Solution theory to semilinear parabolic stochastic partial differential equations with polynomially bounded coefficients

In the present paper we deal with stochastic semilinear partial differential equations of parabolic type with (t,x)-depending coefficients which may admit a polynomial growth with respect to the space variable. Under suitable assumptions on the coefficients of the parabolic operator, on the initial data and on the stochastic noise, we prove existence of a unique (mild) function-valued solution for the associated Cauchy problem.