reversibility and detailed balance
Reversibility is a strong symmetry of a Markov process with respect to an invariant measure: run in equilibrium, the process looks statistically the same forwards and backwards in time. It is much more than mere invariance, and in the operator language it is exactly the SELF-ADJOINTNESS of the transition semigroup on L^2(mu) — which unlocks spectral theory, variational principles, and the cleanest convergence-rate results.
Detailed balance is the kernel-level condition: with respect to a measure mu (with density m, say), the flux from x to y balances the flux from y to x, m(x) p_t(x, y) = m(y) p_t(y, x) for all x, y, t. Summing over y recovers ordinary invariance, so detailed balance implies stationarity but is strictly stronger. In operator terms, mu-reversibility says each P_t is SELF-ADJOINT on the Hilbert space L^2(mu): integral (P_t f) g dmu = integral f (P_t g) dmu for all f, g. Differentiating at t = 0, the generator A is SYMMETRIC (self-adjoint) on L^2(mu): integral (A f) g dmu = integral f (A g) dmu, the so-called integration-by-parts / Dirichlet-form symmetry. For a diffusion with generator A f = (1/2) sum d_i (a_(ij) d_j f) + b . grad f, reversibility w.r.t. mu = e^(-V) dx holds precisely when the drift is a gradient, b = -(1/2) a grad V (modulo divergence-free corrections), i.e. the drift derives from a potential and there is no rotational/circulating part.
Why it matters: self-adjointness means the generator has a real spectrum and an orthonormal eigenbasis, so P_t = e^(tA) decays mode by mode and the convergence rate to equilibrium is governed by the spectral gap (the first nonzero eigenvalue); it also gives the variational Dirichlet-form representation of the generator and the Poincaré and log-Sobolev inequalities that quantify mixing. Honest cautions: most Markov processes are NOT reversible (any genuine circulation — a nonzero probability current at stationarity — breaks detailed balance); reversibility is the assumption underlying the convergence guarantees of Metropolis-Hastings MCMC, deliberately engineered, not generic; and a process can be stationary yet strongly irreversible, with the non-self-adjoint (antisymmetric) part of A encoding the persistent equilibrium currents.
A gradient diffusion dX = -grad V(X) dt + sqrt(2) dB is reversible w.r.t. the Gibbs measure mu proportional to e^(-V(x)) dx: its generator A f = Delta f - grad V . grad f is self-adjoint on L^2(mu), and detailed balance holds. Adding a divergence-free drift dX = (-grad V + b)dt with div(e^(-V) b) = 0 keeps mu invariant but DESTROYS reversibility — a persistent equilibrium current circulates.
Reversibility = self-adjoint generator on L^2(mu); a divergence-free drift keeps invariance but breaks detailed balance.
Detailed balance is strictly stronger than invariance: a process can be stationary yet irreversible (nonzero equilibrium currents), and reversibility is engineered, not generic — it is what MCMC relies on.