This paper develops a tractable analytical channel model for first-hitting-time molecular communication systems under time-varying drift. While existing studies of nonstationary transport rely primarily on numerical solutions of advection--diffusion equations or parametric impulse-response fitting, they do not provide a closed-form description of trajectory-level arrival dynamics at absorbing boundaries. By adopting a change-of-measure formulation, we reveal a structural decomposition of the first-hitting-time density into a cumulative-drift displacement term and a stochastic boundary-flux modulation factor. This leads to an explicit analytical expression for the Corrected-Inverse-Gaussian (C-IG) density, extending the classical IG model to strongly nonstationary drift conditions while preserving constant-complexity evaluation. High-precision Monte Carlo simulations under both smooth pulsatile and abrupt switching drift profiles confirm that the proposed model accurately captures complex transport phenomena, including phase modulation, multi-pulse dispersion, and transient backflow. The resulting framework provides a physics-informed, computationally efficient channel model suitable for system-level analysis and receiver design in dynamic biological and molecular communication environments.
Deep Dive into Corrected-Inverse-Gaussian First-Hitting-Time Modeling for Molecular Communication Under Time-Varying Drift.
This paper develops a tractable analytical channel model for first-hitting-time molecular communication systems under time-varying drift. While existing studies of nonstationary transport rely primarily on numerical solutions of advection–diffusion equations or parametric impulse-response fitting, they do not provide a closed-form description of trajectory-level arrival dynamics at absorbing boundaries. By adopting a change-of-measure formulation, we reveal a structural decomposition of the first-hitting-time density into a cumulative-drift displacement term and a stochastic boundary-flux modulation factor. This leads to an explicit analytical expression for the Corrected-Inverse-Gaussian (C-IG) density, extending the classical IG model to strongly nonstationary drift conditions while preserving constant-complexity evaluation. High-precision Monte Carlo simulations under both smooth pulsatile and abrupt switching drift profiles confirm that the proposed model accurately captures compl
Molecular communication (MC) channels are governed by the stochastic transport of signaling molecules through diffusion and advection [1], [2]. The dominant modeling paradigm in the literature characterizes such channels via concentrationbased impulse responses [3] derived from advection-diffusion partial differential equations [4]. Under steady and uniform drift conditions, these formulations admit compact analytical expressions, and for absorbing receiver settings, the firsthitting-time (FHT) distribution provides a natural description of molecule arrival dynamics. In particular, when drift is constant, the FHT density follows the classical Inverse-Gaussian (IG) distribution [5], which has been widely adopted as a tractable baseline analytical channel model.
In realistic biological and engineered environments, however, transport conditions are rarely stationary. Pulsatile cardiovascular flows [6], [7] and time-varying electrophoretic transport in microfluidic platforms [8], [9] induce explicitly time-dependent drift velocities. Physically, these environments are commonly modeled using oscillatory velocity profiles superimposed on a nonzero mean drift, reflecting periodically driven pressure gradients in vascular systems [10], [11]. Such nonstationary drift reshapes molecule arrival statistics, leading to phase modulation, multi-peak behavior, and transient backflow effects that cannot be captured by stationary channel models.
Despite extensive investigation of time-varying transport, most existing MC studies remain concentration-centric. While recent analytical advances have successfully modeled timevariant MC channels induced by the random Brownian mobility of transceivers [12], these approaches fundamentally assume a static fluid medium governed by isotropic diffusion. Characterizing the nonstationarity induced by explicitly timevarying fluid drift remains a distinctly different and open analytical challenge. Unlike transceiver mobility, which can be resolved by statistically averaging a static channel impulse response over random distances, time-varying advection continuously alters the underlying stochastic trajectory. This triggers directional phenomena, such as transient backflow, that purely diffusion-based mobility models cannot capture. Consequently, time variability in drift is typically handled via numerical solutions of advection-diffusion partial differential equations (PDEs) or simulation-calibrated channel impulse responses (CIRs), where system parameters are adjusted dynamically to fit observed data [13]. While such approaches accurately capture macroscopic concentration evolution, they do not yield closed-form analytical models for trajectory-level arrival statistics at absorbing boundaries [14]. From a stochasticprocess viewpoint, the exact FHT density under time-varying drift can in principle be characterized through Volterra-type integral equations arising from first-passage theory [15], [16]. However, these formulations generally lack closed-form solutions and require recursive numerical evaluation, making them unsuitable for real-time channel modeling and signal processing applications.
To address this challenge, this paper develops a physicsinformed analytical channel model for FHT behavior under time-varying drift. By reformulating stochastic transport through a change-of-measure perspective, we uncover a structural decomposition of the FHT density into a cumulative-drift displacement term and a stochastic boundary-flux modulation factor. This leads to an explicit analytical Corrected-Inverse-
Gaussian (C-IG) density formula that extends the classical IG model to strongly nonstationary drift conditions while preserving constant-complexity evaluation.
The main contributions of this work are summarized as follows.
• Analytical Framework for Nonstationary Transport:
We
Exact first-passage formulations of the FHT density under time-varying drift are analytically intractable. To obtain a tractable representation, we adopt a change-of-measure framework that separates reference diffusion from drift-induced perturbations.
This formulation reveals a natural two-layer structure of the FHT density: an exponential displacement core determined by cumulative drift, and a stochastic boundary-flux modulation factor. The following subsections derive these two components.
Let (Ω, F , {F t } t≥0 , P) be a filtered probability space, and let W t denote a standard one-dimensional Brownian motion adapted to {F t } t≥0 under P. Under the reference measure P, the signaling molecule follows drift-free diffusion,
(2) Under the target measure Q, the dynamics incorporate a deterministic time-varying drift, dX t = µ(t)dt + σdW t . We assume that µ(t) is deterministic and square-integrable on finite intervals. The stopping time to an absorbing boundary ℓ > x 0 is defined as T := inf{t > 0 : X t = ℓ}.
By the Girsanov theorem [17], the Radon-Nikodym derivative evaluated at the stopping time T is
Appl
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