Non-Ergodic Brand Perception: Diffusion Dynamics on Multi-Dimensional Perceptual Manifolds

Brand perception evolves over time, yet no existing brand theory provides a formal dynamical model of this evolution. This paper models the trajectory of an observer’s brand perception as a stochastic process on 𝑆7+, the positive octant of the 7-sphere in Spectral Brand Theory’s (SBT) eight-dimensional perception space. Signal encounters drive Brownian motion on the manifold, signal decay introduces deterministic drift toward a neutral prior, and negative conviction creates absorbing boundaries at the octant boundary where any perceptual dimension reaches zero. We formulate a Stratonovich stochastic differential equation (SDE) on 𝑆7+ and prove four main results. First, the SDE with absorbing boundaries is well-posed with a unique strong solution up to the absorption stopping time (Theorem 1). Second, the survival probability decays as 𝑆(𝑡, 𝑥) ∼ 𝐶(𝑥) exp(−𝜆𝐷,1𝜎2𝑡/2), where 𝐶(𝑥) depends on the initial position’s distance from the boundary (Theorem 2). Third, the Dirichlet spectral gap on 𝑆7+ is 𝜆𝐷,2− 𝜆𝐷,1 = 48, so mixing to the quasi-stationary distribution is faster than on the full sphere: conditioned on survival, perception trajectories are confined to the deep interior of the octant (Theorem 3). Fourth, for the absorbed process, the time average and ensemble average of any non-trivial observable diverge with probability 1 as 𝑡 → ∞, establishing non-ergodicity in the precise sense of Peters (2019) (Theorem 4). Application to five case-study brands shows that the absorption risk ordering – Tesla (C-) > Erewhon (B-) > IKEA (A-) > Patagonia (B+) > Hermes (A+) – nearly matches the SBT coherence grading (Proposition 5), with one inversion (IKEA/Patagonia) explained by coherence-type-dependent drift strength, providing the first formal connection between brand coherence and stochastic stability. The results formalize SBT’s ergodicity coefficient as 𝜀 ∼ 1/𝜏mix, complete the dynamic extension of the static metric framework established in Zharnikov (2026d), and establish that the distinction between vectorized and rasterized brand management is not merely practical but reflects a fundamental mathematical asymmetry between trajectory-level and snapshot-level information.