Θέμα: Γενικό Σεμινάριο Τμήματος Ομιλία 18/12/24

Το Γενικό Σεμινάριο του Τμήματος συνεχίζεται αύριο, Τετάρτη 18/12/24, ώρα 13:00 - 14:00 στην αίθουσα 342 (προσοχή στην αλλαγή της ώρας). Η ομιλία θα γίνει στα ελληνικά.

Τα στοιχεία της ενδέκατης ομιλίας:
Ομιλητής: Konstantinos Dareiotis (University of Leeds)
Τίτλος: Regularisation by Gaussian Rough path lifts of fractional Brownian motions

Περίληψη: Regularisation by noise refers to the phenomenon of certain non-linear dynamical systems behaving better in the presence of a noisy (stochastic) perturbation compared to their deterministic counterpart. In this talk, we will discuss such phenomena for differential equations of the form
$$
dX_t=f(X_t) \, dt, \qquad X_0 =x.
$$
It is well known that the above equation admits a unique solution if $f$ is Lipschitz continuous, and this is essential sharp: if $f$ is only $\alpha$-H"older continuous for some $\alpha \in (0,1)$, one might have infinitely many solutions and if $f$ is not even continuous, it might not have solutions at all. We will consider equations of the form
$$
dX_t=f(X_t) \, dt+ \sigma(X_t) \, dB^H_t, \qquad X_0 =x,
$$
where $B^H$ is a fractional Brownian motion of Hurst parameter $H \in (1/3,1/2)$. We will see that this equation admits a unique solution for quite irregular $f$, provided that $\sigma$ is bounded away from zero. More precisely, $f$ does not even need to be a function but merely a Schwartz distribution of regularity $\alpha > 1-(1/2H)$ (notice that $\alpha$ can be negative) in the Besov scale $\mathcal{B}^\alpha_{\infty, \infty}$. We will discuss what is meant by a solution in this case and we will present the main ideas which rely on the theory of rough paths, Malliavin calculus, and the stochastic sewing lemma. Our result provides a multiplicative noise analogue to a result of Catellier-Gubinelli in 2016. The talk is based on joint work with M. Gerencs\'er, K. L\^e, and C. Ling.


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