Your space $\mathcal F$ consists of constant functions only. The arguments referred to in the post which you quote are somewhat misleading as they use "too much": reduction to the Liouville theorem about bounded classical harmonic functions or recurrence for a compactly supported symmetric random walk in dimension 2. In fact, neither of these two properties is really needed.

One can reformulate your question in the following more general way. Let $\mu$ be a probability measure on a locally compact abelian group $G$ (particular case: $G=\mathbb R^d$). Call a function $\mu$-harmonic if $f(x)=\int f(x+y)\ d\mu(y)$ for any $x\in G$. ** If the closure of the group generated by the support of $\mu$ is the whole group $G$, then all measurable bounded $\mu$-harmonic functions are constant a.e. with respect to the Haar measure (in particular, all continuous bounded $\mu$-harmonic functions are constant). **This result is known as the Choquet-Deny theorem (proved in 1960).