  We introduce the local martingale problem associated to semilinear stochastic
evolution equations driven by a cylindrical Wiener process and establish a
one-to-one correspondence between solutions of the martingale problem and
(analytically) weak solutions of the stochastic equation. We also prove that
the solutions of well-posed equations are strong Markov processes. We apply our
results to semilinear stochastic equations with additive noise where the
semilinear term is merely measurable and to stochastic reaction-diffusion
equations with H\"older continuous multiplicative noise.
