Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion

Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in <i>x</i> axis and branches in <i>y</i> axis. Then, we find the exact analytical solutions for marginal distribution concerning <i>x</i> and <i>y</i> axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in <i>y</i> direction, i.e., <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">〈</mo> <msup> <mrow> <mo>(</mo> <mo>Δ</mo> <mi>y</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">〉</mo> <mo>∝</mo> <mi>t</mi> </mrow> </semantics> </math> </inline-formula>, and a crossover between sub and ballistic diffusion behaviors in <i>x</i> direction, i.e., <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo stretchy="false">〈</mo> <msup> <mrow> <mo>(</mo> <mo>Δ</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">〉</mo> </mrow> <mo>∝</mo> <msup> <mi>t</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo stretchy="false">〈</mo> <msup> <mrow> <mo>(</mo> <mo>Δ</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo stretchy="false">〉</mo> </mrow> <mo>∝</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula> respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure.