Talks

Contractive couplings and entropic curvature of reversible Markov chains (Francesco Pedrotti, Institute of Science and Technology Austria (ISTA))

Ricci curvature lower bounds for Riemannian manifolds have been linked to many functional inequalities: this has motivated the seminal independent works of Sturm [5] and Lott and Villani [3], who extended the notion of curvature lower bound and many of its consequences to a large class of metric measure spaces. In spite of its generality, this theory does not apply to Markov chains on discrete spaces; for this reason, several adapted notions of curvature have been proposed, based on different equivalent characterisations of curvature of Riemannian manifolds. Different notions have different pros and cons: e.g., the entropic curvature of Erbar-Maas [2] is hard to compute in some examples, while Ollivier’s coarse Ricci curvature [4] is not known to imply a modified logarithmic Sobolev inequality.

It is still an open problem to compare these notions. In the present work, adapting arguments of a recent article by G. Conforti [1], we show how contractive couplings (a concept naturally connected to Ollivier’s curvature) can be used to establish entropic curvature lower bounds for some examples of reversible Markov chains.

Monday, December 12 2022, 11.30 Aula Circoscrizione

Pathwise uniqueness for fractional stochastic Volterra equations (Ole Cañadas, Dublin City University)

Motivated by fractional/rough volatility models in Finance, in this talk we investigate fractional stochastic Volterra equations. The question of pathwise uniqueness of solutions to this kind of equation is delicate since we are not dealing with a Markov process or a semi-martingale. Hence, no Ito formulas are applicable. At the moment it is unknown if solutions satisfy a comparison principle. This talk serves as an introduction to the methodology used in the proof of pathwise uniqueness and my current research objective which is the extension to a comparison principle.

Monday, December 12 2022, 12.00 Aula Circoscrizione

Bivariate fractional Ornstein-Uhlenbeck process: basic properties and correlation estimators (Giacomo Giorgio, Roma Tor Vergata)

We define a bivariate fractional Ornstein-Uhlenbeck process (Y^1_t, Y^2_t)_{t∈R} (2fOU). The components of our bivariate process are both two correlated one dimensional fractional Ornstein-Uhlenbeck process, driven respectively by two correlated one dimensional fractional Brownian motions B^{H_1} and B^{H_2}. H1 and H2 are the Hurst indexes of the fBms, and in general H_1 \neq H_2. These fBms are the components of a bivariate fractional Brownian motion (2fBm), defined and explored in [ACLP13].

Starting from the analysis of the covariance structure of the 2fBm in [ACLP13] and from the analysis of the auto-covariance function of

a one dimensional fOU in [CKM], we study the covariance structure of our process 2fOU, proving the power law decay of the cross-covariance

function. We propose an estimator for the correlations parameters that appear in the cross covariance of the 2fBm, then in the cross covariance of the 2fOU, and prove a CLT for these estimators when the Hurst indexes H_1 and H_2 are in a suitable sub-interval of (0, 1).

Monday, December 12 2022, 12.30 Aula Circoscrizione

Equivalence of Sobolev norms in Malliavin spaces (Davide Addona, Università di Parma)

In this talk I present some equivalence of Sobolev norms in Malliavin spaces D^{k,p}, focusing my attention on the L^1-case. In this setting the situation is very challenging since many other techniques (such as Meyer’s inequalities and Wiener chaos decomposition), which work well for p > 1, fail in this limit case. Hence, we provide a different approach relying on a Poincar ́e inequality and a finite dimensional approximations, which give a complete answer when k = 2. The case k ≥ 3 is still open, and we just obtain some improvement of known results.

This is a joint work with Matteo Muratori (Politecnico di Milano) and Maurizia Rossi (Università di Milano Bicocca).

Monday, December 12 2022, 16.30 Aula Circoscrizione

Schauder type theorems for mild solutions to non-autonomous Ornstein-Uhlenbeck equations (Paolo De Fazio, Università di Parma)

TBA

Monday, December 12 2022, 17.00 Aula Circoscrizione

Harnack inequalities with power p ∈ (1, +∞) for transition semigroups associated to dissipative systems (Davide Augusto Bignamini, Università di Pavia)

TBA

Monday, December 12 2022, 17.30 Aula Circoscrizione

A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient (Verena Schwarz, Universität Klagenfurt)

In this talk we present a strong approximation result for the solution of jump-diffusion stochastic differential equations with discontinuous drift, possibly degenerate diffusion coefficient, and Lipschitz jump-diffusion. We construct a transformation-based jump-adapted quasi-Milstein scheme, which has convergence order 3/4 under additional piecewise smoothness assumptions to the drift and diffusion coefficient. To obtain this result we show that under slightly stronger assumptions on the coefficients the jump-adapted quasi-Milstein scheme also has convergence order 3/4.

This is joint work with Paweł Przybyłowicz and Michaela Szölgyenyi.

Tuesday, December 13 2022, 16.30 Aula Circoscrizione

Martingale solutions to the stochastic thin-film equation in two dimensions (Max Sauerbrey, TU Delft)

We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich noise in the physically relevant dimension d=2 and allow in particular for solutions with non full support. The construction relies on a time-splitting scheme, which was recently successfully employed in d=1. The additional analytical challenges due to the higher spatial dimension are overcome using α-entropy estimates and corresponding tightness arguments.

Tuesday, December 13 2022, 17.00 Aula Circoscrizione

Regularization by noise for SDEs with (fractional) Brownian noise and the density of the solution (Lukas Anzeletti, CentraleSupelec - Paris Saclay)

We study existence and uniqueness of solutions to the equation $X_t=b(X_t)dt + dB_t$, where $b$ may be distributional and $B$ is a fractional Brownian motion with Hurst parameter $H\leq 1/2$. We follow two approaches, namely using the stochastic sewing lemma and nonlinear Young integrals in $p$-variation. Furthermore, several results on the density of the solution will be presented. Joint work with Alexandre Richard and Etienne Tanré.

Thursday, December 15 2022, 16.30 Aula Circoscrizione

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation (Florian Bechtold, Universität Bielefeld)

We study pathwise regularization by noise for equations in the plane in the spirit of Catellier and Gubinelli. To this end we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. We then use this concept in order to establish regularization by noise for stochastic differential equations in the plane by providing a space-time regularity estimate for the local time associated with the fractional Brownian sheet. As a further illustration of our results we also prove regularization of 1D wave equations with distributional non-linearity through a noisy boundary.

Based on joint work with Fabian Harang (BI Norwegian Business School) and Nimit Rana (Imperial): https://arxiv.org/abs/2206.05360

Thursday, December 15 2022, 17.00 Aula Circoscrizione