Weak uniqueness for singular stochastic equations driven by fractional Brownian motion
A talk in the Bielefeld Stochastic Afternoon series by
Oleg Butkovsky
| Abstract: | Joint work with Leonid Mytnik (Technion - Israel Institute
of Technology). We consider the stochastic differential equation
$$
dX_t = b(X_t) dt + dB_t^H,
$$,
where the drift $b$ is a Schwartz distribution in the space
$\mathcal{C}^\alpha$, $\alpha < 0$, and $B^H$ is a fractional Brownian
motion of Hurst index $H \in (0, 1/2]$. If $H = 1/2$, both weak and
strong uniqueness theories for this SDE have been developed. However,
the situation is much more complicated if $H \neq 1/2$, as the main
tool, the Zvonkin transformation, becomes unavailable in this setting.
The breakthroughs by Catellier and Gubinelli, and later by Le,
established strong well-posedness of this SDE via sewing/stochastic
sewing arguments. However, weak uniqueness for this SDE remained a
challenge for quite some time, since a direct application of
stochastic sewing alone does not seem very fruitful. I will explain
how a combination of stochastic sewing with certain arguments from
ergodic theory allows to show weak uniqueness in the whole regime
where weak existence is known, that is, $\alpha > 1/2 - 1/(2H)$. If
time permits, we will discuss weak uniqueness for rough SDEs
$$
d X_t = \sigma(X_t) d B_t^H,
$$,
where $\sigma$ is a Hölder continuous function. [1] O. Butkovsky, L. Mytnik (2024). Weak uniqueness for singular stochastic equations. arXiv preprint arXiv:2405.13780. Within the CRC this talk is associated to the project(s): B1 |