Special Session 19: Topics on singular stochastic equations

Wave propagation for 1-dimensional reaction-diffusion equations with nonzero random drift

Hui He Beijing Normal University Peoples Rep of China Co-Author(s):    Dihang Guan, Wenqing Hu, Jiaojiao Yang

Abstract: We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the asymptotic wave fronts propagating to the positive and negative directions are both pushed in the negative direction, leading to the possibility that both wave fronts propagate toward negative infinity. Our proof is based on the Large Deviations Principle for diffusion processes in random environments, as well as an analysis of the Feynman-Kac formula. Such probabilistic arguments also reveal the underlying physical mechanism of the wave fronts formation: the drift acts as an external field that shifts the (quenched) free-energy reference level without altering the intrinsic fluctuation structure of the system.

Fluctuation of heat kernels on random graphs.

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

Abstract: Let $\mathcal{G}$ be a graph equipped with random conductance. We give sufficient conditions for the Markov chain on $\mathcal{G}$ to exhibit large scale fluctuations in its on-diagonal heat kernel. The conditions also imply non-tightness of the height process w.r.t. the quenched measure, while the height process is tight w.r.t. the annealed measure. As an application of the general theory, we prove large scale fluctuations for the simple random walk on $1$-dimensional critical long-range percolation. This is based on a joint work with Zherui Fan and Takashi Kumagai.

Regularization by noise phenomena in stochastic nonlinear PDEs with modulated dispersion

Guopeng Li Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we first consider the Korteweg-de Vries equation (KdV) with a modulated dispersion. We observe the regularization-by-noise effects resulting from this modulation: we establish well-posedness of the modulated KdV on the circle in the regime where the unmodulated KdV is ill-posed. In particular, we show that the modulated KdV on the circle is locally well-posed in Sobolev spaces of arbitrarily low regularity, provided that the modulation is sufficiently irregular. Then I will present more recent results on the stochastic modulated KdV on the circle with multiplicative fractional-in-time noise, where we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that the stochastic modulated KdV is pathwise locally well-posed in Sobolev spaces of arbitrarily low regularity, provided that the modulation is sufficiently irregular. If time permits, I will mention the results when the multiplicative noise is white-in-time. Based on joint works with Khalil Chouk (formerly Edinburgh), Massimiliano Gubinelli (Oxford), Jiawei Li & Tadahiro Oh (Edinburgh), and Andreia Chapouto (CNRS & Monash).

Transportation-cost information inequalities for McKean-Vlasov equations and mean field interacting particle system

Wei Liu Wuhan University Peoples Rep of China Co-Author(s):    Liming Wu

Abstract: TDA

Schrodinger-Follmer Sampler

Yanyan Liu Wuhan University Peoples Rep of China Co-Author(s):    Jian Huang , Yuling Jiao, Lican Kang , Xu Liao, Jin Liu,Yanyan Liu

Abstract: Sampling from probability distributions is an important problem in statistics and machine learning, specially in Bayesian inference when integration with respect to posterior distribution is intractable and sampling from the posterior is the only viable option for inference. In this paper, we propose Schrodinger-Follmer sampler (SFS), a novel approach to sampling from possibly unnormalized distributions. The proposed SFS is based on the Schrodinger-Follmer diffusion process on the unit interval with a time-dependent drift term, which transports the degenerate distribution at time zero to the target distribution at time one. Compared with the existing Markov chain Monte Carlo samplers that require ergodicity, SFS does not need to have the property of ergodicity. Computationally, SFS can be easily implemented using the Euler-Maruyama discretization. In theoretical analysis, we establish nonasymptotic error bounds for the sampling distribution of SFS in the Wasserstein distance under reasonable conditions. We conduct numerical experiments to evaluate the performance of SFS and demonstrate that it is able to generate samples with better quality than several existing methods.

Quantitative Asymptotics for Time-Inhomogeneous L\`evy-Driven SDEs with Asymptotically Vanishing Drifts

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

Abstract: We are concerned with a class of multi-dimensional time-inhomogeneous stochastic differential equations (SDEs) on $\R^d$ driven by pure-jump L\`evy processes, where the drift coefficient $b(t,x)$ satisfies $\lim_{t\to \infty}b(t,x) =0$ for every $x\in \R^d$. Distinguishing three regimes according to the index $\alpha$ of large jumps for the driven L\`evy noise, we investigate the quantitative asymptotic behavior of the corresponding rescaled processes. More precisely, for $\alpha\in (0,2)$, we prove that the rescaled processes, governed by time-inhomogeneous SDEs driven by additive processes, converge with respect to suitably chosen Wasserstein distances to time-homogeneous SDEs driven by symmetric $\alpha$-stable processes. Notably, the driving noise in the limiting SDEs depends only on large jumps of the underlying additive processes. A phase transition occurs for all values of the index $\alpha$, and a diffusive phenomenon arises whenever $\alpha\ge2$.

On reflected SDEs with superlinear coefficients

Jing Wu Sun Yat-sen University Peoples Rep of China Co-Author(s):

Abstract: In this talk we are concerned with some problems of reflected SDEs when both the drift and diffusion coefficients are of superlinear growth, including the the well-posedness and numerical approximations.

Stochastic Differential Equations with Local Growth Singular Drifts

Wenjie Ye School of Mathematics and Statistics at Fujian Normal University Peoples Rep of China Co-Author(s):

Abstract: In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift $b$ and the weak gradient of Sobolev diffusion $\sigma$ are supposed to satisfy $||b(x){1}_{|x|\le R}||_{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})$ and $||\nabla \sigma(x)1_{|x|\le R} ||_{p_1} \le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$ respectively. The main tools for these results are the decomposition of global two-point motions, Krylov`s estimate, Khasminskii's estimate, Zvonkin`s transformation and the characterization for Sobolev differentiability of random fields.