First-passage functionals for Ornstein Uhlenbeck process with stochastic resetting

Abstract

We study the statistical properties of first-passage Brownian functionals (FPBFs) of an Ornstein-Uhlenbeck (OU) 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 $x_R$ 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 of the particle to the center as $t_f$. In this set-up, we investigate the following functionals: (i) local time $T_{loc} = \int _0^{t_f}d τ~ δ(x-x_R)$ i.e., the time a particle spends around $x_R$ until the first passage, (ii) occupation or residence time $T_{res} = \int _0^{t_f} d τ~θ(x-x_R)$ i.e., the time a particle typically spends above $x_R$ until the first passage and (iii) the first passage time $t_f$ 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 first passage time 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 \& -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.

Figures (4)

• Figure 1: Comparison of the first moment of $T_{loc}(x_R)$ for OU process with numerical simulation for different values of (a) stiffness constant $k$ and (b) resetting rate $r$. Here, $x_0=x_R=2.5$. The lines and symbols are analytical and numerical results respectively.
• Figure 2: Comparison of the theoretical expressions for the first moment of (a) local time density $\langle T_{loc}\rangle|_{r = 0}$ and (b) residence time $\langle T_{res}\rangle|_{r = 0}$ as a function of stiffness constant $k$ in the absence of resetting with numerical simulation. The lines and symbols are analytical and numerical simulations results respectively. We have set: $x_0=2.5$.
• Figure 3: Comparison of the first moment of residence time $\langle T_{res}(x_R)\rangle$ for OU process with numerical simulation as a function of (a) stiffness constant $k$ for different values of $r$ and, (b) resetting rate $r$ for different values of $k$. The lines and symbols are analytical and numerical simulations results respectively. (c) Continuous transition of optimal resetting rate $r^c$, which from a finite value transits to zero at the critical point $k^c = 0.22624$. The locus of $r^c$ (solid line) is obtained by numerically minimizing $\langle T_{res}(x_R)\rangle$ for different values of $k$. In particular, the diamond and square markers depict $r^c>0$ and $r^c=0$ respectively obtained by minimizing $\langle T_{res}(x_R)\rangle$ from \ref{['Resi-moms-Fig']}(b) for two different values of $k$. Inset: $r^c$ near the critical point $k^c$ shows a power law behavior. In the simulation, we have set: $x_0=x_R=2.5$ and $D=1$.
• Figure 4: Comparison of the MFPT $\langle t_f(r) \rangle$ for OU process with numerical simulations for different values of (a) resetting rate $r$ and (b) spring constant $k$. The lines and symbols are analytical and numerical simulation results respectively. (c) Second order like phase transition of the optimal resetting rate $r^*$ which continuously reaches to zero from a finite value at the critical point $k^* = 0.1749$. The locus of $r^*$ (solid line) is obtained by numerically minimizing $\langle t_f(r) \rangle$ for different values of $k$. In particular, the diamond and square markers depict $r^*>0$ and $r^*=0$ respectively obtained by minimizing $\langle t_f(r)\rangle$ from \ref{['FPT-moms-Fig']}(b) for two different values of $k$. Inset: Optimal resetting rate $r^*$ close to the critical point $k^*$ shows a power law behavior $r^* \sim |k-k^*|^\alpha$ with $\alpha=1$. (d) Phase diagram for resetting transition and verification of the critical point $k^*$ from the $CV>1$ criterion. The $CV$ as a function of $k$ intersects unity (dashed horizontal line) exactly at the value $k^* = 0.174895$. For $k>k^*$ i.e., when $CV<1$, resetting is detrimental while for $k<k^*$ i.e., when $CV>1$, resetting always remains beneficial. This is in accordance with Fig. \ref{['FPT-moms-Fig']}(c). In the simulation, we have set: $x_0=x_R=2.5$ and $D=1$.