Cosmological Redshift as a Diffusion-Induced Temporal Evolution of Gravitational Potential

We propose a novel mechanism for the observed cosmological redshift ($z$) based on the temporal diffusion of the gravitational potential $\phi$. Unlike standard $\Lambda$CDM models which attribute redshift to the metric expansion of space, or ``Tired Light'' models based on scattering, this model suggests that photons lose energy as they traverse a gravitational potential that evolves via a diffusion-like process. We derive a modified Poisson equation incorporating a first-order time derivative weighted by the Hubble constant $H_0$. We show that for a homogeneous mass density $\rho$, this leads to a redshift $z$ that scales linearly with look-back time, recovering Hubble's Law as a local approximation.