All-times compact support for the discrete SDE system

Determine whether, for solutions X to the spatially discrete infinite-dimensional SDE system 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) with Hölder noise coefficient satisfying σ(x)≈x^γ near zero for γ ∈ (0,1/2), the support of X_t is compact for all times t ∈ [0,T] almost surely (i.e., whether there are no exceptional times with non-compact support).

Background

The main theorem shows that for γ∈(0,1/2) the times with compact support form a set of full Lebesgue measure, but it does not exclude exceptional times when the support might be non-compact.

The authors also prove that the classical compact support property (boundedness of the union of supports over [0,T]) fails and, in a special case, that times when the support is arbitrarily large are dense, but whether compactness holds at every single time remains unresolved.