Abstract
Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.
Highlights
Description of evolution at the kinetic stage is usually expressed in the form of differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes
It is rather obvious that the result is stochastic field theory with fractional derivatives in the dynamic actions
In the analysis of renormalization of the model the important statement is that renormalization produces local counterparts of fractional derivatives with nontrivial scaling dimensions and these generation terms play an important rôle in the renormalization and the subsequent asymptotic analysis of the model
Summary
Description of evolution at the kinetic stage is usually expressed in the form of differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. For instance, due to random advection, random walks with power-law falloff of the step-length distribution or long-tailed distribution of waiting times between consecutive steps. This report is concentrated on the anomalous scaling behaviour in diffusion-limited reactions brought about by powerlike distribution of step length and waiting times. In the position space the fractional power of the Laplace operator ∇2 may be defined with the use of an integral operator (Riesz derivative) In this case the fractional differential operator must be understood in terms of distribution theory [1]. Memory effects may be described by integral operators, which in case of powerlike asymptotic behaviour of waiting-time distribution gives rise to fractional differentiation and integration
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