Detailed balance and time reversal

Let π be a stationary probability on a finite or countable space. The stationary probability of seeing x followed by y is π(x)P(x,y). Conditioning that same pair in the reverse direction defines P*(y,x) by dividing by π(y). On the support Sπ of π, all denominators are positive. Stationarity makes each reversed row sum to one, and the reversed chain has the same stationary law. States of zero stationary mass do not occur in a stationary movie, so this construction does not determine reverse transitions starting from those states.

The support is closed under positive-probability transitions: if π(x)>0 and P(x,y)>0, stationarity forces π(y)>0. Thus restricting to Sπ loses no stationary path. For an irreducible positive recurrent chain the support is the entire state space. For a reducible chain, one must say which stationary law is being reversed, because different mixtures of closed classes can have different supports. No aperiodicity is required for time reversal; a rotating deterministic cycle has a perfectly meaningful reverse rotation.

Detailed balance requires π(x)P(x,y)=π(y)P(y,x) for every pair of states in the entire state space. For a candidate probability law, checking only positive-mass pairs is insufficient unless its support is also closed. It says exactly that P*=P there, and summing over x proves stationarity. The converse fails because stationarity balances total inflow and outflow at a vertex, allowing nonzero currents around cycles. A doubly stochastic finite matrix has uniform stationary law, but it need not be symmetric and hence need not be reversible. Testing a single vertex balance equation cannot establish reversibility of an entire chain.

Reversibility extends from pairs to full finite paths by repeatedly applying the balance identity: the probability of a stationary path equals the probability of its reversed path under the same kernel. In operator language, P is self-adjoint for the inner product weighted by π, making its eigenvalues real in the finite case. A real spectrum alone is not a substitute for checking self-adjointness. In later lessons the weighted structure supplies the correct variances, Dirichlet form and spectral-gap formulas; treating a nonuniform reversible kernel as an ordinary symmetric matrix can give the wrong calculation.

Choose symmetric nonnegative edge weights c(x,y) with finite positive row totals c(x). The walk P(x,y)=c(x,y)/c(x) has invariant measure c(x). If the sum of all row totals is finite, normalizing gives a reversible stationary probability because each stationary edge flow is c(x,y) divided by that common total. On a finite undirected graph this gives degree-proportional mass, not generally uniform mass. Adding identical holding probability at every state preserves this stationary law and reversibility.

This construction separates geometry from normalization and from speed. Scaling every edge weight by the same factor leaves the discrete kernel unchanged, while adding holds changes transition timing. On an infinite graph, symmetric weights can produce an infinite invariant measure rather than a stationary probability. The reverse-kernel formula based on a probability law should then not be presented as a stationary probability movie unless normalization is possible. Reversible invariant measures remain useful analytically, but their infinite mass does not remove the recurrence questions from the previous lesson.