Unique invariant measure for the skew stochastic heat equation

The problem asks whether the skew stochastic heat equation has at most one invariant probability measure. A 2011 paper constructs a canonical Markov solution and an explicit invariant measure, but does not establish uniqueness.

Known results

• The 2011 construction uses a Dirichlet-form method to obtain a canonical weak solution and invariant measure.
• It notes that uniqueness of solutions for general measure-valued drifts appears out of reach.
• Athreya, Butkovsky, Lê, and Mytnik (2021) prove strong well-posedness for a class including $b=\delta_0$, but not uniqueness of invariant measures.

September 11, 2026 equilibrium-limit paper

Labbé, Le Guerch, and Zambotti study convergence at equilibrium from the skew equation to the reflected stochastic heat equation as skewness tends to infinity. Their paper does not address uniqueness of the invariant probability measure.

Current status (as of September 2026): Existence of an invariant probability measure is established, while uniqueness for the Markov semigroup remains open.