Stochastically gated local and occupation times of a Brownian particle

We generalize the Feynman-Kac formula to analyze the local and occupation times of a Brownian particle moving in a stochastically gated one-dimensional domain. (i) The gated local time is defined as the amount of time spent by the particle in the neighborhood of a point in space where there is some target that only receives resources from (or detects) the particle when the gate is open; the target does not interfere with the motion of the Brownian particle. (ii) The gated occupation time is defined as the amount of time spent by the particle in the positive half of the real line, given that it can only cross the origin when a gate placed at the origin is open; in the closed state the particle is reflected. In both scenarios, the gate randomly switches between the open and closed states according to a two-state Markov process. We derive a stochastic, backward Fokker-Planck equation (FPE) for the moment-generating function of the two types of gated Brownian functional, given a particular realization of the stochastic gate, and analyze the resulting stochastic FPE using a moments method recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment-generating function, averaged with respect to realizations of the stochastic gate.

Figure 1

Schematic diagram of Brownian motion for (a) no gate and (b) a switching gate at $x=a$. In the latter case the open (closed) state of the gate is indicated by black (gray) vertical bars. In (a) the local time density $T$ characterizes the amount of time the particle spends in a neighborhood of the point $x=a$, whereas in (b) the stochastically gated local time $\mathcal{T}$ only counts the time around $x=a$ when the gate is open. Note that the gate does not affect the motion of the Brownian particle.

Figure 2

Schematic diagram of two possible trajectories for gated Brownian motion. (a) The gate is open when the Brownian particle reaches the origin and the trajectory crosses over from the left to right domains. (b) The gate is closed when the particle reaches the origin resulting in the particle being reflected.

Figure 3

Comparison of functions $g\left(\tau \right)=2\beta {\rho }_0{\int }_0^{\tau }f(\tau -t)tdt$ and ${\rho }_1\tau$ for $\alpha =\beta ={\rho }_0={\rho }_1=0.5$. (a) $g\left(\tau \right)\ll {\rho }_1\tau$ at relatively short times. (b) On longer time scales $g\left(\tau \right)\to {\rho }_1\tau /2$.

Figure 4

Examples of gated diffusion. (a) The nuclear envelope with inset showing components of a single nuclear pore. 1, Nuclear envelope, 2, outer ring; 3, spokes; 4, basket; 5, filaments. Each of the eight protein subunits surrounding the actual pore (the outer ring) projects a spoke-shaped protein into the pore channel. (Public domain figure from Wikimedia). (b) Picket-fence model of membrane diffusion. The plasma membrane is parceled up into compartments whereby both transmembrane proteins and lipids undergo short-term confined diffusion within a compartment and long-term hop diffusion between compartments. Transmembrane proteins are confined either by the actin-based membrane skeleton or anchored-proteins.