Path-integrals and optimal paths for the fractional Ornstein-Uhlenbeck process

TL;DR

Addresses deriving a path-integral description of the non-Markovian fractional Ornstein-Uhlenbeck process driven by nonstationary RL fGn on finite intervals. Develops two equivalent quadratic actions: one in terms of fractional integrals and another as a nonlocal kernel double integral, yielding $P[x_t]\propto\exp(-S)$. Derives closed-form, covariance-based optimal paths conditioned to reach $X$ at $T'$ and reveals subdiffusive overshoot under strong confinement, with monotone approach in the superdiffusive case. The results provide a memory-aware framework for constrained relaxation applicable to viscoelastic transport and intracellular dynamics.

Abstract

We derive the path-integral representation of the fractional Ornstein-Uhlenbeck process driven by Riemann-Liouville fractional Gaussian noise, for both the subdiffusive and superdiffusive regimes. We express the corresponding action, which is a quadratic functional of individual trajectories of the process, in two alternative but equivalent forms: either as a fractional integral or as a double integral with a nonlocal kernel. Moreover, we determine in closed form the optimal (action-minimizing) paths conditioned to reach a prescribed point at a fixed time moment and discuss their behavior, which appears to be non-intuitive for subdiffusive processes in the presence of a strong confining potential.

Figures (3)

• Figure 1: Mean-square displacement (MSD) and the covariance function as functions of $t$ of the Ornstein-Uhlenbeck process driven by the Riemann-Liouville fractional Gaussian noise. Red curves depict the behavior in the superdiffusive case with $H = 3/4$, while the blue ones - in the subdiffusive case with $H = 1/4$. Black solid curves present the behavior for the standard Ornstein-Uhlenbeck process with $H = 1/2$. Units are such that $D =1/2$ and $\gamma=1$. Top row: The MSD, eq. \ref{['MSD']}, for $\tau^* = 1/3$ (panel (a)) and $\tau^* = 3$ (panel (b)). Bottom row: $\mathop{\mathrm{cov}}\nolimits_x(t,t')$, eq. \ref{['COVX']}, with $\tau^* = 1/3$ (panel (c)) and $\tau^* = 3$ (panel (d)). $t' = 5$ (vertical dashed line).
• Figure 2: Kernels $q_{\mathrm{sub}}(\tau_1,\tau_2)$ and $q_{\mathrm{sup}}(\tau_1,\tau_2)$ in eqs.\ref{['Sub']} and \ref{['Sup']} as functions of $\tau_2$ for $\tau_1 = 7$ (vertical dashed line) and $T = 10$. Panel (a): Subdiffusive case. The kernel $q_{\mathrm{sub}}(\tau_1,\tau_2)$ in eq. \ref{['Sub']} with $H = 1/4$ (blue solid curve) and $H = 1/8$ (blue dashed curve). Panel (b): Superdiffusive case. The kernel $q_{\mathrm{sup}}(\tau_1,\tau_2)$ in eq.\ref{['Sup']} for $H=7/8$ (red dashed curve) and $H = 3/4$ (red solid curve).
• Figure 3: Optimal paths $x^*_t$ with $t \in (0,T)$ (here, $T = 10$) conditioned to be at point $X$ (here, $X = 3$) at time moment $T'$ (here, $T'=6$, vertical dashed line) for superdiffusive regime with $H = 3/4$ (red curves) and subdiffusive regime with $H = 1/4$ (blue curves). Black solid curves depict the corresponding optimal paths for the standard OU process ($H = 1/2$). In four panels we present $x_t^*$ for different values of $\tau^*$: $\tau^* = 1/3$ (panel (a)), $\tau^* = 3$ (panel (b)), $\tau^* = 7$ (panel (c)) and $\tau^*=12$ (panel(d)).