Page 571 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 571
STOCHASTIC EVOLUTION EQUATIONS (MS-68)
value function V is rigorously obtained, which means that V can be viewed as a function on the
Wasserstein space of probability measures on the set of continuous functions valued in Hilbert
space. We then define a notion of pathwise measure derivative, which extends the Wasserstein
derivative due to P.L.Lions (2006), and prove a related functional Itô formula in the spirit of
Dupire (2009) and Wu-Zhang (2020). The Master Bellman equation is derived from the DPP
by means of a suitable notion of viscosity solution. We provide different formulations and sim-
plifications of such a Bellman equation notably in the special case when there is no dependence
on the law of the control.
Stochastic heat equations with distributional drifts
Khoa Le, khoa.le.n@gmail.com
TU Berlin, Germany
Coauthors: Siva Athreya, Oleg Butkovsky, Leonid Mytnik
We consider stochastic heat equations on the real line with space-time white noise and distri-
butional drifts in negative-indexed Besov spaces. It is shown that a weak solution exists if the
regularity index is at least -3/2 while pathwise uniqueness holds of the regularity index is at least
-1. In the particular case when the drift is a Dirac mass at a point, the equation has a unique
strong solution. These results are obtained using the stochastic sewing lemma introduced by the
speaker in 2018. This is a joint work with S. Athreya, O. Butkovsky and L. Mytnik.
Regularization by noise of semilinear stochastic damped wave equations
with Hölder continuous coefficients
Federica Masiero, 19fede75@gmail.com
Università di Milano Bicocca, Italy
We prove that semilinear stochastic abstract wave equations and damped wave equations are
well-posed in the strong sense with an α-Hölder continuous drift coefficient, if α ∈ (2/3, 1).
The uniqueness may fail for the corresponding deterministic PDE and well-posedness is re-
stored by considering an additive pertubation of white noise type which describes an external
random forcing. This shows that a kind of regularization by noise holds for the semilinear wave
equation.
In the proof we adopt an approach based on backward stochastic equations and use non-standard
regularizing properties for the transition semigroup associated to the stochastic wave equation;
these properties are based on control theoretic results.
We finally briefly discuss how our method applies also to stochastic evolution of parabolic type
The talk is based on joint works with D. Addona and E. Priola
569
value function V is rigorously obtained, which means that V can be viewed as a function on the
Wasserstein space of probability measures on the set of continuous functions valued in Hilbert
space. We then define a notion of pathwise measure derivative, which extends the Wasserstein
derivative due to P.L.Lions (2006), and prove a related functional Itô formula in the spirit of
Dupire (2009) and Wu-Zhang (2020). The Master Bellman equation is derived from the DPP
by means of a suitable notion of viscosity solution. We provide different formulations and sim-
plifications of such a Bellman equation notably in the special case when there is no dependence
on the law of the control.
Stochastic heat equations with distributional drifts
Khoa Le, khoa.le.n@gmail.com
TU Berlin, Germany
Coauthors: Siva Athreya, Oleg Butkovsky, Leonid Mytnik
We consider stochastic heat equations on the real line with space-time white noise and distri-
butional drifts in negative-indexed Besov spaces. It is shown that a weak solution exists if the
regularity index is at least -3/2 while pathwise uniqueness holds of the regularity index is at least
-1. In the particular case when the drift is a Dirac mass at a point, the equation has a unique
strong solution. These results are obtained using the stochastic sewing lemma introduced by the
speaker in 2018. This is a joint work with S. Athreya, O. Butkovsky and L. Mytnik.
Regularization by noise of semilinear stochastic damped wave equations
with Hölder continuous coefficients
Federica Masiero, 19fede75@gmail.com
Università di Milano Bicocca, Italy
We prove that semilinear stochastic abstract wave equations and damped wave equations are
well-posed in the strong sense with an α-Hölder continuous drift coefficient, if α ∈ (2/3, 1).
The uniqueness may fail for the corresponding deterministic PDE and well-posedness is re-
stored by considering an additive pertubation of white noise type which describes an external
random forcing. This shows that a kind of regularization by noise holds for the semilinear wave
equation.
In the proof we adopt an approach based on backward stochastic equations and use non-standard
regularizing properties for the transition semigroup associated to the stochastic wave equation;
these properties are based on control theoretic results.
We finally briefly discuss how our method applies also to stochastic evolution of parabolic type
The talk is based on joint works with D. Addona and E. Priola
569
