  We consider the spin boson model with external magnetic field. We prove a
path integral formula for the heat kernel, known as Feynman-Kac-Nelson (FKN)
formula. We use this path integral representation to express the ground state
energy as a stochastic integral. Based on this connection, we determine the
expansion coefficients of the ground state energy with respect to the magnetic
field strength and express them in terms of correlation functions of a
continuous Ising model. From a recently proven correlation inequality, we can
then deduce that the second order derivative is finite. As an application, we
show existence of ground states in infrared-singular situations.
