First-passage functionals for Ornstein–Uhlenbeck process with stochastic resetting

Abstract

Abstract We study the statistical properties of first-passage Brownian functionals (FPBFs) of an Ornstein–Uhlenbeck process in the presence of stochastic resetting. We consider a one dimensional set-up where the diffusing particle sets off from x 0 and resets to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> </mml:msub> </mml:math> at a certain rate r . The particle diffuses in a harmonic potential (with strength k ) which is centered around the origin. The center also serves as an absorbing boundary for the particle and we denote the first passage time (FPT) of the particle to the center as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:msub> </mml:math> . In this set-up, we investigate the following functionals: (i) local time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">l</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>τ</mml:mi> <mml:mtext> </mml:mtext> <mml:mi>δ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> i.e. the time a particle spends around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> </mml:msub> </mml:math> until the first passage, (ii) occupation or residence time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>0</mml:mn> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>τ</mml:mi> <mml:mtext> </mml:mtext> <mml:mi>θ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> i.e. the time a particle typically spends above <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> </mml:msub> </mml:math> until the first passage and (iii) the FPT <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">f</mml:mi> </mml:mrow> </mml:msub> </mml:math> to the origin. We employ the Feynman–Kac formalism for renewal process to derive the analytical expression for the first moment of all the three FPBFs mentioned above. In particular, we find that resetting can either prolong or shorten the mean residence and FPT depending on the system parameters. The transition between these two behaviors or phases can be characterized precisely in terms of optimal resetting rates, which interestingly undergo a continuous transition as we vary the trap stiffness k . We characterize this transition and identify the critical-parameter and -coefficient for both the cases. We also showcase other interesting interplay between the resetting rate and potential strength on the statistics of these observables. Our analytical results are in excellent agreement with the numerical simulations.

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