Weak uniqueness for non-negative solutions when γ∈(0,1/2)

Prove or disprove weak uniqueness (uniqueness in law) for non-negative solutions to (i) the stochastic heat equation ∂_t u(t,x) = (1/2)Δu(t,x) + u(t,x)^γ Ẇ(t,x) on ℝ and (ii) the spatially discrete infinite-dimensional SDEs dX_t(i) = (∑_{j∈ℤ^d} q(i−j)(X_t(j)−X_t(i))) dt + f(X_t(i)) dt + σ(X_t(i)) dB_t(i) (including the special case dX_t(i) = L X_t(i) dt + X_t(i)^γ dB_t(i)), for γ ∈ (0,1/2).

Background

Below the Yamada–Watanabe threshold, the diffusion coefficient is less than 1/2-Hölder at zero, so standard pathwise uniqueness results do not apply. For signed solutions in one dimension, weak uniqueness is known to fail for related equations, but the non-negative case is not settled.

The authors emphasize that, for γ∈(0,1/2), it is not known whether non-negative solutions to these models are unique in law; they explicitly flag this as an open problem.

References

For $\gamma \in (0,1/2)$, it is an open problem to prove or disprove weak uniqueness for solutions to eq_SHE, eq_sys_intro, or the more general spatially discrete system which we introduce in the next section. Thus, it is unknown if the solutions we consider are unique-in-law.

— A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients  (2603.29442 - Hughes et al., 31 Mar 2026) in Introduction