Nested Stochastic Resetting: Nonequilibrium Steady States and Exact Correlations

Stochastic resetting breaks detailed balance and drives the formation of nonequilibrium steady states. Here, we consider a chain of diffusive processes <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>x</a:mi></a:mrow><a:mrow><a:mi>i</a:mi></a:mrow></a:msub><a:mo stretchy="false">(</a:mo><a:mi>t</a:mi><a:mo stretchy="false">)</a:mo></a:mrow></a:math> that interact unilaterally: at random time intervals, the process <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>x</e:mi><e:mi>n</e:mi></e:msub></e:math> stochastically resets to the instantaneous value of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>x</g:mi><g:mrow><g:mi>n</g:mi><g:mo>−</g:mo><g:mn>1</g:mn></g:mrow></g:msub></g:math>. We derive analytically the steady-state statistics of these nested stochastic resetting processes including the stationary distribution for each process as well as its moments. We are also able to calculate exactly the steady-state two-point correlations <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">⟨</i:mo><i:msub><i:mi>x</i:mi><i:mi>n</i:mi></i:msub><i:msub><i:mi>x</i:mi><i:mrow><i:mi>n</i:mi><i:mo>+</i:mo><i:mi>j</i:mi></i:mrow></i:msub><i:mo stretchy="false">⟩</i:mo></i:math> between processes by mapping the problem to one of the ordering statistics of random counting processes. Understanding statistics and correlations in many-particle nonequilibrium systems remains a formidable challenge and our results provide an example of such tractable correlations. We expect this framework will both help build a model-independent framework for random processes with unilateral interactions and find immediate applications, e.g., in the modeling of lossy information propagation.