# Research Areas B1

# Probabilistic modeling and singular stochastic dynamics

## Research Area Leaders: Anton Bovier, Massimiliano Gubinelli

## PIs: Anton Bovier, Massimiliano Gubinelli, Sven Rady, Karl-Theodor Sturm

## Contributions by Sergio Albeverio, Alexey Bufetov, Francesco de Vecchi, Andreas Eberle, Patrik Ferrari, Lorens Imhof, Anna Kraut, Muhittin Mungan, Eveliina Peltola, Jonas Wahl

## Topics and goals

The understanding of large-scale observables of probabilistic models is a central objective of probability theory. Leveraging our expertise on exact computations, asymptotic analysis, potential theory, coupling methods, control theory, multiscale and infinite-dimensional analysis, we will advance the understanding of basic mechanisms that drive large-scale observables. These involve extremal events in highly correlated and high-dimensional random fields and their influence on complex and singular stochastic dynamics. The crucial task here is to identify universal asymptotic structures and to show convergence to these in a variety of particular situations, ranging from spatial branching processes, log-correlated Gaussian fields, and models of interface growth to singular stochastic partial differential equations and, on the applied side, to models of strategic acquisition and transmission of information and Markov Chain Monte Carlo methods.

## State of the art, our expertise

**Asymptotics in branching structures**. Bovier et al. determined the structure of extremal processes in correlated Gaussian environments in high-dimensional spaces and their multiscale structure. In particular, they solved the long-standing problem of determining the extremal process of branching Brownian motion, which is a prototype for a large class of so-called log-correlated fields that also includes the Gaussian free field [ABK13]. In another line of work, they derived rigorous scaling limits of interacting branching particle systems motivated by biological applications [BBC17]. These models are important for the IRUs D3–5. Interface models and Kardar–Parisi–Zhang (KPZ) universality. Universality of growing one-dimensional interfaces is a very active area of research, but many problems remain open, in particular concerning the joint distribution of fluctuations at different times. Some recent progress has been made by rigorous asymptotics and non-rigorous replica methods. Gubinelli et al. [GP17a, GP17b, DGP17] developed new stochastic-analysis tools to prove convergence of microscopic models to mesoscopic singular stochastic PDEs, proving in particular the universality of the largescale fluctuations of growth models with weak asymmetry. Ferrari et al. analyzed the large-time fluctuations for some stochastic growth models in the KPZ universality class, both exploiting integrable structures and developing more robust techniques. The analysis of two-dimensional interface growth is a completely open research direction. Here anisotropic scaling limits are expected to give rise to Gaussian free field behavior. Ferrari et al. [BCF17] obtained preliminary information about these scaling limits for a model which corresponds to the short-time dynamics of the KPZ equation.

**Non-reversible stochastic dynamics and MC methods**. The long-time behavior of non-reversible, degenerate, or time-inhomogeneous Markov processes is not sufficiently well understood. Compared to traditional approaches based on functional inequalities, coupling techniques provide a much more flexible tool for a systematic investigation of the corresponding stability problems. In this context, Eberle [Ebe16] developed new coupling approaches that have been applied to the kinetic Langevin equation.

**Stochastic dynamics and information flow**. A recent major research focus in theoretical economics has been on the dynamics of information acquisition and transmission in strategic contexts, in particular strategic experimentation. In this context, Rady has analyzed continuous-time experimentation games in which the players face two-armed bandit problems. The natural state variable for these problems is the common posterior belief about the bandits’ characteristics, and the analysis of Markovian equilibria requires a precise understanding of the agents’ control problems and of the stochastic differential equations induced by their strategies. Recent work on a restless bandit problem, for example, has led to Tanaka-like stochastic differential equations and belief dynamics with sticky boundary behavior. Imhof et al. [EFI16] developed a framework to analyze the speed of learning in stochastic game dynamics. The approach applies to models for which existing results on mean-field approximations are of limited use.

**Infinite-dimensional singular dynamics**. There has been substantial progress recently in the rigorous understanding of the small-scale structure of stochastic PDEs where nonlinear phenomena require renormalization. These problems are related to quantum field theory and to the wider topic of universality in probability. Gubinelli et al. constructed novel solution theories for such singular stochastic parabolic PDEs, introducing a paradifferential calculus applicable to a large class of equations which are not well-posed in usual functional spaces. They established the uniqueness of energy solutions for the KPZ equation [GP17b] and applied these tools to stochastic quantization and to universality for certain classes of random fields. In collaboration with Koch (RA C4) and Oh,Gubinelli started the investigation of the renormalization problem for singular stochastic hyperbolic PDEs [GKO17]. Sturm et al. [LS18] applied gradient flow techniques to diffusions on singular spaces and on path and configuration spaces.

## Research program

**Log-correlated processes, interfaces and beyond**. Log-correlated processes, such as branching Brownian motion and the two-dimensional Gaussian free field, have received massive attention over the last years. They lie at the borderline where stochastic dependence becomes relevant for the properties of extremes of stochastic processes. Processes with stronger correlations will exhibit even richer structures. Bovier plans to analyze these, starting with the case of variablespeed branching Brownian motion and the Gaussian free field with inhomogeneous variance. Interestingly, an important role is played here by the Ferrari–Spohn process [FS05] and the Airy equation. Technically, there are close relations to renormalization group methods that Müller employs in RA C1. The Gaussian free field is expected to appear also in the large-scale behavior of critical singular stochastic PDEs in two dimensions. Gubinelli and Ferrari will investigate this connection in the setting of the stochastic heat equation and the anisotropic KPZ equation.

**KPZ universality class and its dynamics**. Ferrari plans to study the universal scaling regime of time-time covariance of some processes in the KPZ universality class. In particular, he intends to prove convergence to the KPZ fixed point recently described in [CQR15] using robust probabilistic approaches which rely as little as possible on integrability. An example of this line of attack can be seen in the paper [CFS] on the limit distribution for KPZ growth models with random but non-stationary initial conditions. Another example is recent work by Ferrari–Occelli on the GOE Tracy–Widom universality for flat initial conditions. Further research will address the even more challenging universality problem for generic models in the KPZ class. Gubinelli and Ferrari will investigate the problem of describing the Markov process associated with the KPZ fixpoint [CQR15] via a suitable stochastic dynamics.

**Hamiltonian Monte Carlo methods (HMC)**. These are Monte Carlo Markov chain methods based on discretizations of the Hamiltonian flow combined with momentum refreshments. Eberle plans to extend the robust coupling approach developed in [Ebe16] to analyze HMC and derive quantitative bounds on the distance to the stationary distribution and error bounds for ergodic averages. The goal is to confirm rigorously the improved performance of HMC with respect to reversible dynamics, which is empirically observed in many situations, and to clarify the relations between HMC and recently developed methods such as Event-Chain Monte Carlo or the Bouncy Particle Sampler. The derivation of convergence rates that scale well in the dimension or are even dimension-free will require an extension of coupling methods to a setup with a multiscale structure. These improved methods are of interest also in the context of high-dimensional Langevin equations and mean-field interacting particle systems that are studied by Bovier.

**Stochastic dynamics on complex landscapes**. Sturm intends to push forward the stochastic calculus on manifolds and singular spaces with particular focus on time-dependent geometries. Rady will investigate general multi-agent bandit problems, removing restrictions on the correlation structure between the bandits’ characteristics, as well as allowing for characteristics that change stochastically over time. Imhof plans to extend the analysis of the speed of evolution to more complex games, such as games with iterated dominant equilibria, and to game dynamics in randomly fluctuating environments. In collaboration with Koch, Gubinelli will extend multiscale tools to the analysis of subcritical hyperbolic and dispersive singular SPDEs. He plans also to develop novel techniques in functional analysis and quantum mechanics, for example for the analysis of Hamiltonians and of Kolmogorov operators related to singular stochastic PDEs. Harmonic analysis including paraproducts and multilinear estimates are a major theme in these investigations, and there are close relations with the investigations in RA C4. Eberle and Gubinelli will analyze stochastic algorithms introduced in the physics literature to approximate Feynman amplitudes numerically via diffusions on Lefschetz thimbles. This project has connections with IRU D2 where such diffusions will be studied from the point of view of stochastic quantization.

## Summary

This research area aims to develop new insights into how correlations and rare events build up and propagate on nontrivial landscapes, and to apply this knowledge to a wide spectrum of major problems in probability theory which share common features such as strong correlations or complex dynamics. The analysis of these problems involves a broad variety of methods ranging from stochastic analysis, renormalization, spectral theory and asymptotic analysis to algebraic methods from representation theory. This links this RA methodologically to many other areas of the cluster, in particular RAs A2, C1, C4 and IRUs D2, D3–5.

## Bibliography

[ABK13] L.-P. Arguin, A. Bovier, and N. Kistler. The extremal process of branching Brownian motion. *Probab. Theory Related Fields, 157(3-4):535–574, 2013*.

[BBC17] M. Baar, A. Bovier, and N. Champagnat. From stochastic, individual-based models to the canonical equation of adaptive dynamics in one step.* Ann. Appl. Probab., 27(2):1093–1170, 2017*.

[BCF17] A. Borodin, I. Corwin, and P. L. Ferrari. Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes.* Probab. Theory Related Fields, 2017. DOI:10.1007/s00440-017- 0809-6*.

[CFS] S. Chhita, P. Ferrari, and H. Spohn. Limit distributions for KPZ growth models with spatially homogeneous random initial conditions. *Ann. Appl. Probab. to appear*.

[CQR15] I. Corwin, J. Quastel, and D. Remenik. Renormalization fixed point of the KPZ universality class. *J. Stat. Phys., 160(4):815–834, 2015*.

[DGP17] J. Diehl, M. Gubinelli, and N. Perkowski. The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions. *Commun. Math. Phys., 354(2):549–589, 2017*.

[Ebe16] A. Eberle. Reflection couplings and contraction rates for diffusions. *Probab. Theory Related Fields, 166(3-4):851–886, 2016*.

[EFI16] G. Ellison, D. Fudenberg, and L. Imhof. Fast convergence in evolutionary models: a Lyapunov approach.* J. Econom. Theory, 161:1–36, 2016*.

[FS05] P. L. Ferrari and H. Spohn. Constrained Brownian motion: fluctuations away from circular and parabolic barriers.** ***Ann. Probab., 33(4):1302–1325, 2005*.

[GKO17] M. Gubinelli, H. Koch, and T. Oh. Renormalization of the two-dimensional stochastic nonlinear wave equation. *Trans. Amer. Math. Soc., 2017. DOI:10.1090/tran/7452*.

[GP17a] M. Gubinelli and N. Perkowski. *KPZ Reloaded. Commun. Math. Phys., 349(1):165–269, 2017*.

[GP17b] M. Gubinelli and N. Perkowski. Energy solutions of KPZ are unique. *J. Amer. Math. Soc., 2017. DOI:10.1090/jams/889*.

[LS18] J. Lierl and K.-T. Sturm. Neumann heat flow and gradient flow for the entropy on non-convex domains.* Calc. Var. Partial Differential Equations, 57, 2018. DOI:10.1007/s00526-017-1292- 8.*