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PDEs & Applications Seminar

Friday, September 27th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Zachary Selk (¾ÅÐãÖ±²¥)

Title: Fictitious Densities of the \Phi^4 measure

Abstract: The \Phi^4 stochastic PDE \partial_t u=\Delta u-u^3+\infty u +\xi where \xi is space-time white noise is a prototypical example of a singular stochastic PDE if the space dimension is at least 2. The singularity is due to the nonlinear cubic term, the fact that SPDEs in space dimension at least 2 have distribution valued solutions, and the product of distributions is in general ill-defined - leading to the need of renormalization. The \infty u term represents a renormalization term needed to handle the ill-definedness of the term u^3, and this renormalization procedure is formalized by Hairer's theory of regularity structures (A theory of regularity structures, Inventiones Mathematicae, 2014) or Gubinelli, Imkeller and Perkowski's theory of paracontrolled calculus (Paracontrolled distributions and singular PDEs, Forum of Math. PI, 2015). 
   
   The \Phi^4 SPDE comes from stochastic quantization. In analogy with Langevin dynamics and SODEs, stochastic quantization is a way of constructing measures on infinite dimensional spaces as the invariant measure of a SPDE. The invariant measure of the \Phi^4 SPDE is an important object in mathematical physics. One potential criticism of this stochastic quantization procedure is that everything is asymptotic and the limiting object is hard to characterize. Barashkov and Gubinelli computed its Laplace transform explicitly (A variational method for \Phi_3^4, Duke Math Journal, 2020) in terms of a stochastic control problem. 
   
   In an ongoing joint work with Ioannis Gasteratos (TU Berlin) we are studying the fictitious density of this measure. In particular we have shown that the Onsager-Machlup function (which plays the role of a density function in infinite dimensions) of the invariant measure is what is expected in dimensions 1 and 2, however we suspect that in dimension 3 the Onsager-Machlup function is infinite everywhere, necessitating further generalizations of fictitious densities. We suspect this due to the measure's singularity with respect to every nonzero shift.