Novel Approaches to Renormalization Group Transformations in the Continuum and on the Lattice

This thesis is about new methods of achieving RG transformations, in both a continuum spacetime background and on a lattice discretization thereof. The subject is explored from the point of view of euclidean quantum field theory. As a thesis grounded on the computational method of lattice simulation, I emphasize the role of lattice formulations throughout the work, especially in the first two chapters. In the first, I describe the essential aspects of lattice theory and its symbiosis with RG. In the second, I present a new, continuous approach to RG on the lattice, based on a numerical tool called Gradient Flow (GF). Simulation results from quartic scalar field theory in 2 and 3 dimensions ($\phi^4_d$) and 4-dimensional 12-flavor SU(3) gauge theory will be presented. In the third and fourth chapters, the focus becomes more analytic. Chapter 3 is an introductory review of Functional Renormalization Group (FRG). In chapter 4, I introduce the concept of Stochastic RG (SRG) by working out the relationship between FRG and stochastic processes.