Moving Polymer in a Random Environment (23rit003)

The Banff International Research Station will host the "Moving Polymer in a Random Environment" workshop at the UBC Okanagan campus in Kelowna, B.C., from July 2 to July 16, 2023.

Workshop Report - Click here to download

Recently, we worked on the small ball problem in the setting of the stochastic heat equation (SHE) driven by space-time white noise, that is \be \label{eq:u} \partial_t u = \partial_x^2 u + \sigma(u) \dot W,\;\, t\ge 0, \, x\in [0,1], \ee with periodic boundary conditions and starting with initial profile $u(0,\cdot) \equiv 0$. Let ${\bf P}_0$ be the underlying probability measure on the path space of $u$. The small ball problem concerns asymptotic (as $\epsilon \downarrow 0$) lower and upper bounds on the probability \be {\bf P}_0\left(\sup_{t\in [0,T],\, x\in [0,1]} |u(t,x)|\le \epsilon\right) \ee where $u(t,x)$ is the solution to the SHE. We showed under fairly general conditions on $\sigma$ that the logarithm of the above quantity was of the order $-\epsilon^{-6}$, and the article has now appeared in Annals of Probability.

Our motivation to study the above small ball problem was to understand the behaviour of the moving polymer under the influence of traps or a random field.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).