Speaker
Luca Galimberti – King’s College London
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 \(\mbox{div}_h\, u\) bounded in \(L^p_{t,x}\) for \(p>d+2\), where \(d\) is the dimension of \(M\) (we do not assume \(\mbox{div}_h\, u \in L^\infty_{t,x}\)). 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 (\(a_i\) constant). To our knowledge, this is the first instance of “regularization by noise” phenomena beyond \(\mathbb R^d\). The proof is based on an 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).