(x,y)=|x-y|^\vartheta \wedge 1$, with $\vartheta \in (0,1] \cap (0,\alpha)$.

Singular McKean-Vlasov equations

Xicheng Zhang Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Z. Hao, S. Menozzi, F. Jabir, M. R\ockner

Abstract: We establish the local and global well-posedness of weak and strong solutions for second-order fractional mean-field SDEs. These equations involve singular or distribution interaction kernels and measure initial values, with examples including Newton or Coulomb potentials, Riesz potentials, Biot-Savart law, among others. Our analysis relies on the theory of anisotropic Besov spaces. Building on the well-posedness results of the McKean-Vlasov equations, we investigate the propagation of chaos for moderately interacting particle systems with singular kernels and derive quantitative convergence rates.