Interacting Systems and SPDEs

A conference about stochastic partial differential equations, and their relationship to interacting particle systems.

June 13^{th}-15^{th}, University Of Sheffield

** Invited Speakers **

**Siva Athreya** (Indian Statistical Institute Bangalore)

** Subdiffusivity of a random walk among a Poisson system of moving traps on Z **
* We consider a random walk among a Poisson system of moving traps on Z. We study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. This is joint work with Alex Drewitz and Rongfeng Sun. *

** Márton Balázs** (University of Bristol)

** How to initialise a second class particle? **
*
Hydrodynamics of asymmetric particle systems like exclusion is well known to
develop rarefaction fans from decreasing step initial data.
Second class particles are probabilistic objects that come from coupling two
instances of the models. They are very useful and their behaviour is highly
nontrivial.
The beautiful paper of P. A. Ferrari and C. Kipnis connects the above two: they
proved that the second class particle of simple exclusion chooses a uniform
random velocity when started in a rarefaction fan. The extremely elegant proof
is based, among other ideas, on the fact that increasing the mean of a
Bernoulli distribution can be done by adding or not adding 1 to the random
variable.
For a long time simple exclusion was the only model with an established large
scale behaviour of the second class particle in its rarefaction fan. I will
explain how this is done in the Ferrari-Kipnis paper, then show how to do this
for other models that allow more than one particles per site. The main issue is
that most families of distributions are not as nice as Bernoulli in terms of
increasing their parameter by just adding or not adding 1. To overcome this we
use a signed, non-probabilistic coupling measure that nevertheless points out a
canonical initial probability distribution for the second class particle. We
can then use this initial distribution to greatly generalize the Ferrari-Kipnis
argument. I will conclude with an example where the second class particle
velocity has a mixed discrete and continuous distribution.
Joint work with Attila László Nagy.
*

** Nicolas Dirr** (Cardiff University)

** Gradient flows and interacting particle systems **
* Nonlinear diffusion, but in a certain sense also mean curvature flow, are examples of gradient flows which arise as hydrodynamic limit of interacting particle systems. We will explain recent attempts to connect the macroscopic gradient flow structure, given by a (Lyapunov) functional and a metric, directly to a microscopic interacting particle system. *

** Benjamin Gess** (Max Planck Institute & Universität Bielefeld)

** Well-posedness by noise for scalar conservation laws **
* In certain cases of (linear) partial differential equations random perturbations have been observed to cause regularizing effects, in some cases even producing the uniqueness of solutions. In view of the long-standing open problems of uniqueness of solutions for certain PDE arising in fluid dynamics such results are of particular interest. In this talk we will extend some known results concerning the well-posedness by noise for linear transport equations to the nonlinear case.*

** Elena Issoglio** (University of Leeds)

** FBSDEs with distributional coefficients **
* In this talk I will present some recent results about systems of forward-backward stochastic differential equations (FBSDEs) where some of the coefficients are Schwartz distributions, in particular they are elements of a fractional Sobolev space of negative order (with regularity > -1/2). A notion of virtual solution is introduced in order to make sense of the singular integrals that appear in the FBSDE. One of the key tools we use is a theorem of existence, uniqueness and regularity of the solution of a PDEs with distributional coefficients - a singular PDE that plays the role of the Kolmogorov backward equation. In this setting, we investigate existence and uniqueness of a virtual solution for the singular FBSDE and we also show the validity of the so-called non-linear Feynman-Kac formula.
(This talk is based on a joint work with Shuai Jing (ArXiv:1605.01558).) *

** Cyril Labbé** (Université Paris-Dauphine)

** Localisation of the continuous Anderson Hamiltonian in 1d **
* We consider the Anderson Hamiltonian with a white noise potential on a segment of length L and endowed with Dirichlet boundary conditions. We show that, as L goes to infinity, the (appropriately rescaled and shifted) eigenvalues converge to a Poisson point process on R with an explicit intensity, and that the eigenfunctions converge to Dirac masses located at iid uniform points. Furthermore, we show that the shape of each eigenfunction near its maximum is given by an explicit, deterministic function which does not depend on the corresponding eigenvalue. This is a joint work with Laure Dumaz (Paris-Dauphine).*

** Carl Mueller** (University of Rochester)

** An SDE related to the wave equation with noise**
* This is joint work with A. Gomez, J.J. Lee, E. Neuman, and M. Salins. There are still many challenging questions about the qualitative behavior SPDE. Although we now have good information about the stochastic heat equation with multiplicative noise, much less is known about the stochastic wave equation. For example, basic properies of uniqueness and blowup are still unknown in the latter case. As a preparation for the study of such questions, we look at an SDE which is a toy model or zero-dimensional case for the stochastic wave equation. For this SDE, we can resolve the questions about uniqueness and blowup. Our equation is actually a system of SDE, and it bears repeating that the basic tool for proving uniqueness for SDE is due to Yamada and Watanabe. However, their technique is largely restricted to one dimension.*

** Marcel Ortgiese** (University of Bath)

** Interfaces in the symbiotic branching model **
* The symbiotic branching model describes a spatial population consisting of two types in terms of a coupled system of SPDEs. One particularly important special case is Kimura's stepping stone model in evolutionary biology. Our main focus is a description of the interfaces between the types in the large scale limit of the system. As a new tool we will introduce a moment duality, which also holds for the limiting model. This also has implications for a classification of entrance laws of annihilating Brownian motions.*

** Nicolas Perkowski** (Humboldt-Universität zu Berlin)

** A weak universality result for the parabolic Anderson model **
* We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin. *

** Nadia Sidorova** (University College London)

** Delocalising the parabolic Anderson model **
* The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, where the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions. This is a joint work with Stephen Muirhead and Richard Pymar. *

** Jon Warren** (University of Warwick)

** The stochastic heat equation and total positivity **
* Abstract in pdf form *

** Hendrik Weber** (University of Warwick)

** The dynamic Phi^4_3 model comes down from infinity **
* We prove an a priori bound for the dynamic Phi^4_3 model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean Phi^4_3 field theory on finite volume.
Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.*

** Nikolaos Zygouras** (University of Warwick)

** Marginally relevant disordered systems below and at the critical temperature. **
* In joints works with Francesco Caravenna and Rongfeng Sun we es-tablished that so-called “disorder relevace” is intimately related to the existence of non trivial continuum limits when the strength of the disorder is suitably scaled to zero. At the critical dimension disorder becomes marginal and an interesting structure emerges. Among others, the class of “marginally relevant disordered systems” includes the two dimensional directed polymer model and stochastic heat equation with multiplicative noise. The continuum limit of these marginal models exhibits a phase transition at a certain critical temperature. We will describe the continuum limit and the structure that underlies marginal polymer-type models below the crit- ical temperature, as well as recent progress (in joint work with Caravenna and Sun in understanding the limiting behaviour at the critical temperature. *

** Contributed Talks **

** Peter Embacher** (Cardiff University)

**Computing Transport Coefficients from Particle Models out of Equilibrium**
* A new method is proposed to numerically extract the diffusivity from stochastic particle systems which are macroscopically described by a possibly non-linear diffusion equation. The method allows for the system to be out of equilibrium and is based on the fact, that large particle systems formally obey a stochastic partial differential equation of gradient-flow type satisfying a fluctuation-dissipation relation. The method is showcased over two zero-range-processes, a symmetric simple-exclusion-process and a Kawasaki-type dynamic model. *

** Carina Geldhauser** (Università di Pisa)

** The scaling limit of a particle system with long-range interaction **
* We describe the macroscopic behaviour of a coupled bistable particle system where a large number of particles interact with each other. Due to the properties of the driving force and the noise, the scaling limit does not lead in general to a well-posed equation. We develop conditions on the interaction strength between the particles to ensure existence of solutions to the limiting stochastic PDE. As a corollary we obtain that the metastable behaviour of the system under investigation is described by the Stochastic Allen Cahn equation, which has been analyzed by Barret, Bovier and Meleard, Barret and by Berglund and Gentz. This is joint work with Anton Bovier. *

** Aleksander Klimek** (University of Oxford)

** Selection in a fluctuating environment **
* We consider both spatial and non-spatial \Lambda-Fleming-Viot process describing frequencies of genetic types in a population living in \R^d with two possible genetic types and natural (fecundity) selection favouring one of the types. The favourability depends on the state of the environmental variable, which evolves in time and space, with some correlation. We find a unique diffusion approximation to the model via separation of timescales argument. The limiting equation is an analogue of the Fisher-KPP equation with coloured noise. Supervised by Alison Etheridge (and joint Niloy Biswas) *

** Nikolaos Kolliopoulos** (University of Oxford)

** Stochastic evolution equations for large portfolios of stochastic volatility models **
* We consider a large market model of defaultable assets in which the asset price processes are modelled as Heston stochastic volatility models with default upon hitting a lower boundary. We assume that both the asset prices and their volatilities are correlated through systemic Brownian motions. We are interested in the loss process that arises in this setting and consider the large portfolio limit of the empirical measure for this system. This limit evolves as a measure valued process and we show that it will have a density that satisfies a stochastic partial differential equation of filtering type with Dirichlet boundary conditions. We are able to show uniqueness and regularity of that solution. We employ Malliavin calculus to establish the existence of a regular density for the volatility component and an approximation by models of piecewise constant volatilities combined with a kernel smoothing technique to obtain the desired results for the full two-dimensional problem. *

** Eyal Neuman** (Imperial College London)

** Pathwise uniqueness for the stochastic heat equation with Hölder continuous drift and noise coefficients **
* We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is H ̈older continuous of index γ >3/4, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are H ̈older continuous.*

** Scott Smith** (Max Planck Institute)

** The Boltzmann equation with stochastic kinetic transport **
* The study of stochastic fluid dynamics has seen many developments in recent years. A number of authors have considered the existence problem for Navier-Stokes and Euler equations with random perturbations. IN this talk, we consider a related SPDE, but from the mesoscopic viewpoint of the Boltzmann equation. Namely, we allow for a background, or environmental noise to act on particles in between collisions. We will discuss recent results on the existence of renormalized (in the sense of Di-Perna/Lions) martingale (in the sense of Stroock/Varadhan) solutions to this SPDE. The approach is based on a detailed analysis of weak martingale solutions to linear stochastic transport equations driven by a ranndom source; including criteria for renormalization, velocity averaging, and weak compactness of the soluiton set. This is a joint work with Sam Punshon-Smith (University of Maryland). *

** Umesh Umesh** (King's College London)

**Weak solution of stochastic Cauchy equation driven by cylindrical Levy process**
* A cylindrical Levy process is a generalisation a cylindrical Weiner process. The theory of integration with respect to a cylindrical Levy process was recently introduced and can be used to study the stochastic Cauchy problem. In this talk, we prove the existence and uniqueness of weak solution of the stochastic Cauchy problem driven by a cylindrical Levy process. For this purpose, we prove first a stochastic version of the Fubini theorem. (Joint work with Markus Riedle.)*