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.
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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