Functional renormalization group approach to the Kraichnan model

Abstract

We study the anomalous scaling of the structure functions of a scalar field advected by a random Gaussian velocity field, the Kraichnan model, by means of Functional Renormalization Group techniques. We analyze the symmetries of the model and derive the leading correction to the structure functions considering the renormalization of composite operators and applying the operator product expansion.

Forward–backward SDEs with distributional coefficients

“…It turns out that indeed the BSDEs (25) and (26) are equivalent as shown in the following proposition.…”

“…In recent years the Kraichnan model has been researched by physicists also when the velocity field is a stochastic process, see e.g. [26] or [14] and references therein. An example of b that we can treat in this paper is the formal gradient of the realization of some random field (like fractional Brownian noise cut at infinity, but one could consider also other fields not necessarily Gaussian so long as their realizations are α-Hölder continuous with α > 1/2).…”

“…Once this is done, the next challenge is to investigate existence and uniqueness of the solution. To do so, we introduce a transformation -which in some sense can be regarded as the analogous for BSDEs of a Zvonkin transformation for SDEs-and rewrite the original BSDE as an auxiliary backward SDE which can be treated with classical methods, see equation (26). For the auxiliary BSDE it is then possible to show existence and uniqueness of a strong solution, which leads to the same result for the original BSDE (1) by transforming back the equation, see Theorem 15.…”

Why the Cosmological Constant Seems to Hardly Care About Quantum Vacuum Fluctuations: Surprises From Background Independent Coarse Graining

“…More general composite operators O ĝ µν can be included in the gravitational EAA[34][35][36][37][38] by coupling them to independent sources[39][40][41][42][43][44][45].Frontiers in Physics | www.frontiersin.org…”

Operator product expansion coefficients from the nonperturbative functional renormalization group

“…(n,m) k can be obtained from (30) by taking functional derivatives wrt φ and h. The presence of the source h in addition to the field φ allows one to follow the flow of composite fields, an approach which proved to be useful in tackling a wide range of issues [51,[62][63][64][65][66][67][68][69][70].…”