Probability Seminar: Luca Galimberti - WELL-POSEDNESS OF STOCHASTIC CONTINUITY EQUATIONS ON RIEMANNIAN MANIFOLDS
| Dates: | 18 March 2026 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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Luca Galimberti (King's College London) will speak at the Probability seminar.
Title: WELL-POSEDNESS OF STOCHASTIC CONTINUITY EQUATIONS ON RIEMANNIAN MANIFOLDS
Abstract: We analyze continuity equations with Stratonovich stochasticity on a smooth closed and compact Riemannian manifold $M$ with metric $h$. The velocity field $u$ is perturbed by Gaussian noise terms $\dot W_1(t), \ldots, \dot W_N(t)$ driven by smooth spatially dependent vector fields $a_1(x), \ldots, a_N(x)$ on $M$. The velocity $u$ belongs to $L^1_t W^{1,2}_x$ with $\mathrm{div}_h\, u$ bounded in $L^p_{t,x}$ for $p > d+2$, where $d$ is the dimension of $M$ (we do not assume $\mathrm{div}_h\, u \in L^\infty_{t,x}$). We show that by carefully choosing the noise vector fields $a_i$ (and the number $N$ of them), the initial-value problem is well-posed in the class of weak $L^2$ solutions, although the problem can be ill-posed in the deterministic case because of concentration effects. The proof of this ``regularization by noise'' result reveals a link between the nonlinear structure of the underlying domain $M$ and the noise, a link that is somewhat hidden in the Euclidean case (when the $a_i$ are constant). To our knowledge, this is the first instance of ``regularization by noise'' phenomena beyond $\mathbb{R}^d$. The proof is based on an \emph{a priori} estimate in $L^2$, which is obtained by a duality method, and a weak compactness argument.
This is a joint work with Kenneth Karlsen (UiO).
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