Stochastics Seminar

We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property (CSP) refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

It has been a long-standing goal (of mine) to understand the large-$N$ behavior of eigenvalues of Brownian motion on $\mathrm{GL}(N,\mathbb{C})$. For $N>2$ these random matrices are almost surely non-normal, which makes the task quite difficult due to the unstable properties of pseudo-spectrum. Over the last decade, we have identified what the large-$N$ limit should be, but the tools to prove convergence evaded us. In two recent projects with overlapping authors, we have fully resolved this problem. Moreover, the eigenvalues of a broad family of matrix random walk approximations to Brownian motion have also been fully analyzed, and produced an unexpectedly powerful scaling limit: almost regardless of covariance, if the random walk steps are unitarily bi-invariant, the large-$N$ limit eigenvalue distribution always matches the Brownian motion (and in fact exhibits superconvergence). When the process starts at the identity matrix, the spectrum looks like a lima bean. This is joint work with Tanya Brailovskaya, Nick Cook, Bruce Driver, Brian Hall, Ching Wei Ho, Yuriy Nemish, Vaki Nikitopoulos, and Felix Parraud.

In this talk, we motivate a variant of optimal transport theory, namely causal optimal transport (COT), and its induced (adapted) Wasserstein distance as a tool suitable for stochastic analysis problems. In particular, we demonstrate that these transports between the laws of many commonly encountered processes (Brownian motions, stochastic differential equations and their fractional variants) admit explicit characterizations and are quite rigid. For example, all (bi-)causal transport maps between the laws of Brownian motion are described by the stochastic integral of rotation-valued integrands. As a corollary, we prove that classical results from optimal transport theory remain true in this causal setting (e.g, the density of transport maps among transport couplings) and show how to explicitly solve the (bi-)causal transport problem between laws of scalar stochastic equations and Gaussian processes.

For 0 Nevertheless, many properties of infinite-dimensional Gaussian process can be seen to hold analogously for SLE(κ) loop measures. In this talk, I will discuss a Cameron–Martin type quasi-invariance result for these measures. In place of translations, we consider quasiconformal deformations of SLE loops. These form a natural group action of the Weil–Petersson Teichmüller space on SLE loops via compositions of conformal welding homeomorphisms.

I will recall the stationary measure of the 1D KPZ equation on an interval. I will then describe the fluctuations of the KPZ height in the large time limit. I will obtain the associated cumulants by two methods, one is by a limit from the ASEP, the other via a direct replica method introduced by Brunet and Derrida. If time permits I will indicate how an extension of this second method also allows to obtain large time formulas for the case of two non-crossing polymers on the cylinder. The first works are in collaboration with G. Barraquand (published), the last one with A. De Luca (in preparation).