presentation

2026

1. Talk

   Elliptic Phi_4^4 Model with Boundary Blow-up: Triviality and Renormalization

   PGR Research Day, Durham University, Durham, U.K., May 2026

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   Phi^4 model refers to a class of stochastic PDEs related to construction of Phi^4 Euclidean quantum field theory. They involve cubic non-linearity, driven by white noise and usually called stochastic quantization equations. There has been extensive research on parabolic and hyperbolic Phi_2^4 models, where the subscript "2" means 2-dimensional domains. In this talk, I will present my recent work for elliptic models on the 4-dimensional hypercube, imposed with an infinity boundary condition. Without renormalization, the regularised equations will converge to something deterministic as white noise recovers, which is the triviality property shared with parabolic and hyperbolic settings. The limit is a PDE if noise intensity is weak, and is zero if noise strength is strong. To find a meaningful stochastic object solving elliptic models, we will prove existence of renormalized solutions. This requires renormalizing suitably the non-linear term in equations, and yields non-trivial limits as noise regularisation is removed.

2025

1. Talk

   Why People Get Lazy in the Long Run? A Perspective from Ergodic Control

   PGR Research Day, Durham University, Durham, U.K., May 2025

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   Being lazy seems inevitable as humans, and we are not tireless robots after all. In this talk, I will defend ourselves by considering an ergodic control problem. It can be proved that an optimal control turns out of a lazy type. This provides mathematical evidence for our daily behaviour.

2024

1. Talk

   Local Dynamics of McKean-Vlasov Processes in Double-Well Landscapes

   47th Research Students Conference in Probability and Statistics, University of Exeter, Exeter, U.K., Aug 2024

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   For a McKean-Vlasov process confined by a double-well potential, there is a phase transition on the number of invariant measures. When the noise is strong, the process admits a unique invariant measure. When the noise is weak, it has exactly three invariant measures, which coincides with the number of equilibria of its ODE counterpart. One may wonder if the McKean-Vlasov process behaves like the ODE solution under the smallness of noise. In this talk, I will give an affirmative answer to this question. To be precise, consider the semigroup, associated to the process, in the 2-Wasserstein space. I will show that the central invariant measure is unstable, there are connecting orbits from the central one to the other two invariant measures, and these peripheral two invariant measures attract their own open neighbourhoods respectively. The method is based on the theory of monotone dynamical systems, and can apply to asymmetric or multi-well potentials.

2. Talk

   Why Is It Impossible to Stand a Pencil with Tip Down?

   SIDur PGR Symposium, Durham University, Durham, U.K., Oct 2024

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3. Talk

   Gradient Flows on Wasserstein Spaces

   Gandalf Talk, Durham University, Durham, U.K., Nov 2024

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   This talk serves as a gentle introduction to gradient flows on Wasserstein spaces. We will recall the definition of Wasserstein distances as optimal cost of transportation. And the transporting dynamics, or say the Benamou-Brenier formula, give rise to ”Riemannian structures” on Wasserstein spaces. Then we can define their tangent spaces and formulate gradient flows on them. There are also some equivalent objects from PDE and stochastic analysis, corresponding to Wasserstein gradient flows. For example, choosing the entropy functional, the gradient flow turns out to be the heat equation, and gives the distribution of a Brownian motion. The contraction rate of this entropy gradient flow can be related to the curvature lower bound of the base space. Other examples and some generalization will be discussed.

2023

1. Seminar

   Two-Dimensional Stochastic Navier-Stokes Equations

   Research Group Seminar, 2023

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