  Bootstrap is an idea that imposing consistency conditions on a physical
system may lead to rigorous and nontrivial statements about its physical
observables. In this work, we discuss the bootstrap problem for the invariant
measure of the stochastic Ising model defined as a Markov chain where
probability bounds and invariance equations are imposed. It is described by a
linear programming (LP) hierarchy whose asymptotic convergence is shown by
explicitly constructing the invariant measure from the convergent sequence of
moments. We also discuss the relation between the LP hierarchy for the
invariant measure and a recently introduced semidefinite programming (SDP)
hierarchy for the Gibbs measure of the statistical Ising model based on
reflection positivity and spin-flip equations.
