Regularization by noise via stochastic sewing with random controls

Speaker: Dr Oleg Butkovsky (Weierstrass Institute, Berlin)

Abstract: (Joint work with Siva Athreya, Khoa Le, and Leonid Mytnik)

I would like to present a new tool in stochastic analysis, stochastic sewing with random controls, which seems to be quite powerful for studying regularization by noise of SDEs and SPDEs.
Our first example is the stochastic differential equation
dX_t=\delta_0(X_t)dt + d B_t^H,
where $\delta_0$ is the Dirac delta function, $B^H$ is a fractional Brownian motion, $H\in(0,1)$. For $H=1/2$ this equation becomes a skew Brownian motion and its strong existence and uniqueness was shown in [2] using the Zvonkin-Veretennikov transformation method. If $H\neq1/2$, then the Zvonkin-Veretennikov transformation method is not applicable. On the other hand, one can conjecture from simple general considerations, that this equation should have a unique strong solution for $H<1/2$ as well. This conjecture was proven in [3,4] but only for $H<1/4$. Stochastic sewing with random controls allowed us to obtain strong existence and uniqueness for $H<1/3$.

If time permits, I will show how the same proof technique and related ideas allows to show existence and uniqueness of solution of stochastic heat equation with $L_p$ drift for $p\ge1$:
\partial_t u =\Delta u +b(u) +\dot{W),
where $\dot{W}$ is a space-time white noise. This extends [5], where this result was obtained only for $p>2$. At the end of the talk I will formulate a number of open problems related to regularization by noise.

[1] S. Athreya, O. Butkovsky, K. Le, L. Mytnik (2020). Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation. arXiv:2011.13498.
[2] J. Harrison, L. Shepp (1981). On skew Brownian motion. Annal. Probab., 9, 309-313.
[3] R. Catellier, M. Gubinelli (2012). Averaging along irregular curves and regularisation of ODEs. arXiv:1205.1735.
[4] D. Banos, S. Ortiz-Latorre, A. Pilipenko, F. Proske (2017). Strong solutions of SDE’s with generalized drift and multidimensional fractional Brownian initial noise. arXiv:1705.01616
[5] I. Gyongy, E. Pardoux (1993). On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations. Probab. Theory Related Fields, 97, 211–229.  



The talk will take place online over zoom. If you are interested in attending, you can write us an email here or here and we will send you the meeting ID and password.