Abstract
In this thesis convergence of the three-dimensional dynamic Ising-Kac model to Φ43 is proved. The analogous two-dimensional result was proved in [MW17], where the authors exploited the so called Da Prato - Debussche trick to find a solution for the continuous equation. Our goal here is to extend the same convergence result to dimension 3, where however it becomes necessary to use newer and different techniques, like the Theory of Regularity Structures introduced by [Hai14], to manage the more irregular solution of the Φ4 equation.Additionally, we also need to work with a discretised version of the Theory of Regularity Structures, as introduced in [EH19]. Discrete models and discrete modelled distributions are the two ingredients introduced in [EH19] that allow a local description of solutions of Interacting Particle Systems (IPSs). In the first part of this work, we provide a theorem for bounding p-moments of iterated integrals with respect to a general class of martingales and we use it to get boundedness results for p-moments of integrals of discrete kernels with respect to the same family of martingales. In many IPSs, these iterated integrals of discrete kernels are called discrete trees and they make up the discrete models; therefore our outcomes allow to get a bound on the norm of the model, as well as to get the convergence results on the norm of the difference of two models.
To prove convergence of the three-dimensional Ising-Kac model to Φ43, we first write the discrete solution as a local expansion, in the sense of [EH19], using the discrete trees as building blocks. The solution theory for the Φ4 equation described in [Hai14] produces another local description which is formally similar to the discrete one, except that in the discrete solution we have the appearance of error terms; moreover, [EH19] and [Hai14] provide us with tools for bounding the difference of two local expansions and with stability arguments with respect to the convergence of the models. Applying our result on the norm of the models and using these arguments, we are therefore able to get our final convergence result.
We would like to stress that the theory we developed for the Ising-Kac model has a more general scope. Alongside with the results coming from [EH19], our theory can be applied to a broad class of IPSs, thus contributing to the establishment of a general and complete framework for the convergence of IPSs to Singular Stochastic Partial Differential Equations (SPDEs).
| Date of Award | 13 Sept 2023 |
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| Original language | English |
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| Supervisor | Hendrik Weber (Supervisor) & Sarah Penington (Supervisor) |