Generalized arcsine laws for a sluggish random walker with subdiffusive growth

Abstract We study a simple one dimensional sluggish random walk model with subdiffusive growth. In the continuum hydrodynamic limit, the model corresponds to a particle diffusing on a line with a space dependent diffusion constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and a drift potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">|</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> parameterizes the model. For α = 0 it reduces to the standard diffusion, while for α &gt; 0 it leads to a slow subdiffusive dynamics with the distance scaling as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> at late times with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mtext>⩽</mml:mtext> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . In this paper, we compute exactly, for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the full probability distributions of three observables for a sluggish walker of duration T starting at the origin: (i) the occupation time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:math> denoting the time spent on the positive side of the origin, (ii) the last passage time <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mtext>l</mml:mtext> </mml:msub> </mml:mrow> </mml:math> through the origin before T , and (iii) the time t M at which the walker is maximally displaced on the positive side of the origin. We show that while for α = 0 all three distributions are identical and exhibit the celebrated arcsine laws of Lévy, they become different from each other for any α &gt; 0 and have nontrivial shapes dependent on α . This generalizes the Lévy’s three arcsine laws for normal diffusion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) to the subdiffusive sluggish walker model with a general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Numerical simulations are in excellent agreement with our analytical predictions.