Equations with Boundary Noise

Abstract

In 1993, Da Prato and Zabczyk showed that if one considers the heat equation on the interval (0,1) with white noise Dirichlet boundary conditions then a function-valued mild solution cannot be obtained. In this thesis, we revisit this 'Dirichlet boundary noise problem'. First, we return the deterministic situation and extend the approach of Lasiecka and Balakrishan to the Banach setting to provide a point of comparison for the stochastic case. We then construct an equivalent stochastic theory and show that we cannot obtain Lp- valued solutions. Next we extend to higher-dimensions the idea of Alos and Bonaccorsi: function-valued solutions can be obtained on the half-line if considered in an appropriate weighted Lp space. We also show that a dichotomy is obtained: either a process is obtained on the weighted space or boundary values are understood in terms of traces, but not both. We also study the properties of the Dirichlet heat semigroup on weighted spaces in an attempt to build a foundation for the semigroup approach. Next, from the desire to understand the 'Dirichlet map' of random boundary data we move to the unit disk and perform a stochastic extension of some classic harmonic analysis results. This leads us to work with harmonic Hardy spaces and to consider radial and non-tangential convergence towards the boundary, to obtain an interesting representation theorem, and to finally reconnect our results with weighted spaces. Returning to our problem, we conclude that harmonic Hardy spaces are 'too small' to consider the white noise boundary data. Therefore, what is the appropriate space? We consider this question from two angles. First we take a larger space, specifically the Bloch space, and show that it may be an appropriate place to consider these dynamics. We then extend Makarov's LIL theorem to our stochastic case which gives a rate of blow-up near the boundary of our harmonic random field. We conclude that although the random Bloch dynamics are very interesting, we still do not have clear relationship with the boundary data but only an "inside-out" viewpoint. This inspires a "outside-in" approach whereby, due to a simple representation, we start with the white-noise data on the boundary and moving inwards we show rates of blow-up near typical and exceptional points. Finally, we show how these concepts may be extended to more general situations to provide a framework for understanding the local spatial dynamics of SPDE.