Periodic Reporting for period 1 - GE4SPDE (Global Estimates for non-linear stochastic PDEs)
Periodo di rendicontazione: 2022-10-01 al 2025-03-31
The project is concerned with the global behaviour of solutions to Stochastic Partial Differential Equations (SPDEs) from Mathematical Physics which arise e.g. in the description of scaling limits of interacting particle systems and in the analysis of Quantum Field Theories. The equations contain noise terms which describe random fluctuations and act on all length scales. In this situation the presence of a non-linear term can lead to divergencies. A subtle renormalisation procedure, which amounts to removing infinite terms, is needed.
Over the last years the understanding of non-linear SPDEs has been revolutionised and a systematic treatment of the renormalisation procedure has been achieved. This led to a short-time well-posedness theory on compact domains for a large class of highly relevant semi-linear SPDEs. In this project, I will describe the global - both in time and over infinite domains - behaviour of solutions of some of the most prominent examples, by combining PDE techniques for the non-linear equations without noise and the improved understanding of the subtle small-scale stochastic cancellations. I have already pioneered such a programme in an important special case, the dynamic Phi-4 model.
The project has three specific strands: A) Proving estimates for the stochastic quantisation equations of the Sine-Gordon and Liouville Quantum Gravity models and eventually Gauge theories, and giving a PDE-based approach to the celebrated 1-2-3 scaling of the KPZ equation, B) giving PDE-based constructions of Phi-4 models in fractional dimension and describing phase transitions in terms of mixing properties of the dynamics, C) treating degenerate parabolic equations and exploring if systems that fail to satisfy a fundamental "sub-criticality" scaling assumption can still be treated using SPDE techniques.
Over the last years the understanding of non-linear SPDEs has been revolutionised and a systematic treatment of the renormalisation procedure has been achieved. This led to a short-time well-posedness theory on compact domains for a large class of highly relevant semi-linear SPDEs. In this project, I will describe the global - both in time and over infinite domains - behaviour of solutions of some of the most prominent examples, by combining PDE techniques for the non-linear equations without noise and the improved understanding of the subtle small-scale stochastic cancellations. I have already pioneered such a programme in an important special case, the dynamic Phi-4 model.
The project has three specific strands: A) Proving estimates for the stochastic quantisation equations of the Sine-Gordon and Liouville Quantum Gravity models and eventually Gauge theories, and giving a PDE-based approach to the celebrated 1-2-3 scaling of the KPZ equation, B) giving PDE-based constructions of Phi-4 models in fractional dimension and describing phase transitions in terms of mixing properties of the dynamics, C) treating degenerate parabolic equations and exploring if systems that fail to satisfy a fundamental "sub-criticality" scaling assumption can still be treated using SPDE techniques.
A priori bounds that show non-explosion of solutions were shown for a non-linear parabolic Anderson equation. This equation was among the very first treated within the framework of regularity structures, and the local-in-time solution theory was established in Hairer’s original work on regularity structures. Among the applications global solutions to the stochastic quantisation equation for the Sine-Gordon model were constructed for a certain parameter range (including construction of the invariant measure) and construction of stochastic flows for several classical SPDEs with multiplicative noise.
The stochastic quantisation equation for the (\varphi^4)_{3,\epsilon} model was treated. The equation is given by a non-linear fractional heat equation with a driving space-time white noise. Strong a priori estimates that exclude the possibility of finite time blow-up were established. To this end, a new Regularity-Structures Schauder theory for the fractional heat operator was developed and ideas developed for the (local) heat operator were adapted to the fractional case.
A method to rigorously derive non-linear stochastic PDEs as scaling limits of interacting particle systems was developed. More specifically, iterated stochastic integrals with respect to discrete jump-martingales are analysed with the aim to derive discrete counterparts of the continuous model bounds encountered in the theory of regularity structures and this method was used to rigorously establish the convergence of a certain long-range Ising model to the \varphi^4 SPDE in 3 dimensions. In a forthcoming work, similar techniques are used to give a direct, regularity structure-based proof of the classical convergence result of a weakly asymmetric simple exclusion process to the KPZ equation.
Substantial effort has been made towards developing a theory of quasilinear degenerate singular SPDEs and towards understanding equations which are critical in the sense of regularity structures. Results in both directions are forthcoming.
The stochastic quantisation equation for the (\varphi^4)_{3,\epsilon} model was treated. The equation is given by a non-linear fractional heat equation with a driving space-time white noise. Strong a priori estimates that exclude the possibility of finite time blow-up were established. To this end, a new Regularity-Structures Schauder theory for the fractional heat operator was developed and ideas developed for the (local) heat operator were adapted to the fractional case.
A method to rigorously derive non-linear stochastic PDEs as scaling limits of interacting particle systems was developed. More specifically, iterated stochastic integrals with respect to discrete jump-martingales are analysed with the aim to derive discrete counterparts of the continuous model bounds encountered in the theory of regularity structures and this method was used to rigorously establish the convergence of a certain long-range Ising model to the \varphi^4 SPDE in 3 dimensions. In a forthcoming work, similar techniques are used to give a direct, regularity structure-based proof of the classical convergence result of a weakly asymmetric simple exclusion process to the KPZ equation.
Substantial effort has been made towards developing a theory of quasilinear degenerate singular SPDEs and towards understanding equations which are critical in the sense of regularity structures. Results in both directions are forthcoming.
Previous works that show a priori bounds for non-linear SPDEs in the framework of regularity structures had relied strongly on a non-linear damping term. The work on the non-linear parabolic Anderson model described above shows how to derive a priori bounds for a non-linear equation without such a damping mechanism. The work builds on the (unexpected) observation that the non-linear terms one encounters can be re-interpreted as transport terms, which in turn permits to invoke a maximum principle that yields an L^\infty bound.
The work on the fractional phi^4 model relies on novel regularity estimates for fractional heat operators in the context of regularity structures. It was well known in the community that the fractional heat operator does not satisfy the assumptions from the original theory of regularity structure by Hairer. We proved an optimal multi-level Schauder estimate for this operator using a new argument thereby closing this gap.
The work on Scaling limits develops a new method to systematically derive regularity structure model bounds in the context of discrete particle systems. This opens the door to a systematic study of more complicated systems (including the forthcoming work on the KPZ equation mentioned above).
The work on the fractional phi^4 model relies on novel regularity estimates for fractional heat operators in the context of regularity structures. It was well known in the community that the fractional heat operator does not satisfy the assumptions from the original theory of regularity structure by Hairer. We proved an optimal multi-level Schauder estimate for this operator using a new argument thereby closing this gap.
The work on Scaling limits develops a new method to systematically derive regularity structure model bounds in the context of discrete particle systems. This opens the door to a systematic study of more complicated systems (including the forthcoming work on the KPZ equation mentioned above).