Extreme value statistics of first-passage trajectories of resetting Brownian motion in an interval

Abstract In this work, we consider one-dimensional resetting Brownian motion confined in a unit interval with absorbing boundaries at both ends. The Brownian particle is reset to the starting position x 0 at random times but with a constant rate r , and the process terminates whenever the particle touches any of absorbing boundaries. We study the statistics of the span S , defined as the size of the visited domain during the first-passage process, and the time t m at which the maximum (minimum) displacement is achieved when the particle is absorbed by the left (right) end. We analytically obtain the distribution of S and its mean value. The distribution of S can be significantly altered by the resetting, and the mean value of S can increase, decrease, or remain insignificant as r increases, depending on the value of x 0 . Moreover, we find that the mean value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> has a highly nontrivial dependence on r . When x 0 is close to the boundary, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> can be reduced by the resetting, and a nonzero resetting rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi>opt</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> exists for which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> attains its minimum. Otherwise, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> </mml:math> has its minimum when the resetting is absent. We observe that the transition occurs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>0.180</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mn>67</mml:mn> </mml:mrow> </mml:math> (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>0.819</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mn>33</mml:mn> </mml:mrow> </mml:math> ). Such a transition is first-order, characterized by an abrupt jump of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>r</mml:mi> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi>opt</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> from a nonzero value to zero at the transition point.