In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with continuous sample paths. For more details see the page about Itō_diffusion.
A sample path of a diffusion process mimics the trajectory of a molecule, which is embedded in a flowing fluid and at the same time subjected to random displacements due to collisions with other molecules, i.e. Brownian motion. The position of this molecule is then random; its probability density function is governed by an advection-diffusion equation.
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“The moralist and the revolutionary are constantly undermining one another. Marx exploded a hundred tons of dynamite beneath the moralist position, and we are still living in the echo of that tremendous crash. But already, somewhere or other, the sappers are at work and fresh dynamite is being tamped in place to blow Marx at the moon. Then Marx, or somebody like him, will come back with yet more dynamite, and so the process continues, to an end we cannot foresee.”
—George Orwell (1903–1950)