The conference will be held in room Aula Dini of Palazzo del Castelletto, Via del Castelletto 17/1.
Day 1 - Aula Dini
Registration 8:30
8:50 - 9:00
[Luigi Ambrosio]
9:00- 9:40
[K. David Elworthy]
The aim is to look at the possibilty of obtaining growth estimates of derivatives of solutions of heat equations in terms of the underlying geometry of the equation, as in the Bakry-Emery theory for first derivatives. A potentially complete description has been given for the heat equation on spheres, and the approach should work for compact Lie groups and other compact symmetric spaces. The technique for this comes from earlier work with Xue-Mei Li and Yves LeJan. After reviewing those results, I will raise the question of whether they do give a "complete description" even for long time behaviour, and also show the difficulties which arise when trying to adapt the method to more general compact manifolds.
The positive results about spheres are taken from : Elworthy, K. David, Higher order derivatives of heat semigroups on spheres and Riemannian symmetric spaces, in Geometry and invariance in stochastic dynamics, Springer Proc. Math. Stat., volume 378, 113–136, 2021
9:45- 10:25
[Annie Millet]
We study the convergence for the spectral Galerkin approximation applied to the stochastic Burgers equation driven by additive trace-class noise. The strong convergence is well-understood and we fill a gap by proving weak error estimates. We show that the weak rate of convergence is twice the strong rate. The main ingredients of the proof are original regularity results on the solutions of the associated Kolmogorov equations. This is joint work with Charles-Edouard Bréhier and Sonja Cox.
Coffee Break 10:30 - 11:00
11:00 - 11:40
[Gianmario Tessitore]
The talk is devoted to the optimal control of a system with two time-scales, in a regime when the limit equation is not of averaging type but, in the spirit of Wong-Zakai principle, it is a stochastic differential equation for the slow variable, with noise emerging from the fast one. We prove that it is possible to control the slow variable by acting only on the fast scales. The concrete problem, of interest for climate research, is embedded into an abstract framework in Hilbert spaces, with a stochastic process driven by an approximation of a given noise. The principle established here is that convergence of the uncontrolled problem is sufficient for convergence of both the optimal costs and the optimal controls. This target is reached using Girsanov transform and the representation of the optimal cost and the optimal controls using a Forward Backward System. A challenge in this program is represented by the generality considered here of unbounded control actions.
Joint work with: F. Flandoli, G. Guatteri, U. Pappalettera
11:45 - 12:25
[Roberto Triggiani]
We consider the MHD system in a bounded d-domain, d=2,3, with homogeneous BC and subject to external sources assumed to create instability. Initial conditions of both fluid and magnetic equations are of low regularity. We then seek to uniformly stabilize such MHD-system in the vicinity of an unstable equilibrium pair, by means of explicitly constructed, static, feedback controls, in minimum number, localized in an arbitrarily small interior subdomain. Joint work with I.Lasiecka and B.Priyasad. To appear in Research in Mathematical Sciences
Lunch 12:30 -14:30
14:30 - 15:10
[Francesco Russo]
Stochastic differential equations (SDEs) in the sense of McKean are stochastic differential equations, whose coefficients do not only depend on time and on the position of the solution process, but also on its marginal laws. Often they constitute probabilistic representation of conservativ PDEs, called Fokker-Planck equations;
In general Fokker-Planck PDEs are well-posed if the initial condition is specified. Here, alternatively, we consider the inverse problem which consists in prescribing the final data: in particular we give sufficient conditions for existence and uniqueness.
We also provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.
The research is motivated by some application consisting in representing some semilinear PDEs (typically Hamilton-Jacobi-Bellman in stochastic control) fully backwardly.
This work is based on a collaboration with L. Izydorczyk (Mazars), N. Oudjane (EDF), G. Tessitore (Milano Bicocca).
15:15 - 15:55
[Abdelaziz Rhandi]
In this talk we study evolution equations that are perturbed at the boundary by both noise and an unbounded perturbation. First, using the theory of regular linear systems, we prove the existence of solutions to this equation. Second, we investigate the long-time behavior of the solutions, such as the absolute continuity and the existence of an invariant measure.
Coffee Break 16:00 - 16:30
16:30 - 17:10
[Stefano Bonaccorsi]
In this talk we discuss some results in the recent literature on equations with boundary noise. We start with a chronological perspective that, we hope, can help to classify the results thematically.
Day 2 - Aula Dini
9:00 - 9:40
[Zdzislaw Brzezniak]
We will consider deterministic and stochastic nonlinear heat equation subject to the constriant that the $L^2$-norm is preserved. The problem is well posed in the space $H^1\cap L^p$ for an appropriate $p$. We consider $p$ large so that $H^1\notsubset L^p$ The large deviations principle will also be discussed.
9:45 - 10:25
[Irena Lasiecka]
A model of shallow shell with a dissipation on the boundary will be discussed.
It will be shown taht under suitable "shallowness" hypothesis the dynamics converges to a. coherent structure described by a finite dimensional attractor. The main tool of the analysis are quasistable estimates within the framework of differential geometry.
Coffee Break 10:30 - 11:00
11:00 - 11:40
[Robert Dalang]
We consider an SPDE driven by a parabolic second order partial differential operator with a nonlinear random external forcing defined by a Gaussian noise that is white in time and has a spatially homogeneous covariance. We prove existence and uniqueness of a random field solution to this SPDE. Our main result concerns the space-time sample path regularity of its solution.
11:45 - 12:25
[Andrea Pugliese]
Several papers have investigated the connection between within-host infection dynamics and infection transmission between individuals. For instance, Hoyer-Leitzer et al. [3] study how the interactions of the pathogen with the immune system shape the potential for re-exposure during a single epidemic outbreak.
This presentation is based on the model by Gandolfi et al. [2] where it is assumed that the viral load at exposure is related to the viral load of the exposure, and affects the subsequent within-host dynamics. The model is presented as a system of transport-type PDEs with non-local boundary conditions, that can be formulated also as a renewal equation, but in that paper is mainly illustrated through numerical simulations. Here (see also [4]) I prove the well-posedness of the problem as a Volterra integral equation in L1 coupled with an ODE; moreover, I show the conditions for the stability of the Disease-Free equilibrium and its connection with the existence of a positive equilibrium.
Finally, I extend the problem to the case where immunity to infection wanes (similarly to [1]) which makes the problem more complex and with a richer dynamics.
References
[1] Barbarossa, M. V., & Röst, G. (2015). Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting. J. Math. Biol., 71(6–7), 1737–1770. https://doi.org/10.1007/s00285-015-0880-5
[2] A. Gandolfi, A. Pugliese and C. Sinisgalli. (2015) Epidemic dynamics and host immune response: a nested approach, J Math Biol. 70: 399-435, DOI: 10.1007/s00285-014-0769-8
[3] Hoyer-Leitzel A, Iams SM, Haslam-Hyde AJ, Zeeman ML, Fefferman NH (2023). An immuno-epidemiological model for transient immune protection: A case study for viral respiratory infections. Infect Dis Model. 8(3):855-864. doi: 10.1016/j.idm.2023.07.004..
[4] Pugliese, A. (2024). Within-Host and Between-Hosts Epidemic Dynamics: A Journey with Alberto. In: d'Onofrio, A., Fasano, A., Papa, F., Sinisgalli, C. (eds) Problems in Mathematical Biophysics. SEMA SIMAI Springer Series, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-031-60773-8_11
Lunch 12:30 - 14:30
14:30 - 15:10
[Hélène Frankowska]
Dynamical systems involving large number of agents can be approached by considering moving sets of agents instead of union of vector-valued time-dependent paths of individual agents. In some social sciences models, like the evacuation one, it may be interesting to assign to sets their measures and to work with metric spaces of Borel probability measures, the so called Wasserstein spaces. Here translation of vectors is replaced by the push-forward operation on measures leading to natural extensions of notions of nonsmooth analysis to this metric space.
The dynamics of measures can be described then via controlled continuity (transport) equations. For Lipschitz kind dynamics, some cornerstone results of classical control theory known in the Euclidean framework have their analogues in Wasserstein spaces.
In this talk I will discuss some necessary and sufficient conditions for the existence of solutions to state-constrained control systems on Wasserstein spaces.
15:15 - 15:55
[Daniela Sforza]
Beginning with my time in Pisa as a PhD student under the supervision of Giuseppe Da Prato, I would like to share some insights into the integro-differential problems I focused on during my research. These problems are often found in fields such as control theory, mathematical physics and stochastic processes, where they arise in the modeling of systems influenced by both past states (through the integral terms) and instantaneous rates of change (through the differential terms).
My initial works addressed the case where the integral operators are defined using regular kernels. The research explored the existence, uniqueness, regularity and long-term behavior of solutions to equations involving these types of integral operators.
This work laid the foundation for further studies, where singular kernels could be considered, leading to broader generalizations of the problems I initially investigated.
Integrable singular kernels are often characterized by power-law memory, which naturally leads to time-fractional differential equations, a growing area of research with increasing importance in mathematics and its applications.
In this talk, I will primarily focus on recent results (see [1,2]) concerning Caputo and Riemann-Liouville fractional equations, obtained by applying the multiplier method along with specific techniques from interpolation theory.
References
[1] P. Loreti, D. Sforza, Fractional diffusion-wave equations: hidden regularity for weak solutions. Fract. Calc. Appl. Anal. 24 (2021), no.4, 1015-1034.
[2] P. Loreti, D. Sforza, Trace operators for Riemann-Liouville fractional equations, work in progress.
Coffee Break 16:00 - 16:30
16:30 - 17:10
[Giorgio Metafune]
We study elliptic and parabolic problems governed by singular elliptic operators under Dirichelet or Neumann boundary condition, in the half-space. We prove elliptic and parabolic Lp-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that L generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.
Day 3 - Aula Dini
9:00 - 9:40
[Jerzy Zabczyk]
TBA
9:45 - 10:25
[Diego Pallara]
We discuss regularity properties of some classes of generalised Mehler semigroups and their generators.
Coffee Break 10:30 - 11:00
11:00 - 11:40
[Istvan Gyongy]
Existence, uniqueness and stability of the solutions to linear stochastic evolution equations are investigated. The results are applied to linear second order stochastic partial differential equations with singular lower order coefficients, to obtain theorems on existence uniqueness and regularity of their solutions in L_p-spaces.
The talk is based on a joint work with Nicolai Krylov.
11:45- 12:25
[Mimmo Iannelli]
The COVID-19 pandemic experience has highlighted the complex shape of threatening epidemic outbreaks that will represent the main challenge for future preparedness activities.
In this talk, I will summarise published and progress work on complex epidemics relying on a number of variants of the classical Kermack and McKendrick epidemic model with age of infection (1927), which still represents the basis of any attempt to understand infectious diseases spread and their control/mitigation interventions.
The model provides an adequate reproduction of the overall story of the COVID-19 pandemic until the spread of omicron variant by using a minimal parametrization. Moreover, an optimal control variant of the model is used to seek optimal combinations of social distancing and vaccination.
Lunch 12:30 - 14:30
14:30- 15:10
[Pierangelo Marcati]
We show the existence of global in time, finite energy weak solutions to the Quantum Euler-Maxwell system, also knwon as quantum magnetohydrodynamics (QMHD), for a large class of initia data that are slightly more regular than being of finite energy. Hydrodynamic models of this type have been used by Feynman and Ginzburg, Landau, Lifschits and Pitaevskii in the study of superconductivity. It is also prototypical model for a quantum plasma, arising for instance in the description of dense astrophysical objects such as white dwarf stars.
We first introduce some preliminary results we will need throughout the talk, as well as the precise notion of finite energy weak solutions then we prove suitable dispersive estimates for weak solutions to the non-linear Maxwell-Schrödinger system. These estimates play a crucial role in order to rigorously derive the continuity and momentum equations via the Madelung type transformations. Hence as a byproduct, the Lorentz force is well defined. Therefore we show non trivial local-smoothing estimates for the Maxwell-Schrödinger system and we use them we prove our main results concerning the QMHD system.
18:00 - 19:30
19:45
Concert in Sala Azzurra (A. Caraceni, D.Trevisan, G.Zanco)
Buffet in Sala degli Stemmi
Day 4 - Aula Dini
9:00 - 9.40
[Michael Röckner]
In this talk we shall present the construction of a stochastic process, which is related to the parabolic p-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) Laplacian.
Joint work with:
1) Viorel Barbu, Al.I. Cuza University and Octav Mayer Institute of Mathematics of Romanian Academy, Ia¸si, Romania
2) Marco Rehmeier, Faculty of Mathematics, Bielefeld University, Germany
9:45 - 10:25
[Fausto Gozzi]
In this talk we present some recent results on the solution of the mean field game system of Hamilton-Jacobi-Bellman and Fokker-Planck-Kolmogorov PDE when the state space of the agents is an infinite dimensional Hilbert space.
We expose results on the linear quadratic case and on a non-linear case with strong smoothing assumptions.
We also discuss examples of applications and some further related work.
Coffee Break 10:30 - 11:00
11:00- 11:40
[Marco Fuhrman]
We present recent and new results on optimal control problems for McKean-Vlasov stochastic differential equations, i.e. where the coefficients depend on the marginal laws of the state. These models describe for instance the behavior of a representative agent maximizing her utility when interacting with a large collection of other homogeneous agents. The basic method of dynamic programming leads to the Hamilton-Jacobi-Bellman equation (HJB), whose solution is expected to coincide with the value function of the optimal control problem. In the McKean-Vlasov case it is natural to write HJB as a partial differential equation on the Wasserstein space of probability measures on Euclidean space with finite second moment. Several variants of HJB equations will be presented, including models for the collective behaviour of heterogeneous agents. This is joint work with A De Crescenzo, I. Kharroubi, H. Pham and work in progress with S. Rudà.
11:45- 12:25
[Etienne Pardoux]
We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual.
We establish a functional law large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics.
In the FLLN, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly.
In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection rate functions in addition to the two white noises mentioned above. The solution to the SPDE can be also explicitly characterized, given this auxiliary process. The uniqueness of the SPDE solution is established under stronger assumptions (density and its derivative being locally bounded) on the distribution function of an infectious duration.
Lunch 12:30 - 14:30
14:30 - 15.10
[Enrico Priola]
This is a joint work with Alessandro Bondi (Luiss).
We prove a Dynamic Programming Principle (DPP) in a strong formulation for a stochastic control problem involving controlled SDEs with jumps driven by a Brownian motion and a stationary Poisson point process with values in ${R}^d$.
We consider arbitrary predictable controls $a $ with values in a closed convex set $C \subset R^{l}$.
The coefficients of the SDE satisfy linear growth and Lipschitz-type conditions in the $x-$variable, and are continuous in the control variable.
Moreover, we consider the value function
$ v(s,x)$ $= \sup_{a} \,{E}\big[\int_{s}^{T} h (r,X_r^{s,x,a}, a_r )dr $ $ + j(X_T^{s,x,a})\big]$,
assuming that $h$ and $j$ are bounded and continuous; here $X_r^{s,x,a}$ is the solution to the controlled SDE. To prove the DPP we show the existence of a regular stochastic flow for the SDEs when the coefficients are independent of the control $a$. Notably, this regularity result is new for jump diffusions even when there is no large-jumps component (cf. Kunita's recent book on stochastic flows and jump diffusions).
The proof of the DPP is completed by introducing an approach that relies on a suitable subclass of finitely generated step controls.
These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves; we believe that this novel method is of independent interest.
15:15 - 15.55
[Mark Veraar]
In this talk I will give an overview of several recent developments on quasi- and semi-linear stochastic PDEs in critical spaces. I will present a new method to prove local and global well-posedness results, and new bootstrap method to show higher order regularity of the solution. In the talk several applications to reaction diffusion equations will be discussed in details. In particular, the new setting allows to prove global well-posedness for several systems which do not satisfy classical coercivity estimates.
The talk is based on joint work with Antonio Agresti.
Coffee Break 16:00 - 16:30
16:30 - 17:10
[Nicolai V. Krylov]
Note: This seminar will be online on Zoom.
We present a new short proof of one of Adams’s theorems about estimating the $L_{p}$-norm of $bDu$ and discuss some issues related to Morrey spaces such as interpolation inequalities and embedding theorems.
Day 5 - Aula Dini
9:00 - 9.40
[Arnaud Debussche]
Stochastic fluid model with transport noise are popular, the transport noise models unresolved small scales. The main assumption in these models is a very strong separation of scales allowing this representation of small scales by white - ie fully decorrelated - noise. It is therefore natural to investigate whether these models are limits of models with correlated noises. Also, an advantage of correlated noises is that they allow classical calculus. In particular, it allows to revisit the derivation of stochastic models from variational principle and allows to derive equation for the evolution of the noise components. The advantage of having such equations is that in most works, the noise components are considered as given and stationary with respect to time which is non realistic. Coupling stochastic fluid models with these gives a more realistic systems.
9:45 - 10:25
[Francesca Bucci]
Proving the well-posedness of appropriate Riccati equations is a crucial step in the study of the optimal control problem with quadratic functionals for important classes of linear partial differential equations. In the case of evolution equations with finite memory, a Riccati-based approach to the optimization problem -- geared towards attaining not only a closed-loop representation of the optimal control, but also its synthesis by way of solving a suitable system of operator equations -- has been spurred by recent progress in a finite dimensional context. The aim of this talk is to describe the study carried out on two distinct model equations (in Hilbert spaces) which display memory terms; the most recent advances concern the case when the memory pertains to the control actions, to wit, the evolution depends also on past inputs. While the latter problem appears interesting enough in itself, addressing one challenge at a time is meant to eventually lead to a full understanding (and solution) of the optimal boundary control of significant integro-differential equations.
(This is an ongoing joint work with Paolo Acquistapace, Univ. di Pisa (Ret.))
Coffee Break 10:30 - 11:00
11:00 - 11:40
[Luca Lorenzi]
In this talk, we consider Young and rough evolution equations
dy(t)=Ay(t)dt+\sum_{i=1}^dF_i(y(t))dx_i(t)
where A is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space E, x is a nonsmooth path, which is \eta-H\"older continuous for some \eta\in (1/3,1/2] (rough equations) and \eta\in (1/2,1) (Young equations). In the first case F_i are non-smoothing linear operator on E, whereas in the second case F_i are Lipschitz continuous functions. We show that the Cauchy problem associated with the previous equation admits a unique mild solution and the solution increases the regularity of the initial datum as soon as time evolves. We also show that the mild solution is also an integral solution and this allows to prove a It\^o formula.
This talk is based on a couple of papers in collaboration with D. Addona (University of Parma) and G. Tessitore (University of Milan Bicocca).
11:45 - 1225
[Luciano Tubaro]
A mild Girsanov formula
Lunch 12:30 - 14:30