# PDE and Randomness

#### 01/09/2021 - 10/09/2021

Organisers:

Professor Hendrik Weber
Dr Andris Gerasimovics

#### EVENT OVERVIEW

PDE and Randomness symposium, 1st - 10th of September 2021, University of Bath, UK

summary

The last few years have witnessed a number of exciting developments at the interface of Probability theory and the theory of PDE. Among these are

• Singular stochastic PDEs: starting with the discovery of regularity structures / paracontrolled distributions a systematic theory of white noise driven SPDEs from mathematical Physics was developed.

• New results on the large-scale behaviour of discrete models from Statistical Mechanics have emerged.

• Stochastic Homogenization: the classical theory of stochastic homogenization has been completely rewritten and quantified.

• Dispersive PDEs with random data have been shown to display a much better solution theory than naive regularity considerations would suggest.

These a priori quite different topics share a number of common features, among them the question of renormalization / removal of infinite terms as well as the prominent use of tools from regularity theory / harmonic analysis.

The aim of this symposium is to bring together some of the leading protagonists in these discoveries, to showcase their results, similarities and differences in between them and to prepare the grounds for future developments.

Due to the spread of the Delta variant in UK both school and the workshop will be held online.

## Speaker List

#### The Coleman correspondence at the free fermion point.

Abstract: Two-dimensional statistical and quantum field theories are
special in many ways. One striking instance of this is the equivalence
of certain bosonic and fermionic fields, known as bosonization. I will
first review the idea of this correspondence in the explicit instance of
the massless Gaussian free field and massless Euclidean Dirac
fermions. I will then present a result that extends this
correspondence to the massless' sine-Gordon field on $\R^2$ at
$\beta=4\pi$ and massive Dirac fermions. This is an instance of
Sidney Coleman's prediction that the massless' sine-Gordon model and the massive Thirring model are equivalent. We use this correspondence to show that correlations of the massless'
sine-Gordon model decay exponentially for $\beta=4\pi$. This is joint
work with C. Webb.

See profile

#### Fluctuating Boltzmann equation and large deviations for a hard sphere gas.

Abstract: In this talk, we will consider the hard sphere dynamics and
we will show the convergence of the fluctuation field to the fluctuating
Boltzmann equation. We will also show that the occurence of atypical
evolutions can be quantified by a large deviation principle. Even
though the microscopic dynamics is deterministic, the corrections to
the Boltzmann equation, in the Boltzmann-Grad limit, behave as if the
particle system was driven by a (stochastic) Kac's process.

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#### Edwards-Wilkinson fluctuations for the Anisotropic KPZ in the weak coupling regime.

Abstract: In this talk, we present recent results on an anisotropic
variant of the Kardar-Parisi-Zhang equation, the Anisotropic KPZ
equation (AKPZ), in the critical spatial dimension d=2. This is a
singular SPDE which is conjectured to capture the behaviour of the
fluctuations of a large family of random surface growth phenomena
but whose analysis falls outside of the scope not only of classical
stochastic calculus but also of the theory of Regularity Structures and
paracontrolled calculus. We first consider a regularised version of the
AKPZ equation which preserves the invariant measure and prove the
conjecture made in [Cannizzaro, Erhard, Toninelli, "The AKPZ
equation at stationarity: logarithmic superdiffusivity"], i.e. we show
that, at large scales, the correlation length grows like t1/2 (log t)1/4
up to lower order correction. Second, we prove that in the so-called
weak coupling regime, i.e. the equation regularised at scale N and the
coefficient of the nonlinearity tuned down by a factor (log N)-1/2, the
AKPZ equation converges to a linear stochastic heat equation with
renormalised coefficients.
This is joint work with D. Erhard and F. Toninelli.

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#### Stochastic Quantization of Yang Mills.

Abstract: I will discuss the construction of stochastic dynamics and
corresponding state spaces coming from the stochastic quantisation
of 2 and 3 dimensional non-abelian gauge theories.
This is joint work with Ilya Chevyrev, Martin Hairer, and Hao Shen.

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#### Path functions and homogenisation.

Abstract: In this talk, I will introduce a geometric way to solve
differential equations with jumps based on the notion of path
functions. This method generalises Marcus solutions to stochastic
differential equations and has a natural extension to rough path
space. As an application, I will show how homogenisation theorems
for fast-slow dynamical systems can be stated and proven using path
functions.

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#### Well-posedness of the Dean-Kawasaki equation with correlated noise.

Abstract: The Dean-Kawasaki equation, and more generally certain
singular stochastic PDEs with conservative space-time white noise,

arise formally in fluctuating hydrodynamics and macroscopic
fluctuation theory to describe far from equilibrium behavior in
physical systems such as the fluctuations of an interacting particle
system, like the zero range process, about its hydrodynamic limit. The
treatment of these SPDEs presents a significant mathematical
challenge, due both to their supercriticality and to their degenerate /
singular coefficients. In this talk, which is based on joint work with
Benjamin Gess, I will discuss a well-posedness theory for such
equations with correlated noise. The introduction of smooth noise is
justified by the fact that discrete microscopic systems often have a
natural de-correlation scale such as the grid-size and by the fact that,
along appropriate scaling limits, the solutions accurately describe the
particle system in terms of a hydrodynamic limit / law of large
numbers, central limit fluctuations, and large deviations. The
methods treat general nonlinearities that are only locally 1/2-Hölder
continuous, and solve several open problems including the
well-posedness of the Dean-Kawasaki and nonlinear
Dawson-Watanabe equations with correlated noise.

See profile

#### Optimal-order estimates in stochastic homogenization: Beyond linear equations and smooth data.

Abstract: In recent years, a well-rounded theory of stochastic
homogenization of linear elliptic PDEs has been developed. In
particular, optimal homogenization error estimates have been
obtained by Gloria and Otto as well as by Armstrong, Kuusi, and
Mourrat.
In this talk, we consider two settings that go beyond the case of linear
elliptic PDEs with smooth data: In the first part of the talk, we
establish optimal-order convergence rates for nonlinear random
elliptic PDEs with monotone nonlinearity. Our rate of convergence
improves substantially upon earlier results by Armstrong, Smart, and
Mourrat; however, we make no attept to show optimal stochastic
integrability. In the second part of the talk, we consider the
fluctuating Dirichlet problem for random linear elliptic PDEs. In
particular, we prove decay estimates for boundary layers and give an
application of our estimates in the analysis of the method of
representative volumes.
Joint works with Stefan Neukamm respectively with Peter Bella, Marc Josien, and Claudia Raithel.

See profile

#### Stochastic thin film equations.

Abstract: The stochastic thin film equation arises in the evolution of
thin films of fluids. It captures the evolution of the height of the fluid in terms of a degenerate, quasilinear stochastic PDE of forth order.
This makes the problem of the existence and uniqueness of solutions
challenging. In this talk we first recall the motivation, derivation and
basic facts about the stochastic thin film equation. We then focus on
the construction of (weak) solutions in the case of spatially correlated
noise for quadratic and cubic mobility.

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#### From random dynamics to fractional PDEs with several boundary conditions.

Abstract: In this seminar I will describe the derivation of certain laws
that rule the space-time evolution of the conserved quantities of
stochastic processes. The random dynamics conserves a quantity (as
the total mass) that has a non-trivial evolution in space and time. The
goal is to describe the connection between the macroscopic
(continuous) equations and the microscopic (discrete) system of
random particles. The former can be either PDEs or stochastic PDEs
depending on whether one is looking at the law of large numbers or
the central limit theorem scaling; while the latter is a collection of
particles that move randomly according to a transition probability
and rings of Poisson processes. I will focus on a model for which we
can obtain a collection of (fractional) reaction-diffusion equations
given in terms of the regional fractional Laplacian with different types
of boundary conditions.

See profile

#### Long time behaviors of KPZ on torus.

Abstract: I will present a joint work with Tomasz Komorowski on the
long time behaviors of the KPZ equation on torus.

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#### The Allen-Cahn equation with generic initial condition.

Abstract: We consider the Allen-Cahn equation with a rapidly mixing
Gaussian field as initial condition. We show that provided that the
amplitude of the initial condition is not too large, the equation
generates fronts described by nodal sets of the Bargmann-Fock
Gaussian field, which then evolve according to mean curvature flow.
Joint work with K. Lê and T. Rosati.

See profile

#### The SHE as the scaling limit of the voter model with diffusion.

Abstract: In dimensions up to three, we derive the stochastic heat
equation as the scaling limit of a voter model on which agents move
following a stirring (exclusion) dynamics. In contrast with previous
results in the literature, our voter model has local interactions. Joint
work with Claudio Landim (Rio de Janeiro).

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#### Interface fluctuations for the Widom-Rowlinson model.

Abstract: This is a rare case (one of two) of a model with interacting
particles on continuum for which the ''liquid/vapour'' phase
transition has been rigorously proven.
I will discuss microscopic fluctuations of the coexistence interface for
the two-dimensional Widom-Rowlinson model at low temperatures.
When applied
to the evaluation of the low temperature asymptotics of the surface
tension and the mean crossover time from the metastable state for the
dynamical version of the model,
it leads to a correction term of the order $\beta^{1/3}$. The
Widom-Rowlinson model will be introduced and the main ideas
behind the emergence of these correction terms will be explained.
Based on joint works with Frank den Hollander, Sabine Jansen, and
Elena Pulvirenti.

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#### Segal's axioms and bootstrap for Liouville theory.

Abstract: Graeme Segal presented in the 80's an axiomatic framework
for 2d quantum field theory where QFT correlations on a compact
surface are viewed as compositions of operators assigned to a
decomposition of the surface to discs, annuli and pairs of pants. I will

explain how these axioms can be given a probabilistic realisation for
Liouville conformal field theory and how this leads to an expression
for the correlation functions in terms of simple data: the structure
constants and spectrum of the theory. Joint work with C. Guillarmou,
R. Rhodes and V. Vargas.

See profile

#### Propagation of randomness, Gibbs measures and random tensors for NLS.

Abstract: We review recent work, joint with Yu Deng and Haitian Yue,
about the Gibbs measure for the periodic 2D NLS and 3D Hartree NLS
as well as the theory of random tensors, a powerful
new framework which allows us to unravel the propagation of
randomness under the nonlinear flow beyond the linear evolution of
random data. This enables us in particular, to show the existence and
uniqueness of solutions to the periodic NLS in an optimal range
relative to what we define as the probabilistic scaling.

See profile

#### Stochastic quantization of the $\Phi^3_3$-model.

Abstract: We study the construction of the $\Phi^3_3$-measure and
complete the program on the (non-)construction of the focusing Gibbs
measures, initiated by Lebowitz, Rose, and Speer (1988). This
problem turns out to be critical, exhibiting a phase transition:
normalizability in the weakly nonlinear regime and non-convergence
of the truncated $\Phi^3_3$-measures in the strongly nonlinear
regime.
We also go over the dynamical problem for the canonical stochastic
quantization of the $\Phi^3_3$-measure, namely, the
three-dimensional stochastic damped nonlinear wave equation with a
(= the hyperbolic $\Phi^3_3$-model). We first discuss briefly the
paracontrolled approach for local well-posedness. In the globalization part, we introduce a new,
conceptually simple and straightforward approach, where we directly work with the (truncated) Gibbs measure, using the variational
formula and ideas from theory of optimal transport.
This is a joint work with Mamoru Okamoto (Osaka University) and
Leonardo Tolomeo (University of Bonn).

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#### Multi-index based regularity structures: Stochastic estimate of the model.

Abstract: Starting point is a regularity structure with an index set not
given by trees, but by partial derivatives w.~r.~t.~the Taylor
coefficients of the nonlinear function defining the singular SPDE,
which here is a quasi-linear parabolic equation driven by white noise.
Renormalization can naturally be accommodated in this approach,
which is top-down and analytic, instead of bottom-up and
combinatorial for the tree-based approach. A benefit are the a priori
fewer counter terms for this more greedy index set.
In this talk, we give optimal stochastic estimates on the model.
For this, we estimate the Malliavin/noise derivative of the model,
which we treat as a modelled distribution. The modeledness of degree
$2-$ allows for a regular reconstruction. The spectral gap assumption
(which also allows for a non-Gaussian noise) then provides control of
the variance of the model itself.
This approach is well suited for the BPHZ choice of renormalization,
which provides control of the expectation of the model.
The argument proceeds by induction in an automated way.
Our analysis is guided by exact scaling; we thus work in the whole
space, formulate the estimates in an annealed (as opposed to
quenched) way, and embed white noise in a marginally rougher family
of noises. The gap between $C^\alpha$-based Schauder theory for

modelled distributions and the $L^2$-based Malliavin calculus is
bridged by weighted $L^2$-norms.
This is joint work with P.~Linares, M.~Tempelmayr, and P.~Tsatsoulis.

See profile

#### Large N via stochastic quantization.

Abstract: We consider a coupled system of N interacting Phi4
equations, known as the O(N) linear sigma model. In 2d we show
convergence to a mean-field singular SPDE, also proved to be globally
well-posed. Moreover, we show tightness of the invariant measures in
the large N limit, and for large mass or small coupling they converge
to the (massive) Gaussian free field, in both 2d and 3d. We also
consider fluctuations and tightness for certain O(N) invariant
observables in 2d, and derive exact formulas for their limiting
correlations. Based on joint work with Scott Smith, Rongchan Zhu and
Xiangchan Zhu.

See profile

#### Phase transitions of the focusing Φ^p_1 measures.

Abstract: We study the behaviour of the focusing Φ^p_1 measures on
the one-dimensional torus, initiated by Lebowitz, Rose, and Speer
(1988). Because of the focusing nature of the measure, it is necessary
to introduce a mass cutoff K, so that the measure is formally given by
the expression
Z^{-1} χ(\int |φ|^2 \le K) \exp( β/p \int |φ|^p - 1/2 \int |φ|^2 -
\int |φ’|^2) d φ.
We exhibit the following phase transitions:
- When p = 6, the measure is normalisable for K \le K_0( β), and it is
not normalisable for K > K_0(β). The endpoint result K = K_0 answers
an open question of the original paper by Lebowitz, Rose, and Speer.
This is a joint work with T. Oh and P. Sosoe.
- When p < 6, we consider the limit case of a big torus with weak potential, i.e. K = LK_0, β=L^{-γ}, where L is the size of the torus. We show that the limiting behaviour depends on the value of γIn particular, when γ > γ_0(p), we show that the Φ^p_1 converges to the
Ornstein–Uhlenbeck measure on the real line. When γ < γ_0(p), the measure converges to the \delta measure concentrated in 0 (and it actually concentrates around a single soliton, appropriately rescaled). This extends a result of Rider (2002). This is a joint work with H. Weber. See profile

#### Diffusion in the curl of the 2-dimensional Gaussian Free Field.

Abstract: I will discuss the large time behaviour of a Brownian
diffusion in two dimensions, whose drift is divergence-free, ergodic
and given by the curl of the 2-dimensional Gaussian Free Field.
Together with G. Cannizzaro and L. Haundschmid, we prove the
conjecture by B. Toth and B. Valko that the mean square displacement
is of order $t \sqrt{\log t}$. The same type of superdiffusive
behaviour has been predicted to occur for a wide variety of
(self)-interacting diffusions in dimension d = 2: the diffusion of a
tracer particle in a fluid, self-repelling polymers and random walks,
Brownian particles in divergence-free random environments, and,
more recently, the 2-dimensional critical Anisotropic KPZ equation.
To the best of our authors’ knowledge, ours is the first instance in
which $\sqrt{\log t}$ superdiffusion is rigorously established in this
universality class.

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#### Quenched CLT for random walk in divergence-free random drift field.

Abstract: I prove the quenched version of the central limit theorem for
the displacement of a random
walk in doubly stochastic random environment, under the
$H_{−1}$-condition, with slightly stronger ($L^{2+\epsilon}$ rather
than $L^2$) integrability condition on the stream tensor. The proof
relies on an improved sector condition within
Kipnis-Varadhan-theory and on an extension of Nash’s moment
bound. Talk based on Ann. Probab. v 45, pp 4307-4347 (2017) and
Ann. Probab. v 46, pp 3558–3577 (2018).

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#### Stochastic estimates of the model for quasi-linear SPDEs via spectral gap

Abstract: We consider the renormalised model for singular parabolic
quasi-linear SPDEs. Instead of a tree-based approach, the model is
indexed by partial derivatives with respect to the Taylor coefficients of
the non-linearity, allowing to organise elements of the same noise
homogeneity in linear combinations.
In this talk we will explain an alternative approach to obtain
stochastic estimates of the model. The main ingredients are a spectral
gap assumption on the underlying noise, which also allows for
non-Gaussian drivers, and an estimate on the "Malliavin" derivative of
the model, which satisfies a controlled path condition'' of degree
$2-$ (in contrast to the model itself which is only H\"older
continuous). These two ingredients provide an estimate on the
variance of the model which is compensated by the BPHZ
renormalisation condition, since the latter yields an estimate
on the expectation of the model. In order to accommodate a
H\"older-based approach on the level of the `Malliavin'' derivative of
the model, we consider weighted $L^2$-norms which allow us to
merit from the gain of regularity in Sobolev scales.
This is a joint work with Pablo Linares, Felix Otto and Markus
Tempelmayr. The talk will be related to the one by
Felix Otto, focusing on different aspects of the project.

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#### Transport of Gaussian measures under the flow of Hamiltonian PDE's.

Abstract: The transport of gaussian measures under transformations
in an infinite dimensional space is a delicate issue as shown by the
classical Cameron-Martin theorem (1944). We will show that a
natural class of gaussian measures, invariant under the flow of linear
Hamiltonian PDE's, remain quasi-invariant under nonlinear
(Hamiltonian) perturbations. We will also identify the
Radon-Nikodym derivative by using some hidden cancellations. We
will make an attempt to keep the presentation at a nontechnical level
by probably sacrificing a little rigor.

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#### Singular HJB equations with applications to KPZ on the real line.

Abstract: This talk is devoted to studying Hamilton-Jacobi-Bellman
equations with distribution-valued coefficients, which are not
well-defined in the classical sense and are understood by using the
paracontrolled distribution method introduced by Gubinelli, Imkeller
and Perkowski. By a new characterization of weighted H\"older
spaces and Zvonkin's transformation we prove some new a priori
estimates, and therefore establish the global well-posedness for
singular HJB equations. As applications, we obtain global
well-posedness in polynomial weighted H\"older spaces for KPZ type
equations on the real line, as well as modified KPZ equations for
which the Cole-Hopf transformation is not applicable.
The talk is based on joint work with Xicheng Zhang and Xiangchan
Zhu.

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#### Singular kinetic equations and applications.

Abstract: In this talk we study singular kinetic equations on
$\mathbb{R}^{2d}$ by the paracontrolled distribution method
introduced by Gubinelli, Imkeller and Perkowski.
We first develop paracontrolled calculus in the kinetic setting, and use
it to establish the global well-posedness for the linear singular kinetic
equations under the assumptions that the products of singular terms
are well-defined. We also demostrate how the required products can
be defined in the case that singular term is a Gaussian random forcing
by probabilistic calculation. Interestingly, although the terms in the
zeroth order Wiener chaos are not zero, they converge in suitable
weighted Besov spaces and no renormalization is required. As
applications, the global well-posedness for a nonlinear kinetic
equation with singular coefficients is obtained by the entropy method.
Moreover, we also solve the martingale problem for nonlinear kinetic
distribution dependent stochastic differential equations with singular
drifts. This talk is based on joint work with Zimo Hao, Xicheng Zhang
and Rongchan Zhu.

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#### The critical 2d stochastic heat flow.

Abstract: Abstract: We consider directed polymers in random
environment in the critical dimension two, focusing on the
intermediate disorder regime when the model undergoes a phase
transition. We prove that, at the critical temperature the diffusively
rescaled random field of partition functions has a unique scaling limit
; a universal process of random measures on R^2 with logarithmic
correlations, which we call the Critical 2d Stochastic Heat Flow. This is
the natural candidate for the long sought solution of the critical 2d
Stochastic Heat Equation with multiplicative space-time white noise.
Based on a joint work with Francesco Caravenna and Rongfeng Sun.

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#### Title: Mean-field spin glasses: Beyond Parisi's formula?

Abstract: Spin glasses are models of statistical mechanics encoding
disordered interactions between many simple units. One of the
fundamental quantities of interest is the free energy of the model, in
the limit when the number of units tends to infinity. For a restricted
class of models, this limit was predicted by Parisi, and later rigorously
proved by Guerra and Talagrand. I will first show how to rephrase this
result using an infinite-dimensional Hamilton-Jacobi equation. I will
then present partial results suggesting that this new point of view
may allow to understand limit free energies for a larger class of
models, focusing in particular on the case in which the units are
organized over two layers, and only interact across layers.