Renormalization theory in statistical physics and stochastic analysis

Abstract

In this thesis we study the theory of renormalization from different perspectives. For the first per- spective, we study the long distance behavior of a model from statistical physics, more precisely the classical dipole gas. We develop a rigorous renormalization group method based on conditional expec- tations and harmonic extensions, and show that the dipole interactions result in renormalized Gaussian behavior at large scales. This large scale Gaussian behavior allows us to control functional integrals associated with the model; for instance we can study the scaling limit of generating functional. Our new renormalization group method is implemented purely in real space, as contrast to earlier meth- ods based on decomposition of Gaussian covariances which usually resort to Fourier space. It has some advantages than earlier methods such as simpler norms. Estimates for decay of Poisson kernels and (derivatives of) Green's functions play the essential role. We can generalize the method to deal with slightly spatially-inhomogeneous situations, such as systems with a boundary. The main result is that the scaling limit of the generating function with smooth test function is equal to the generating function for the the renormalized Gaussian free field.

For the second part, we are concerned with short scale behavior of stochastic partial differential equations (SPDEs). These SPDEs are of parabolic type and with additive white noises, which are very singular random inputs as spatial dimension becomes higher. The main problem is to interpret the nonlinearity at presence of these noices. Renormalization is required to remove the small scale singu- larities in these cases. We perform a systematic study of renormalized powers of Gaussian processes associated with the linearized equations. As an example, we study the Ginzburg-Landau equations, improve the regularity results in earlier works in two dimension, and show local well-posedness for Ginzburg-Landau equation with quadratic nonlinearity in three dimension. This part is a minor mod- ified version of a joint work by E, Jentzen and me.

Then we proceed to discuss the shear flow problem modelled by an SPDE. We use exact RG arguments to recover previous results in all different scaling regimes by Avellaneda and Majda. This example shows that the RG method, if implemented exactly instead of performing drastic truncations, can be a powerful tool to obtain the correct large scale behaviors of such systems.