Solvability and exact moment asymptotics for the interpolated stochastic heat and wave equation and the existence of an invariant measure for a special case.
Type of DegreePhD Dissertation
Mathematics and Statistics
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This thesis will consist of two main projects, Chapters 2 and 4, and a smaller project in Chapter 3. We will be studying a general space-time fractional stochastic partial differential equation in Chapters 2 and 3 and the stochastic heat equation in Chapter 4, which is a special case of the just mentioned space-time fractional equation. The aim of this thesis is to handle the following: solvability of the equations, deriving exact moment asymptotics and proving the existence of an invariant measure. In Chapter 2, we study a class of space-time fractional stochastic partial differential equa- tions subject to some time-independent multiplicative Gaussian noise. We derive sharp con- ditions, under which a unique global LppΩq-solution exists for all p ě 2. In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al [BCC22]. In the case when there exists only a local solution, we determine the precise deterministic time, T2, before which a unique L2pΩq-solution exists, but after which the series corresponding to the L2pΩq moment of the solution blows up. By properly choosing the pa- rameters, results in this chapter interpolate the known results for both stochastic heat and wave equations. In Chapter 3, we will again be studying the space-time fractional equation but driven by a space-time white noise. The goal of this project is to show the global existence of the solution when the diffusion term has super-linear growth. The work follows closely a recent work by Millet and Sanz-Solé [MS21]. Chapter 4 deals with the long term behavior of the solution to the nonlinear stochastic heat equation with no drift term that is driven by a Gaussian noise that is white in time and colored in space. Using the theory of the stochastic integral laid out by John Walsh, we provide conditions which will guarantee the existence of an invariant measure for a broad range of initial conditions, which includes bounded L2ρ functions as well as the Dirac delta distribution δ0.