## prove

#### 2017

### Lunedi 18 Dicembre

###### sala seminari

- 15:00 M. Maurelli (WIAS & TU Berlin): A McKean-Vlasov SDE with reflecting boundaries

McKean-Vlasov SDEs are SDEs where the coefficients depend on the law of the solution to the SDE. Their interest is in the links with nonlinear PDEs on one side (the SDE-related Fokker-Planck equation is nonlinear) and with interacting particles on the other side (the McKean-Vlasov SDE can be approximated by a system of weakly coupled SDEs). In this talk, motivated by a model of lithium-ion batteries, we introduce a McKean-Vlasov SDE constrained on a bounded domain. The novelty lies in the particular interplay between the mean field interaction and the boundary conditions. We study well-posedness and particle approximation. Our analysis combines techniques for SDEs with Neumann boundary terms (Skorokhod equation, Lions-Sznitman) and ideas from a recent approach to McKean-Vlasov SDEs (Cass-Lyons).

### Giovedi 7 Dicembre

###### aula cantimori (palazzo d'ancona, sns)

- 14:00 M. Jara (IMPA): Entropy methods in interacting particle systems
- 15:00 M. Gubinelli (Bonn Universität): Weak universality of singular SPDEs

We develop a variational formula to estimate the moment generating function of additive functionals of Markov processes. Combined with a related variational formula for the evolution of the relative entropy of the law of the process with respect to suitable reference measures, this variational formula serves as a substitute for the martingale method pioneered by Kipnis-Varadhan in the case of out-of-equilibrium interacting particle systems.

I will discuss the problem of controlling the large scale limit of weakly non-linear SPDEs in the regime where the non-linearity survives at the macroscopic scale giving rise to singular non-linear SPDEs.

### Martedi 26 Settembre

###### sala seminari

- 10:00 N. Perkowski (Humboldt Universität): 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.

### Martedi 4 Luglio

###### sala seminari

- 11:00 D. Cappelletti (University of Wisconsin-Madison: Chemical reaction networks: deterministic and stochastic models

Chemical reaction networks are mathematical models used in biochemistry, as well as in other fields. Specifically, the time evolution of a system of biochemical reactions are modelled either deterministically, by means of a system of ordinary differential equations, or stochastically, by means of a continuous time Markov chain. It is natural to wonder whether the dynamics of the two modelling regimes are linked, and whether properties of one model can shed light on the behavior of the other one. In this talk some connections will be shown, and both classical and recent results will be discussed. However, many open questions remain. For example, it is not known under what assumptions some qualitative properties of the deterministic model imply positive recurrence, non-explosiveness or absorption events for the stochastic model.

### Martedi 27 Giugno

###### sala seminari

- 11:00 R. Catellier (Université de Nice Sophia-Antipolis): Averaging along irregular curves and regularization of ODEs
- 12:15 M. Nassif (ENS Rennes): Introduction to interacting particle systems

Paths of some stochastic processes such as fractional Brownian Motion have some amazing regularization properties. It is well known that in order to have uniqueness in differential systems such as

*b* needs to be quite regular. However, the oscillations
of a stochastic process added to the system will guarantee uniqueness
for really irregular *b*. In this talk we will show how to
solve the perturbed differential system with a certain stochastic
averaging operator. As an application, we show that the stochastic
transport equation driven by fractional Brownian motion has a unique
solution when *u _{0}∈L^{∞}* and

*b*is a possibly random

*α*-Hölder continuous function for

*α*large enough.

This is a joint work with Massimiliano Gubinelli.

Interacting particle systems is a recently developed field in the theory of Markov processes with many applications: particle systems have been used to model phenomena ranging from traffic behaviour to spread of infection and tumour growth. We introduce this field through the study of the simple exclusion process. We will construct the generator of this process and we will give a convergence result of the spatial particle density to the solution of the heat equation. We will also discuss a variation of the simple exclusion process with proliferation.

### Martedi 6 Giugno

###### sala seminari

- 11:15 M. Leocata (Università di Pisa): A particle system approach to cellular aggregation model

In this talk I will present a model for cellular aggregation based on a system of PDE and I will investigate a microscopic derivations. Description of local interaction is given by the notion of moderate interactions in the sense of K. Oelshchlager.

### Martedi 23 Maggio

###### sala seminari

- 11:15 C. Olivera (Universidade Estadual de Campinas): Regularization by noise in (2x 2) hyperbolic systems of conservation laws

In this talk we study a non-strictly hyperbolic system of conservation law by stochastic perturbation. We show existence and uniqueness of the solution. We do not assume $-regularity for the initial conditions. The proofs are based on the concept of entropy solution and on the method of charactteristics (under the influence of noise). This is the first result on the regularization by noise in hyperbolic systems of conservation law.

### Martedi 16 Maggio

###### sala seminari

- 11:15 D. Luo (Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing): Constantin and Iyer's representation formula for the Navier--Stokes equations on manifolds

In this talk, we will present a probabilistic representation formula for the Navier-Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case. On a Riemannian manifold, there are several different choices of Laplacian operators acting on vector fields. We shall use the de Rham-Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose it as a sum of the square of Lie derivatives. This is a joint work with Shizan Fang.

### Martedi 9 Maggio

###### sala seminari

- 11:15 F. Grotto (SNS Pisa): Regularisation for stationary Stochastic Burgers' Equation

Stochastic Burgers' Equation (SBE) in 1+1 dimensions is the formal space derivative of Kardar-Parisi-Zhang (KPZ) equation, and shares with the latter its ill-posed nature. In the stationary case, works of M. Jara and P. Goncalves, and later M. Gubinelli and N. Perkowski, have shown how to exploit the regularising properties of the linear part of SBE to give meaning to its quadratic drift, deriving the notion of Energy Solution for stationary SBE, and consequently for KPZ equation. We will address how this procedure is carried out, and moreover how regularisation yields existence of energy solutions by means of convergence of SDEs modelling surface growth, among them the ones considered by M. Hairer and J. Quastel in their weak universality result for KPZ. Uniqueness of the energy solution, which implies a weak KPZ universality result in our context, and the equivalence of our solution to the physical Hopf-Cole solution, will be addressed in another future talk.

### Martedi 2 Maggio

###### sala seminari

- 11:15 K. Matetski (U. Toronto): Convergence of general weakly asymmetric exclusion processes

In my ongoing work with J. Quastel we consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite jump ranges and general jump rates. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. In contrast to the celebrated result by L. Bertini and G. Giacomin (in the case of the nearest neighbour interaction) and its extension by A. Dembo and L.-C. Tsai (for jumps of sizes at most three) we do not use the Hopf-Cole transform and work with the KPZ equation using regularity structures. The price which we have to pay for this approach is a non-trivial renormalization which has not been observed before for equations with stationary noises.

In my talk i will give a general review of the Hopf-Cole solution to the KPZ equation and the results of the aforementioned authors. After that I will introduce a general weakly asymmetric exclusion process and explain the difficulty with renormalization.

### Martedi 11 Aprile

###### aula Mancini (SNS)

- 14:30 M. Röckner (Universitä Bielefeld): Global solutions to random 3D vorticity equations for small initial data

One proves the existence and uniqueness in
*(L ^{p}(R^{3}))^{3}*,

*3/2< p < 2*, of a global mild solution to random vorticity equations associated to stochastic 3D Navier-Stokes equations with linear multiplicative Gaussian noise of convolution type, for sufficiently small initial vorticity. This resembles some earlier deterministic results of T. Kato and are obtained by treating the equation in vorticity form and reducing the latter to a random nonlinear parabolic equation. The solution has maximal regularity in the spatial variables and is weakly continuous in

*(L*with respect to the time variable. Furthermore, we obtain the path-wise continuous dependence of solutions with respect to the initial data.

^{3}∩L^{3p/(4p-6)})^{3}This is joint work with Viorel Barbu.

### Martedi 4 Aprile

###### sala seminari

- 11:15 G. Antonelli (SNS, Pisa): Calcolo di Malliavin - II

Proseguirò il lavoro del seminario precedente enunciando altre regole di calcolo per la derivata di Malliavin, in particolare la regola di commutazione fra derivata di Malliavin e integrale di Ito. Vedrò come queste consentono di ottenere un risultato di derivabilità secondo Malliavin di una soluzione forte di una SDE, sotto certe ipotesi di regolarità dei coefficienti.

Dopo aver definito gli opportuni spazi *D ^{k,p}* delle variabili aleatorie derivabili

*k*volte secondo Malliavin e

*p*-sommabili, noterò come una maggiore regolarità dei coefficienti produce soluzioni forti derivabili infinite volte secondo Malliavin.

### Martedi 28 Marzo

###### sala seminari

- 11:15 G. Antonelli (SNS, Pisa): Calcolo di Malliavin - I

Lo scopo della prima parte del seminario sarà introdurre il concetto di derivata di Malliavin e investigarne le proprietà base. Seguendo l'approccio di Nualart (The Malliavin Calculus and related topics), definirò l'operatore di derivata su una classe smooth di funzionali, notando la chiudibilità. Proseguirò enunciando alcune proprietà importanti: la regola della catena per la derivata di Malliavin di una funzione composta e la commutazione fra speranza condizionale e derivata di Malliavin.

Introdurrò l'operatore aggiunto della derivata di Malliavin, l'integrale di Skorokhod, notando che sui processi progressivamente misurabili coincide con il classico integrale di Ito; enuncerò la formula di Clark-Ocone e la formula di commutazione fra derivata di Malliavin e integrale di Ito.

Nella seconda parte inizierò ad introdurre le applicazioni di questo potente strumento: dirò sotto quali condizioni la soluzione di una SDE è derivabile secondo Malliavin e vedrò come la matrice delle covarianze di Malliavin fornisce una regola di integrazione per parti.

### Martedi 21 Marzo

###### sala seminari

- 11:15 M. Pratelli: La topologia di Meyer-Zheng sullo spazio delle funzioni "cadlag"

Se si considera sullo spazio delle funzioni cadlag la topologia della convergenza in probabilità anziché l'usuale topologia di Skorohod si ottengono (mediante l'immersione dello spazio delle funzioni in uno spazio di probabilità) delle condizioni di tensione molto agevoli da verificare per martingale e supermartingale.

Lo scopo del seminario è introdurre brevemente questa topologia assieme ad alcuni problemi che è stato agevole risolvere con questi metodi.

### Mercoledi 8 Marzo

###### sala seminari, dipartimento di matematica

- 11:00 F. R. Nardi (Università di Firenze): Metastability for general dynamics with rare transitions: escape time and critical configurations
- 12:00 V. De Mattei (Università di Pisa): Mean field games with mean field and moderate interactions

Metastability is an ubiquitous physical phenomenon in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions for Markov chains. For Metropolis chains associated with statistical mechanical systems, this phenomenon has been described in an elegant way through a path-wise approach in terms of the energy landscape associated to the Hamiltonian of the system. In the seminar we will first explain the main results and ideas of this approach and compare it with other existing ones. Then we will provide a similar description in the general rare transitions setup that can be applied to irreversible systems as well.

Besides their theoretical content, we believe that our results are a useful tool to approach metastability for non-Metropolis systems such as Probabilistic Cellular Automata. Moreover, we will describe results pertaining to exponential hitting times which range of applicability includes irreversible systems, systems with exponentially growing volumes and systems with a general starting measure.

(joint work with Emilio N. M. Cirillo, R. Fernandez, F. Manzo, E. Scoppola and J. Sohier).

We consider a mean field game where *N* players interact
both in the cost functionals and in the dynamics. The new feature is that the
dynamical interaction is of the so called ``moderate'' type, intermediate between
the more classical mean field game scheme is presented. Convergence to a system
of PDEs, the mean field equation for our system, is discussed

#### 2016

### Martedi 5 Luglio

###### sala seminari, dipartimento di matematica

- 11:30 A. Hocquet (TU Berlin): Finite-time singularities of the stochastic harmonic map flow on surfaces

A ferromagnetic material possesses a magnetization, which, out of equilibrium, satisfies the Landau-Lifshitz-Gilbert equation (LLG). Thermal fluctuations are taken into account by Gaussian space-time white noise.

At least in the deterministic case, there is an important parallel
between this model and the so-called Harmonic Map Flow (HMF). This was
originally used by geometers (in the early sixties) as a tool to build
harmonic maps between two manifolds *u:M→N*. The case where *M* is two
dimensional is critical, in the sense that the natural energy barely
fails to give well-posedness.

We do not address here the problem of the solvability of LLG driven by space-time white noise. Instead, we consider a spatially correlated version. We show that oppositely to the deterministic case, blow-up of solutions happens no matter how we choose the initial data.

### Venerdi 20 Maggio

###### sala seminari, dipartimento di matematica

- 11:00 F. Russo: Probabilistic representation of a generalized porous media equation. The deterministic and stochastic case

### Giovedi 21 Aprile

###### aula magna, dipartimento di matematica

- 11:00 G. Zanco (IST Austria): A brief introduction to rough paths theory - II

We will introduce basic concepts and tools from rough paths theory,
motivated by the need to establish a solution theory for differential equations
driven by irregular signals. In particular we will discuss controlled rough paths,
integration against rough paths, we will compare it to stochastic integration
theories and show how classical results can be extended to the rough paths framework.
We will mainly consider α-Hölder signals with *α∈(1/3,1/2]*;
this choice allows to simplify many concepts of the theory and, although restrictive,
provides enough instruments to deal with interesting problems, like nonlinear SPDEs
driven by space-time white noise.

### Mercoledi 20 Aprile

###### sala seminari, dipartimento di matematica

- 11:00 G. Zanco (IST Austria): A brief introduction to rough paths theory - I

We will introduce basic concepts and tools from rough paths theory,
motivated by the need to establish a solution theory for differential equations
driven by irregular signals. In particular we will discuss controlled rough paths,
integration against rough paths, we will compare it to stochastic integration
theories and show how classical results can be extended to the rough paths framework.
We will mainly consider α-Hölder signals with *α∈(1/3,1/2]*;
this choice allows to simplify many concepts of the theory and, although restrictive,
provides enough instruments to deal with interesting problems, like nonlinear SPDEs
driven by space-time white noise.

### Mercoledi 13 Aprile

###### sala seminari, dipartimento di matematica

- 11:30 G. Cannizzaro (TU Berlin): Calcolo di Malliavin per Strutture di Regolarità: il caso di gPAM

Le strutture di Regolarità, introdotte da M. Hairer in
*A theory of Regularity Structures*, hanno permesso di risolvere
in modo robusto una ricca classe di equazioni alle derivate parziali stocastiche
(SPDEs) mal poste. In questa presentazione vogliamo mostrare come sia possibile
utilizzare tecniche di calcolo di Malliavin al fine di indagare proprietà
probabilistiche delle soluzioni di tali equazioni. Ci concentreremo su un
esempio standard della teoria, l'equazione parabolica di Anderson generalizzata
(gPAM), e vedremo come si possa dimostrare l'esistenza della densità rispetto
alla misura di Lebesgue per la sua soluzione valutata ad un punto dello spazio tempo.

### Mercoledi 9 Marzo

###### sala seminari, dipartimento di matematica

- 11:30 R. Giuliano (Università di Pisa): Some examples of the interplay between Probability and Number Theory

Probability theory have been often used to study problems in number theory: for instance the so--called "probabilistic method" is a powerful tool which traces back to P. Erdős. More recently also the large deviations theory has been applied in number theoretical settings; In this talk I shall illustrate the main results of some recent papers of mine (in collaboration with C. Macci) which are in the same circle of ideas.

### Mercoledi 2 Marzo

###### sala seminari, dipartimento di matematica

- 11:00 M. Gubinelli (Bonn Universität): Weak universality of the stationary KPZ equation
- 12:00 M. Coghi (SNS Pisa): Mean field limit of interacting filaments and vector valued non linear PDEs

I will discuss a notion of solution for the KPZ equation which has been introduced by Jara and Gonçalves (2010) and later improved by Jara and myself (2013) and goes under the name of energy solutions. In a recent work in collaboration with Perkowski we have recently obtained a uniqueness results for energy solutions. Using energy solutions is possible to prove weak universality results for KPZ: namely that a wide class of one dimensional microscopic interacting particle models with weak asymmetry and in non-equilibrium stationary states have large scale fluctuations described by the KPZ equation.

Families of N interacting curves are considered, with long range, mean-field type, interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as N goes to infinity, to a mean field current, solution of a nonlinear, vector valued, partial differential equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero. This set-up is inspired from vortex filaments in turbulent fluids, although for technical reasons we have to restrict to smooth interaction, instead of the singular Biot-Savart kernel. All these results are based on a careful analysis of a nonlinear flow equation for 1-currents, its relation with the vector valued PDE and the continuous dependence on the initial conditions.

### Mercoledi 24 Febbraio

###### sala seminari, dipartimento di matematica

- 11:00 M. Neklyudov (Università di Pisa): A particle system approach to cell-cell adhesion models I
- 12:00 A. Caraceni (SNS Pisa): Le mappe esternamente planari e il loro limite di scala

We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlager.

La modellizzazione di oggetti combinatorici discreti (grafi, alberi, mappe), al crescere della loro complessità, con oggetti di natura continua, è una tecnica sviluppata a partire dai lavori di Aldous, che sta godendo in anni recenti di crescente interesse grazie ai suoi legami con combinatoria, analisi stocastica, meccanica statistica e gravitazione quantistica. Il limite di scala di numerose classi di mappe è la cosiddetta ``Mappa Browniana''; sotto determinate ipotesi, tuttavia, è possibile assistere a comportamenti asintotici differenti, e alcune classi di mappe planari con una unica faccia macroscopica ammettono il CRT (continuum random tree) di Aldous, o un suo multiplo, come limite di scala. Vedremo come sia questo il caso in particolare delle mappe ``esternamente planari, una classe di mappe di naturale interesse in combinatoria, soggette alla condizioneche tutti i vertici siano adiacenti alla faccia ``esterna''. Grazie a una bigezione di Bonichon, Gavoille e Hanusse mostreremo come una variabile aleatoria uniforme sull'insieme delle mappe esternamente planari a n vertici, riscalata di un fattore 1/\sqrt{n}, converga in legge (in senso Gromov-Hausdorff) verso 7\sqrt{2}/9 volte il CRT.

### Mercoledi 10 Febbraio

###### aula M1, polo Fibonacci

- 11:45 D. Trevisan (Università di Pisa): A particle system approach to cell-cell adhesion models II

We investigate micro-to-macroscopic derivations in two models of living cells, in presence to cell-cell adhesive interactions. We rigorously address two PDE-based models, one featuring non-local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlager.

### Mercoledi 3 Febbraio

###### sala seminari, dipartimento di matematica

- 11:30 V. Capasso (ADAMSS, Università di Milano): Modelli matematici per l'angiogenesi tumorale

Nella modellazione matematica della angiogenesi tumorale, il forte accoppiamento tra i processi stocastici di diramazione-elongazione-morte di vasi, e i campi biochimici dovuti alla massa tumorale, è causa di forte complessità dal punto di vista sia analitico che computazionale. Al fine di ridurre tale complessità, si cerca di rendere completamente deterministiche le equazioni di evoluzione dei campi, sostituendo in esse i termini stocastici derivanti dalla evoluzione delle rete di vasi, con una loro approssimazione di campo medio. In tal modo i parametri cinetici dei processi (stocastici) di formazione della rete divengono deterministici. Purtroppo a causa della anastomosi, non è possibile garantire le condizioni di applicabilità tipiche della propagazione del caos, che quindi viene messa in discussione

Una possibile derivazione di equazioni di evoluzione deterministiche per i campi fa ricorso alla media su molte repliche dei processi coinvolti, secondo leggi classiche dei grandi numeri. Simulazioni numeriche incoraggiano l'adozione di questo approccio.

### Mercoledi 27 Gennaio

###### sala seminari, dipartimento di matematica

- 11:30 V. De Mattei (Università di Pisa): An introduction to the theory of mean field games

### Venerdi 15 Gennaio

###### sala seminari, dipartimento di matematica

- 15:00 A. Caraceni (SNS, Pisa): Introduzione alla teoria dei limiti di scala

#### 2014

### Lunedi 9 Giugno

###### sala seminari, dipartimento di matematica

- 11:00 W. Neves (Universidade Federal do Rio de Janeiro): Stochastic (intrinsic) partial differential equations
- [1]
- L. Ambrosio,
*Transport equation and Cauchy problem for BV vector fields*, Invent. Math. 158, 2004. - [2]
- R. DiPerna, P. L. Lions,
*Ordinary differential equations, transport theory and Sobolev spaces*, Invent. Math. 98, 1989. - [3]
- F. Fedrizzi , F. Flandoli.
*Noise prevents singularities in linear transport equations*, J. Funct. Anal. 264, 2013. - [4]
- F. Flandoli, M. Gubinelli, E. Priola,
*Well-posedness of the transport equation by stochastic perturbation*, Invent. Math. 180, 2010. - [5]
- P. L. Lions , B. Perthame, P. E. Souganidis,
*Scalar conservation laws with rough (stochastic) fluxes*, SPDE: Anal. and Comp. 1(4), 2013. - [6]
- W. Neves, C .Olivera
*Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition*, arXiv:1307.6484. - [7]
- W. Neves, C .Olivera
*Stochastic transport equation in bounded domains*, in preparation.

We present some results concerning stochastic linear transport equations and quasilinear scalar conservation laws, where the additive noise is a perturbation of the drift. Due to the introduction of the stochastic term, we may prove for instance well-posedness for continuity equation (divergence-free), Cauchy problem, meanwhile uniqueness may fail for the deterministic case, see [1,2,6]. Also for the transport equation, Dirichlet data, we established a better trace result by the introduction of the noise, see [7]. We introduce the study of stochastic hyperbolic conservation laws, in a different direction of [5], applying the kinetic-semigroup theory. Joint work with Christian Olivera (Universidade Estadual de Campinas).

### Martedi 13 Maggio

###### sala seminari, dipartimento di matematica

- 09:30 Z. Brzezniak (University of York): Invariant measures for stochastic Navier-Stokes equations in unbounded domains via bw-Feller property

In this talk I will describe a general result on the existence of invariant measure for Markov processes having the bw-Feller property and will show how this can be applied to stochastic Navier-Stokes equations in unbounded domains. This talk is based on joint works with M. Ondrejat and Ela Motyl. The results presented are in some sense generalisations of related results for stochastic nonlinear beam and wave equations (where a Pritchard-Zabczyk trick plays an essential role) obtained in a joint work with M. Ondrejat and J. Seidler.

### Mercoledi 7 Maggio

###### aula riunioni, dipartimento di matematica

- 11:00 A. Debussche (ENS Rennes): Invariant measures for stochastic conservation laws

### Lunedi 28 Aprile

###### sala seminari, dipartimento di matematica

- 14:30 M. Erbar (Bonn Universität): Gradient flows of the entropy for jump processes

In the last decade optimal transport has proven very successful in the study of diffusion processes and the associated PDEs.The aim of this talk is to present a link between the theory of jump processes and non-local operators and techniques from optimal transport. We introduce a new transport distance between probability measures that is built from a Levy jump kernel via a non-local variant of the Benamou-Brenier formula. We study various properties of this distance. As the main result we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance. This applies e.g. to the fractional heat equation and provides a non-local analogue of the celebrated result by Jordan--Kinderlehrer--Otto.

### Lunedi 14 Aprile

###### aula riunioni, dipartimento di matematica

- 09:30 F. Delarue (Université de Nice Sophia-Antipolis): Mean field games II

### Mercoledi 9 Aprile

###### aula riunioni, dipartimento di matematica

- 11:00 F. Delarue (Université de Nice Sophia-Antipolis): Mean field games I

### Mercoledi 2 Aprile

###### aula riunioni, dipartimento di matematica

- 11:00 N. Touzi (Ecole Politechnique, Paris): Viscosity solutions of path-dependent PDEs

### Venerdi 28 Febbraio

###### sala seminari, dipartimento di matematica

- 11:00 A. Swiech (GeorgiaTech): Introduction to regularity of solutions of 2
^{nd}order Hamilton-Jacobi-Bellman equations - II

### Mercoledi 26 Febbraio

###### sala seminari, dipartimento di matematica

- 11:00 A. Swiech (GeorgiaTech): Introduction to regularity of solutions of 2
^{nd}order Hamilton-Jacobi-Bellman equations - I

### Martedi 11 Febbraio

###### sala seminari, dipartimento di matematica

- 10:30 O. Aryasova (Institute of Geophysics, National Academy of Sciences of Ukraine): A representation for the derivative with respect to the initial data of the solution of an SDE with non-regular drift
- [Ary12]
- O. V. Aryasova and A. Yu. Pilipenko,
*On properties of a flow generated by an SDE with discontinuous drift*, Electron. J. Probab. 17:no. 106, 1--20, 2012. - [Att10]
- S. Attanasio,
*Stochastic flows of diffeomorphisms for one-dimensional {SDE} with discontinuous drift*, Electron. Commun. Probab. 15:no. 20, 213--226, 2010. - [Dyn63]
- E. B. Dynkin,
*Markov Processes*, Fizmatlit, Moscow, 1963. [Translated from the Russian to the English by J. Fabius, V. Greenberg, A. Maitra, and G. Majone. Academic Press, New York; Springer, Berlin, 1965. vol. 1, xii + 365 pp.; vol. 2, viii + 274 pp.]. - [Fed13]
- E. Fedrizzi and F. Flandoli,
*Hölder flow and differentiability for {SDE}s with nonregular drift*, Stochastic Analysis and Applications, 31(4):708--736, 2013. - [Moh12]
- S. E. A. Mohammed, T. Nilssen, and F. Proske,
*Sobolev differentiable stochastic flows of SDE's with measurable drift and applications*,`arXiv:1204.3867`. - [Ver81]
- A. Y. Veretennikov,
*On strong solutions and explicit formulas for solutions of stochastic integral equations*, Math. USSR Sborn, 39(3):387--403, 1981. - 11:30 E. Orsingher (Roma Sapienza): Random flights

We consider a multidimensional SDE with an identity diffusion matrix and a drift vector being a bounded measurable vector field.

According to [Ver81] there exists a unique strong solution to such an equation.

Recently the Sobolev differentiability of the solution with respect to the initial data was proved under rather weak assumptions on the drift (c.f. [Fed13,Moh12]). If the drift coefficient is smooth the derivative can be represent as a solution of an integral equation. For non-regular drift in dimension one such a representation was obtained using the local time of the initial process (see [Ary12,Att10]).

It is well known that the solution does not have a local time at a point in multidimensional situation. We obtained the representation of the derivative using the theory of continuous additive functionals developed by Dynkin [Dyn63]. This method can be considered as a generalization of the local time approach to the multidimensional case.

In this talk we present different types of random flights and examine their dynamics, probability laws and governing equations. We first consider the telegraph process (a random flight on the line), discuss its distribution, the connections with the telegraph equations, the first-passage time and the limiting case. We consider also the asymmetric telegraph process and its reduction to the symmetric one by means of relativistic transformations.

Planar motions with a finite number of directions (in particular, four orthogonal directions) and an infinite number of directions, chosen at Poisson times with uniform law, are examined and several explicit distributions derived. A particular attention is devoted to the second model, where conditional and unconditional distributions are presented and the related equation of damped planar vibrations probabilistically derived.

Random flights in * R^{d}* are subsiquently
considered and characteristic functions of the position of moving
particles performing the random flights, obtained. The cases

*d = 2*,

*d = 4*, are investigated in detail.

We present random flights in * R^{d}*
with Dirichlet joint distribution for displacements, hyperspherical
uniform law for the orientation of steps and with
a fractional Poisson number of changes of direction.

Two types of fractional extensions of the above material are presented. The first one is obtained by considering Dzerbayshan-Caputo types of time- fractional derivatives. The second fractional extension is obtained by considering fractionalisation of Klein-Gordon equations and by applying the McBride approach to fractional powers of Bessel-type operators.

### Mercoledi 22 Gennaio

###### sala seminari, dipartimento di matematica

- 10:05 F. Russo (ENSTA ParisTech): On Kolmogorov type equations associated with frames of diffusion processes in relation with calculus via regularization

#### 2013

### Lunedi 1 Luglio

###### sala seminari, dipartimento di matematica

- 11:30 N. Glatt-Holtz: On inviscid limits for the stochastic Navier-Stokes equations and related systems

One of the original motivations for the development of stochastic PDEs traces its origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows.

In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings.

This is joint work with Peter Constantin, Vladimir Sverak and Vlad Vicol.

### Giovedi 6 Giugno

###### sala seminari, dipartimento di matematica

- 11:00 C. Bernardin (ENS Lyon): Anomalous diffusion in Hamiltonian systems perturbed by a conservative noise

I will discuss a class of Hamiltonian systems perturbed by a conservative noise in the spirit of models considered in Basile-Bernardin-Olla '06-'09. For exponential interactions I will show that the transport of energy is super diffusive.

### Venerdi 31 Maggio

###### aula riunioni, ex dipartimento di matematica applicata

- 16:00 M. Röckner (Bielefeld Unversität): Stochastic nonlinear Schrödinger equations with linear multiplicative noise

We present well-posedness results for stochastic nonlinear
Schrödinger equations with linear multiplicative Wiener noise
including the non-conservative case. Our approach is different from the
standard literature on stochastic nonlinear Schrödinger equations. By
a rescaling transformation we reduce the stochastic equation to a
random nonlinear Schrödinger equation with lower order terms and treat
the resulting equation by a fixed point argument, based on
generalizations of Strichartz estimates proved by J. Marzuola, J.
Metcalfe and D. Tataru in 2008. This approach allows to improve earlier
well-posedness results obtained in the conservative case by a direct
approach to the stochastic Schrödinger equation. In contrast to the
latter, we obtain well-posedness in the full range *(1, 1 + 4/d)* of
admissible exponents in the non-linear part (where *d* is the dimension
of the underlying Euclidean space), i.e. in exactly the same range as
in the deterministic case.

### Martedi 28 Maggio

###### sala seminari, dipartimento di matematica

- 11:00 C. Mueller (University of Rochester): Particle systems and stochastic PDEs - III

### Venerdi 17 Maggio

###### sala seminari, dipartimento di matematica

- 11:00 C. Mueller (University of Rochester): Particle systems and stochastic PDEs - II

### Mercoledi 15 Maggio

###### sala seminari, dipartimento di Economia

- 11:00 P. Guasoni (Boston University and Dublin College University): Habit, Loss Aversion, and Retirement

We solve a model of consumption and investment for an agent who is averse to declines in the standard of living, assuming constant investment opportunities and an infinite horizon. The optimal policy is to consume at a constant rate, as long as this rate remains between two endogenous fractions of wealth. In good times, consumption rises to keep the consumption/wealth ratio above a minimum level. In bad times, consumption declines below the desired level, as to keep the portfolio solvent. Loss aversion reduces substantially the exposure to risky assets, implying an effective risk aversion much higher than the agent's true risk aversion.

### Martedi 7 Maggio

###### sala seminari, dipartimento di matematica

- 11:00 H. Weber (Warwick University): Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size
- 12:00 C. Mueller (University of Rochester): Particle systems and stochastic PDEs - I
- 14:45 M. Gubinelli (Paris Dauphine): Paracontrolled distributions

Using tools from functional analysis we introduce a calculus which applies to quite general stochastic objects, like space-time white noise or similar random distributions and provide a framework where to give a meaning and solve few classes of non-linear stochastic partial differential equations which were previously not know how to handle. Joint work with P. Imkeller and N. Perkowski.

### Martedi 12 Marzo

###### sala seminari, dipartimento di matematica

- 11:30 D. Blömker (Augsburg Universität): Accuracy and Stability of the Continuous-Time 3DVAR Filter for 2D Navier-Stokes Equation

We consider a noisy observer for an unknown solution of a deterministic model. The observer is a stochastic model arising in the limit of frequent observations in filtering, where noisy observations of the low Fourier modes together with knowledge of the deterministic model are used to track the unknown solution.

We establish stability and accuracy of the filter by studying this for the stochastic PDE describing the observer.

### Venerdi 15 Febbraio

###### aula seminari, ex dipartimento di matematica applicata

- 11:00 A. Bevilacqua (SNS Pisa): Local time and some related topics
- 12:00 M. Maurelli (SNS Pisa): Dirichlet forms: basis and examples - I
- 12:30 D. Trevisan (SNS Pisa): Dirichlet forms: basis and examples - II

### Mercoledi 30 Gennaio

###### Economia, terzo piano, corridoio a dx, seconda porta sulla sx

- 11:00 P. Tankov (Paris VII): Asymptotics for sums of log-normal random variables and applications to finance
- systematic construction of stress scenarios for stress testing large portfolios in the multi-dimensional Black-Scholes model;
- approximation of local volatility of a stock index and of implied volatility of an index option for small strike values.

*n*correlated log-normal random variables. Despite the simplicity of the problem formulation and the importance of this result for applications, the full solution has been absent from the literature so far. As applications, we consider two natural problems from financial mathematics

### Giovedi 17 Gennaio

###### aula seminari, ex dipartimento di matematica applicata

- 14:30 E. Gautier (ENSAE ParisTech): High dimensional linear models in statistics and econometrics

#### 2012

### Giovedi 12 Luglio

###### aula 238, dipartimento di statistica e matematica applicata

- 14:00 B. Acciaio (Università di Perugia e di Vienna): Robust pricing via mass transport and pathwise inequalities

### Giovedi 5 Luglio

###### aula seminari, dipartimento di matematica applicata

- 14:30 C. Macci (Università di Roma Tor Vergata): Large deviations for fractional Poisson processes
- 15:30 S. Cerrai (University of Maryland): On the averaging principle for SPDEs

### Giovedi 21 Giugno

###### aula seminari, dipartimento di matematica applicata

- 15:00 H. Pham (Université Paris Diderot): Optimal high frequency trading in limit order book
- 16:00 Z. Brzezniak (York University): Stochastic geometric heat equations

*H*as a state space since in this case the embedding of the energy space into the Banach space

^{1,2}(**S**^{1},**R**^{d})*C(*of continuous functions holds. Some techniques that have been developed by the speaker in collaboration with Goldys and Jegaraj (2010) are essential. Let us point out a difference between our proof of the global existence and the one in the deterministic case by Eells-Sampson (1964) and Hamilton (1975). While in the latter papers the crucial step is to prove that the energy density solves certain scalar parabolic equation, in our case the crucial step is to prove an inequality for the

**S**^{1},**R**^{d})*L*-norm of the gradient of the solution which is based on certain geometric property of the target manifold

^{2}*M*. Based on a joint work of the speaker with B. Goldys and M. Ondreját.

### Venerdi 15 Giugno

###### sala seminari, dipartimento di matematica

- 15:00 L. Zambotti (UPMC): SPDEs with non-convex potential

### Giovedi 24 Maggio

###### sala seminari, dipartimento di matematica

- 15:00 A. Swiech (Georgia Institute of Technology): Viscosity solutions of integro-PDE in Hilbert spaces

### Giovedi 3 Maggio

###### sala seminari, dipartimento di matematica

- 15:00 F. N. Proske (University of Oslo): Construction and Malliavin differentiability of strong solutions of SDEs with merely measurable drift
- O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske, T. Zhang,
*A variational approach to the construction and Malliavin differentiability of strong solutions of SDE*, Preprint series, University of Oslo, No. 9, June 2011. - A. Y. Veretennikov,
*On the strong solutions of stochastic differential equations*, Theory Probab. Appl. 24, 354-366 (1979). - 16:00 H. Bessaih (University of Wyoming): Invariant measures of Gaussian type for some stochastic shell models

In this talk we present a new approach to the construction of strong solutions of stochastic
equations with merely measurable coefficients [1]. We aim at demonstrating the principles
of our technique by analyzing stochastic differential equations driven by a Brownian motion.
A rather surprising consequence of our method which is based on Malliavin calculus is that
the solutions derived by A. Y. Veretennikov [2] for Brownian motion with bounded and
measurable drift in ^{d} are Malliavin differentiable. Moreover,
it is conceivable that our approach which does not rely on a pathwise uniqueness argument
is also applicable to the construction of strong solutions of stochastic equations in
infinite dimensions.

### Giovedi 26 Aprile

###### aula riunioni, dipartimento di matematica

- 15:00 D. Tsagkarogiannis (Bonn Universität): Current reservoirs in the simple exclusion process
- 16:00 L. Baglioni: An introduction to the stochastic Loewner evolutions - II

### Giovedi 19 Aprile

###### aula 238, dipartimento di statistica e matematica applicata

- 15:00 J. Bion-Nadal (Ecole Polytechnique): Dynamic no good deal pricing measures
- 16:00 M. Bambi (York University): Does habit formation always increase the desire to smooth consumption over time?

In the literature, habit formation has been often introduced to enhance the agents' desire to smooth consumption over time. This characteristic was found particularly useful in solving the equity premium puzzle and in matching several stylized facts in growth, and business cycles theory as, for example, the high persistence in the U.S. output volatility. In this paper we propose a definition of habit formation, which is ``general'' relative to the assumptions on the intensity, persistence, and lag structure, and we unveil two mechanisms which point to the opposite direction: habits may reduce the desire of smoothing consumption over time and then may potentially decrease the power of a model in explaining the previously mentioned facts. More precisely, we propose a complete taxonomy of the rich dynamics which may emerge in an AK model with external addictive habits for all the feasible combinations of the intensity, persistence and lag structure characterizing their formation and we point out to the region in the parameters space coherent with less smoothing in consumption. An economic explanation of these mechanisms is suggested and the robustness of our results in the case of internal habits verified. Finally and crucially habit formation always reduces the desire of consumption smoothing once the model is calibrated to match the average U.S. output and utility growth rates observed in the data.

*Keywords*: Habit formation; endogenous fluctuations, delayed functional differential equations.

### Giovedi 29 Marzo

###### sala seminari, dipartimento di matematica

- 15:00 A. A. Majewski: Stochastic analysis for Levy processes
- 16:00 L. Baglioni: An introduction to the stochastic Loewner evolutions - I

### Venerdi 23 Marzo

###### sala seminari, dipartimento di matematica

- 14:00 A. A. Majewski: Introduction to Levy processes
- 15:00 F. Flandoli: Uniqueness, singularities and zero-noise limit for stochastic equations - III

### Giovedi 15 Marzo

###### aula seminari, dipartimento di matematica applicata

- 15:00 M. Röckner (Bielefeld Universität): Regularization of ordinary and partial differential equations by noise
- [1] N.V. Krylov, M. Roeckner,
*Strong solutions of stochastic equations with singular time-dependent drift*, Probab. Theory Rel. Fields**131**(2005), no. 2, 154-196. - [2] G. Da Prato, F. Flandoli, E. Priola, M. Roeckner,
*Strong uniqueness for stochastic evolution equations in Hilbert spaces with bounded measurable drift*, preprint, 2011. - 16:00 F. Flandoli: Uniqueness, singularities and zero-noise limit for stochastic equations - II

It is a well-known phenomenon that an ordinary differential equation becomes ``more regular'', if one adds a noise term, as e.g. a stochastic differential given by a Brownian motion. On the level of the associated Fokker-Planck-Kolmogorov equations (FPKE), whose solutions are just the transition probabilities of the resulting solution process, this becomes more or less obvious, since the FPKE becomes elliptic, if the noise is not degenerate. From a purely analytic point of view, this regularizing property of the noise is most impressively manifested by the fact that noise can ``produce'' (existence and, in particular) uniqueness of solutions . Indeed, e.g. a classical result of A. Yu. Veretennikov (see [1] and the references therein), tells us that, given an initial condition, any two corresponding solutions of an ordinary differential equation in d-dimensional Euclidean space given by a just measurable bounded vector field and perturbed by the differential of a d-dimensional Brownian path, coincide for almost every such path. In contrast to this, in the deterministic case, neither existence nor uniqueness of solutions hold in such a case.

The purpose of this talk is to present recent results of the same type, but for partial differential equations perturbed by noise, i.e. for the infinite dimensional analogue of the situation described above.

### Giovedi 8 Marzo

###### sala seminari, dipartimento di matematica

- 15:00 M. Röckner (Bielefeld Universität): A new view of Fokker-Planck equations in finite and infinite dimensional space
- 16:00 E. Priola (Università di Torino): On the uniqueness theorem of Stroock and Varadhan for non-degenerate diffusions

Fokker-Planck and Kolmogorov (backward) equations can be interpreted as linearisations of the underlying stochastic differential equations (SDE). It turns out that, in particular, on infinite dimensional spaces (i.e. for example if the SDE is a stochastic partial differential equation (SPDE) of evolutionary type), the Fokker-Planck equation is much better to analyze than the Kolmogorov (backward) equation. The reason is that the Fokker-Planck equation is a PDE for measures. Hence e.g. existence of solutions via compactness arguments is easier to show than for PDE on functions. On the other hand uniqueness appears to be much harder to prove.

In this talk we first give a quite elaborate introduction into the relations between S(P)DE, Fokker-Planck and Kolmogorov equations. Subsequently, we shall sketch a new method to prove uniqueness of solutions for Fokker-Planck equations.

**R**

^{n}with continuous and bounded coefficients and multiplicative Brownian noise. We will examine the proof and show in particular the relation between the probabilistic result and the theory of singular integrals by Calderon and Zygmund.

### Giovedi 1 Marzo

###### sala seminari, dipartimento di matematica

- 15:00 G. Zanco: The Ito formula for functionals of paths of a continuous semimartingale and some of its consequences - II
- 16:00 F. Flandoli: Uniqueness, singularities and zero-noise limit for stochastic equations - I

### Giovedi 23 Febbraio

###### aula seminari, dipartimento di matematica applicata

- 16:00 J. Maas (Bonn Universität): Gradient flows of the entropy for finite Markov chains
- 17:00 G. Zanco: The Ito formula for functionals of paths of a continuous semimartingale and some of its consequences - I

**R**

^{n}, as the gradient flow of the entropy in the Wasserstein space of probability measures. In this talk, I will present a discrete counterpart to this result: given a reversible Markov kernel on a finite set, there exists a Riemannian metric on the space of probability measures, for which the law of the continuous time Markov chain evolves as the gradient flow of the entropy.

### Giovedi 9 Febbraio

###### aula magna, dipartimento di matematica

- 15:00 M. Romito: Hausdorff dimension of random curves - III
- 16:00 M. Neklyudov (Tübingen Universität): The role of noise in finite ensembles of nano-magnetic particles

### Lunedi 30 Gennaio

###### aula seminari, dipartimento di matematica applicata

- 15:00 M. Romito: Dimensione di Hausdorff di curve aleatorie - II
- 16:00 D. Trevisan: Disuguaglianze isoperimetriche con i semigruppi - II

### Giovedi 26 Gennaio

###### sala seminari, dipartimento di matematica

- 11:00 M. Romito: Dimensione di Hausdorff di curve aleatorie - I
- 12:00 D. Trevisan: Disuguaglianze isoperimetriche con i semigruppi - I

### Lunedi 16 Gennaio

###### sala seminari, dipartimento di matematica

- 16:00 F. Delarue (Université de Nice Sophia-Antipolis): Weak solvability of SDEs and related PDE theory: smoothing properties and viscosity solutions - I
- 17:00 F. Delarue (Université de Nice Sophia-Antipolis): Weak solvability of SDEs and related PDE theory: smoothing properties and viscosity solutions - II

#### 2011

### Martedi 20 Dicembre

###### sala seminari, dipartimento di matematica

- 16:00 M. Gubinelli (Paris Dauphine): Large Deviation, Gamma convergence and Entropy - I
- 17:00 M. Gubinelli (Paris Dauphine): Large Deviation, Gamma convergence and Entropy - II

### Giovedi 15 Dicembre

###### sala riunioni, dipartimento di matematica

- 11:00 M. Pratelli: Calcolo di Malliavin - IV
- 12:00 M. Maurelli: Grandi deviazioni - IV

### Lunedi 5 Dicembre

###### sala seminari, dipartimento di matematica

- 16:00 M. Maurelli: Grandi deviazioni - III
- 17:00 M. Pratelli: Calcolo di Malliavin - III

### Giovedi 1 Dicembre

###### sala seminari, dipartimento di matematica

- 11:00 M. Pratelli: Calcolo di Malliavin - II
- 12:00 M. Maurelli: Grandi deviazioni - II

### Lunedi 21 Novembre

###### sala seminari, dipartimento di matematica

- 16:00 M. Pratelli: Calcolo di Malliavin - I
- 17:00 M. Maurelli: Grandi deviazioni - I

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###### Stochastic analysis day, March 3, 2015

###### Stochastic analysis day, July 25, 2016

###### Stochastic analysis day, February 27, 2017

###### Workshop for Maurizio, June 29, 2018

###### Speakers cloud

Acciaio Antonelli Aryasova Baglioni Bambi Bernardin Bessaih Bevilacqua Bion-Nadal Blömker Brzezniak Cannizzaro Capasso Cappelletti(UniversityofWisconsin-Madison Caraceni Catellier Cerrai Coghi DeMattei Debussche Delarue Erbar Flandoli Gautier Giuliano Glatt-Holtz Grotto Guasoni Gubinelli Hocquet Jara Leocata Luo Maas Macci Majewski Matetski Maurelli Mueller Nardi Nassif Neklyudov Neves Olivera Orsingher Perkowski Pham Pratelli Priola Proske Röckner Romito Russo Swiech Tankov Touzi Trevisan Tsagkarogiannis Weber Zambotti Zanco