We describe a probabilistic construction of $H^s$-regular solutions for the
spatially periodic Burgers equation by using a characterization of this
solution through a forward-backward stochastic system.
The classical Chernoff's theorem is a statement about discrete-time
approximations of semigroups, where the approximations are consturcted as
products of time-dependent contraction operators strongly differentiable at
zero. We generalize the version of Chernoff's theorem for semigroups proved in
a paper by Smolyanov et al., and obtain a theorem about descrete-time
approximations of backward propagators.