Arcsine laws for Brownian motion with Poissonian resetting

TL;DR

This work generalizes the classical arcsine laws for Brownian motion to the case of Poissonian resetting at rate $r$ to $x_r=0$. It derives closed-form PDFs for the first arcsine variable $T_r$ and the second arcsine variable $L_r$ by conditioning on the number of resets $N(1)$ and exploiting Beta and mixture structures, and it provides numerical analysis for the third law variable $M_r$. The results show that $T_r|N(1)=k$ is Beta$((k+1)/2,(k+1)/2)$ and yield explicit forms for $p_{T_r}(t)$, while $p_{L_r}(t)$ is a convex mixture of the classical arcsine density and an auxiliary gamma-based term, with computable moments and asymptotic behavior. For $M_r$, an analytical expression remains open due to inter-reset interval dependence, but numerical evidence indicates convergence toward a Uniform$(0,1)$ distribution as $r$ increases, with a nontrivial mean peak around $r oughly 3.5$.

Abstract

We analyze the equivalents of the celebrated arcsine laws for Brownian motion undergoing Poissonian resetting. We obtain closed-form formulae for the probability density functions of the corresponding random variables in the cases of the first and second arcsine law. Furthermore, we obtain numerical results for the third law.

Figures (7)

• Figure 1: Monte-Carlo estimated distribution of $T_r|N(t)=k$\ref{['eq:Tr']}, i.e. the time the Wiener process under resetting spends above the $X$-axis when $k$ resets occurred, with sample size $10^6$ compared to the theoretical result \ref{['eq:TKFinal']}. The theoretical PDF seems to perfectly fit the simulations. With increasing number of resets $k$ we can observe the convergence of the distribution to a point-mass concentrated at $1/2$.
• Figure 2: Monte-Carlo estimated distribution of $T_r$\ref{['eq:Tr']}, i.e. the time the Wiener process under resetting with rate $r$ spends above the $X$-axis, using Wiener process trajectories with sample size $10^5$ and timestep $10^{-4}$. In red we see the theoretical result \ref{['eq:TrResult']}. The quality of the fit visually seems perfect. In accordance with the case presented on figure \ref{['fig:Tk']}, when the resetting rate increases, the distribution approaches a point-mass at $1/2$.
• Figure 3: Monte-Carlo estimated $j$th moments of $\tilde{T}_r$\ref{['eq:Tr']} (crosses), their theoretical values \ref{['eq:TrMom']} and the standard deviation (whiskers) of the Monte Carlo simulations. We can notice a substantial decrease in the standard deviation due to $\abs{\tilde{T}_r}\leq 1$.
• Figure 4: Monte-Carlo estimated $j$th moments of $L_r$\ref{['eq:Lr']} (crosses), their theoretical values \ref{['eq:Lmom']} (circles) and the standard deviation (whiskers) of the Monte Carlo simulations.
• Figure 5: Monte-Carlo estimated distribution of $L_r$\ref{['eq:Lr']}, i.e. the last zero of the Wiener process under resetting with rate $r$, using Wiener process trajectories with sample size $10^5$ and timestep $10^{-4}$ compared to the analytical result \ref{['eq:LrPDF']} on a logarithmic scale. Again, we observe a very good fit.