Program

Timetable

Please notice that on Thursday the conference will take place in Aula Savi, at the Orto Botanico, while on Friday and Saturday will be at the Mathematics Department. Under Travel and accomodation you'll find two maps with the places marked.

Talks

Titles and abstracts

Zdzislaw Brzezniak: Stochastic nonlinear Schrödinger equation on 3d compact manifolds

I will speak about two recent works with Fabian Hornung and Lutz Weis (Karlsruhe). In particular, I will speak about the existence of a global solution to the stochastic nonlinear Schrödinger equation (SNSL) on a 3-dimensional manifold with multiplicative Gaussian (and jump) noise. I will also speak about the uniqueness for such equations in case of Gaussian bilinear noise the proof of which is based on novel Strichartz estimates and Littlewood-Paley and decomposition in time. I will conclude by mentioning the Large Deviations Principle for SNSL.

Michele Coghi: Rough nonlocal diffusions

We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.

Hans Crauel: Minimal Random Attractors

In the theory of random dynamical systems it is common to investigate random attractors which attract "large sets", as for instance bounded sets, deterministic or also random. In this situation the random attractor, once it exists, is unique. Furthermore, it is determined uniquely by attracting deterministic compact sets already. Attractors for "small sets", for instance finite ones, are not unique in general. For deterministic dynamical systems it is not difficult to establish existence of a unique minimal attractor for any family of sets, provided an attractor for this family of sets exists. The topic of this lecture is a corresponding result for random attractors. Here one has to take into account that there are three different kinds of random attractors, namely pullback attractors, forward attractors, and weak attractors. The main result is the existence of a unique minimal pullback attractor and of a unique minimal weak attractor, which in general do not have to coincide -- the minimal weak attractor may be strictly smaller than the minimal pullback attractor. This applies to completely arbitrary families of sets to be attracted, random or deterministic, provided that a pullback or weak, resp., attractor exists. What forward attractors are concerned one gets a negative result. There are examples of random dynamical systems which do have forward attractors, however none of them is minimal.

Joint work with Michael Scheutzow, Berlin

Giuseppe Da Prato: Existence and uniqueness results for continuity equations in Hilbert spaces

Let H be a Hilbert space, F:[0,T]×H→ H a Borel vector field and 𝜁 a Borel probability measure on H. We are concerned with the following continuity equation,

(1) ∫_0^T ∫_H [D_tu+⟨D_x u ,F⟩ ] d𝜈_t dt=- ∫_H u(0,∙) d𝜁, ∀ u ∈ 𝓔,

where the unknown 𝜈 =(𝜈_t)_{t∈[0,T]} is a probability kernel such that 𝜈_0=𝜁. Moreover, D_x represents the gradient operator and u belongs to a suitable space 𝓔 of regular vector fields depending on a finite number of variables.

It is well known that problem (1) in general admits several solutions even when H is finite dimensional. So, it is natural to look for well posedness of (1) within the special class of measures (𝜈_t)_{t∈[0,T]} which are absolutely continuous with respect to a given reference measure 𝛾. In this case, denoting by 𝜌 (t,∙) the density of 𝜈_t with respect to 𝛾, 𝜈_t(dx)=𝜌(t,x) 𝛾 (dx), t ∈ [0,T], equation (1) becomes

(2) ∫_0^T ∫_H [D_tu + ⟨D_x u ,F⟩ ] 𝜌 d𝛾 dt =- ∫_H u(0,∙)𝜌_0 d𝛾 , ∀ u ∈ 𝓔.

Here 𝜌_0:=𝜌 (0,∙) is given and 𝜌 (t,∙), t ∈ [0,T] is the unknown.

In this talk we present some new existence and uniqueness results following the forthcoming paper: Giuseppe Da Prato, Franco Flandoli and Michal Röckner, Absolutely continuous solutions for continuity equations in Hilbert spaces, J. Math. Pure Appl. (to appear)

Arnaud Debussche: From random Boltzmann to the stochastic Navier-Stokes equations through hypo-coercivity estimate

Franco Flandoli: Numerical experiments with Kolmogorov equations in high dimensions

After a brief introduction to some aspects of mathematical research on Climate Change, the problem of a direct (non Monte Carlo) simulation of Kolmogorov equations in high dimension is addressed, giving the theoretical details of a scheme based on Gaussian analysis and showing the performances on a first series of numerical tests. A large number of open problems arise, which will be outlined in the last part of the talk. It is a joint work with Dejun Luo and Cristiano Ricci.

Tadahisa Funaki: Three different time stages in stochastic mass-conserving Allen-Cahn equation

Recently, F-Yokoyama (AP 2019) studied the sharp interface limit for stochastically perturbed mass-conserving Allen-Cahn equation. The reaction term is bistable and satisfies the balance condition. The initial value was assumed to be close to one of the two stable values, and stochastic volume-conserving motion by mean curvature was derived in the limit. For general initial values, before reaching this stage, the dynamics undergoes two different time stages in much shorter time span. The first is an ODE stage and then the second is Huygens stage.

This is ongoing project with D. Antonopoulou, G. Karali and S. Yokoyama.

Benjamin Gess: Optimal regularity for the porous medium equation

We prove optimal regularity for solutions to porous media equations in Sobolev spaces in time and space, based on velocity averaging techniques. In particular, the obtained regularity is consistent with the optimal regularity in the linear limit. This improves previous results by Tadmor and Tao [Tadmor, Tao; CPAM, 2007].

Francesco Grotto: Poissonian Solutions of Damped Stochastic 2D Euler equation

We consider a generalisation of the Euler Point Vortices system, in which vortex intensities are exponentially damped, and a Poisson point process acts as noise introducing new vortices at random times. The stationary solution converges under Central Limit scaling to a stationary solution of damped stochastic 2D Euler equation with additive cylindrical noise. The interest for both the Poissonian and Gaussian models is motivated as an attempt towards a rigorous description of experimental behaviours in 2D turbulence.

Massimiliano Gubinelli: The generators of some singular SPDEs

In recent years solution theories for singular SPDEs like the KPZ equation or the $\Phi^4_3$ equation have been developed using ideas from rough path theory, essentially from a pathwise point of view. In this talk I will introduce a martingale problem formulation of some of these singular SPDEs like for example the Burger’s equation and show that it provides an alternative solution theory with some advantages and some disadvantages with respect to the pathwise method. To achieve this goal one has to develop a detailed analysis of the formal generator of these equations, a problem which has been open for a long time.

Joint work with Nicolas Perkowski.

Martin Hairer: SPDEs in one dimension

Marta Leocata: A global existence result for a quadratic system of stochastic reaction diffusion equations

We study the existence, and the eventual regularity, of solutions to systems of stochastic reaction diffusion equations under some specific structure hypotheses (which imply preservation of positivity and control of an entropy in particular). Joint work with Julien Vovelle.

Dejun Luo: Some scaling limits of the 2D Euler equation with transport noises

We consider the vorticity form of 2D Euler equation on the torus perturbed by transport type noises, with a suitable scaling of the noises. In the framework of stationary solutions (i.e. the solutions have white noise as marginal law), we show that the solutions of the Euler equations converge weakly to the unique stationary solution of the 2D Navier-Stokes equation driven by space-time white noise. In the framework of L^2regular solutions, we show that the weak solutions converge weakly to the unique solution of the deterministic 2D Navier-Stokes equation.

Mario Maurelli: Existence of nonnegative vortex sheets for 2D stochastic Euler equations

In his 1991 paper, J.-M. Delort proved existence of solutions to the 2D Euler equations with H^{-1}-valued nonnegative vorticity; this includes the case of initial nonnegative vorticity concentrated on a line (vortex sheet). Here we prove the analogue result for the stochastic case, with transport noise on the vorticity. Namely, we consider 2D stochastic Euler equations in vorticity form:

∂_t ξ + u · ∇ξ + ∑_k σ_k · ∇ξ ◦ dW_k = 0, ξ = const + curl u,

where σ_k are given (divergence-free) regular vector fields and W_k are independent Brownian motions. Our main result is existence of a weak (in the probabilistic sense) H^{−1}-valued nonnegative solution ξ.

Francesco Morandin: Turbulence, shell models and critical exponents for dissipation

Shell models of turbulence are nonlinear dynamical systems inspired by fluid dynamics. They are idealized and simplified, but tailored to exhibit the same energy cascade behaviour of three dimensional Euler and Navier-Stokes equations. A typical feature of these models is in fact anomalous dissipation of energy, which in finite time "escapes" to infinity, yielding a blow-up and instantaneous loss in regularity. A dissipative term corresponding to viscosity can recover regularity, for some of these models, but in total generality one needs hyper-dissipation, with an exponent larger than the physical one. Recent results hint that in the more refined framework of tree (hierarchical) models the required exponent may be actually lower.

Enrico Priola: Schauder estimates for drifted fractional operators in the supercritical case

Abstract: We consider a non-local operator L_𝛼 which is the sum of a fractional Laplacian 𝛥^{𝛼/2} , 𝛼 ∈ (0,1), plus a first order term which is measurable in the time variable and locally 𝛽-Hölder continuous in the space variables. Importantly, the fractional Laplacian 𝛥^{𝛼/2} does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition 𝛼+𝛽 >1. We also discuss the extension of Schauder estimates to more general non-local operators like the relativistic 𝛼-stable operator. We can also consider the singular cylindrical operator, i.e., when 𝛥^{𝛼/2} is replaced by a sum of one dimensional fractional Laplacians ∑_{k=1}^d (∂_{x_k x_k}^2 )^{𝛼/2} assuming 𝛼 ∈ (1/2,1). Applications to uniqueness for singular SDEs will be also presented.

This is a joint work with Paul-Eric Chaudru de Raynal and S. Menozzi.

Cristiano Ricci: The Navier-Stokes-Vlasov-Fokker-Planck system as a scaling limit of particles in a fluid

The PDEs system Navier-Stokes-Vlasov-Fokker-Planck (NSVFP) is a model describing particles in a fluid, where the interaction particles-fluid is described by a drag force called Stokes drag force. In this talk I will present a particle system interacting with a fluid that converges in a suitable probabilistic sense to the (NSVFP) system in the two dimensional case. Moreover some ideas related to the three dimensional case will be given. The talk is based on a joint work with F. Flandoli and M. Leocata.

Michael Röckner: A natural extension of Markov processes and applications to singular SDEs

We develop a general method for extending Markov processes to a larger state space such that the added points form a polar set. The so obtained extension is an improvement on the standard trivial extension in which case the process is made stuck in the added points, and it renders a new technique of constructing extended solutions to S(P)DEs from all starting points, in such a way that they are solutions at least after any strictly positive time. Concretely, we adopt this strategy to study SDEs with singular coefficients on an infinite dimensional state space (e.g. SPDEs of evolutionary type), for which one often encounters the situation where not every point in the space is allowed as an initial condition. The same can happen when constructing solutions of martingale problems or Markov processes from (generalized) Dirichlet forms, to which our new technique also applies.

Joint work with Lucian Beznea and Iulian Cîmpean.

Francesco Russo: Recent developments in stochastic calculus via regularizations with jumps and applications to BSDEs

The aim of this talk consists in mentioning recent developments about stochastic calculus via regularizations. F. Flandoli and other coauthors have significantly contributed to this topic and to various applications. We recall that a weak Dirichlet process X with respect to a given underlying filtration is the sum of a local martingale and a process A such that [A,N] = 0 for every continuous local martingale. We introduce the notion of special weak Dirichlet process; whenever such a process is a semimartingale, then it is a special semimartingale. We will provide conditions on a function u: [0,T] × ℝ^d →ℝ and on an adapted cadlag process S such that u(t,S_t) is a special weak Dirichlet process. Two applications will be discussed.

1. The existence of a solution to a (strong) solution of a BSDEs with distributional driver, with underlying Brownian filtration (with Elena Issoglio, Leeds).
2. Consider the case a BSDE driven by a random measure: a solution is a triplet (Y,Z,K) where K is a random field. The function u(s,x):= Y^{s,x}_s is deterministic. If u has some minimal regulartiy, the calculus will allow to link Z, K to u (with Elena Bandini, Milano Bicocca).

Giovanni Zanco: Spatial interaction in mean-field models with discontinuous coefficients

We will consider stochastic models for particles whose interaction depends on two variables (one typically being their position in space and the other a quantity that depends on the problem at hand) in a discontinuous way. Our main example will be a system of neurons that follow an integrate-and-fire dynamics and interact via their action potential and their position. As the number of particles goes to infinity, the behaviour of the system is described by a Fokker-Planck PDE, for which we will prove existence and uniqueness of solutions showing directly the convergence of the discrete model to the continuous one. We will also comment on the relation between such PDE and the associated McKean-Vlasov SDE, which shows some differences with the standard case. Finally we will show how the techniques employed here can be used to investigate mean-field games models with spatial interaction.

Posters

Titles and abstracts

Nicolò Cangiotti: Notes on the Ogawa integrability and a condition for convergence in the multidimensional case

The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated by exploiting Ramer’s functional, proving that it cannot hold in general without the introduction of a “renormalization term”. Explicit examples are provided.

Kistosil Fahim: Rough Paths Perturbation Of The Equation of Semilinear Volterra Equations

In the classical theory of thermodynamics, thermal signals propagate with infinite speed, local actions and cumulative behaviour are neglected; the history, even the very recent history is not taken into account. A way to introduce a memory is first to introduce a so called memory function. By this memory function β the history will be taken into account by taking a averaging the past with β and leads to a so-called Volterra equation. The theory of rough path introduced by Terry Lyons in his seminal work as an extension of the classical theory of controlled differential equations. In this work we show existence and uniqueness of mild solution for an infinitesimal semilinear Volterra equations driven by a rough path perturbation. The first step we give some maximal regularity results of the Ornstein Uhlenbeck process with memory term driven by a rough path using the Nagy Dilation Theorem.

Florian Huber: Weak martingale solutions of stochastic cross-diffusion systems

The so-called cross-diffusion systems appear in a multitude of different applications ranging from thermodynamics, population dynamics of interacting species to modelling the evolution of cancer. They are systems of strongly coupled parabolic equations whose diffusion matrix doesn't need to be positive semi-definite or symmetric. We considered a subclass of these equations and introduced a multiplicative (Stratonovich) noise term to account for random influences of the environment as well as intrinsic uncertainties,

\[ du(t)=\text{div}\left(A(u)\nabla u\right)\:dt+\sigma (u)\circ\:dW_{t}. \]

The structure arising due to interactions at macroscopic level pose the main mathematical difficulty. To balance the loss of the positive-semi-definiteness of $A$, we require a certain (entropy) structure. Although our results apply to more general equations, we focus on the case of a biofilm model. We obtain global in time weak solutions, in stochastic and PDE sense, by combining results from both stochastic-and deterministic world through a Wong-Zakai type approximation of the driving noise. The existence proof is based on so called entropy estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski's generalization of the Skorokhod theorem.

Sandra Kliem: The one-dimensional KPP-equation driven by space-time white noise

The one-dimensional KPP-equation driven by space-time white noise,

\[ \partial_t u = \partial_{xx} u + \theta u - u^2 + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 \]

is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial nonnegative finite-mass conditions. If $\theta$ is below a critical value $\theta_c$, solutions die out to 0 in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$, then there exist stochastic wavelike solutions which travel with nonnegative linear speed. For initial conditions that are ``uniformly distributed in space'', the corresponding solutions are all in the domain of attraction of a unique nonzero stationary distribution. We show that for all parameters above the critical value, there exist stochastic wavelike solutions which travel with a deterministic positive linear speed. We further give a sufficient condition on the initial condition of a solution to attain this speed. The existence of traveling wave solutions can be used to prove that the support of any solution to the SPDE is recurrent. Lastly, we discuss the relevance of traveling wave speeds to determine the limiting behavior of solutions with arbitrary initial conditions.

Verena Köpp: Dynamics and synchronization of stochastic lattice equations

We consider a stochastic lattice differential equation on the d-dimensional lattice with infinitely many neighbor interactions, a nonlinear reaction term and additive noise at each node. We prove that the given equation generates a random dynamical system and introduce conditions under which there exists a random attractor. Then we study a coupled system of two such equations and show that synchronization occurs under suitable conditions.

Jakub Slavik: On stochastic 3D primitive equations

New results on the stochatic 3D primitive equations driven by multiplicative white noise will be presented. In particular we will discuss well-posedness under less restrictive assumptions on the noise.

So Takao: Well-Posedness by noise for the linear advection of k-forms

The linear advection equation of differential k-forms form an important class of systems in fluid dynamics. For instance, the advection of a volume form gives rise to the continuity equation and the advection of a two-form can be used to model the transport of a magnetic field in MHD. We discuss the well-posedness of these systems when we add a noise of transport-type, and show that it admits a global weak solution when the drift velocity is merely Holder continuous. This extends well-known results on the "well-posedness by noise" for the linear transport equation and the continuity equation.

Rico Weiske: The BDF2-Maruyama Method for Stochastic Evolution Equations

We study the numerical approximation of stochastic evolution equations with a monotone drift driven by an infinite dimensional Wiener process. To discretize the equation, we combine a drift-implicit two-step method for the temporal discretization with an abstract Galerkin scheme for the spatial discretization. We establish a convergence rate of the strong error under suitable Lipschitz conditions. We also illustrate our theoretical results through numerical experiments and compare the performance of the BDF2-Maruyama method to the backward Euler-Maruyama method.

This is a joint work with Raphael Kruse (TU Berlin).