This thesis considered the connections between parabolic partial differential equations of the diffusion type and Gaussian-Markov stochastic processes, in particular the Wiener process. A method has been developed by which certain Wiener integrals of the type∫C0[0,1] exp{t/a ∫1/0 e[t(1-s), 2 √(t/a) x(s) =ξ] ds} o [2√(t/a) x(1) – ξ] dwxHave been obtained as solutions of non-homogeneous heat equations. In the appendix the method has been extended to the evaluation of Wiener integrals of the type,∫C0 [0,t] exp {∫t/0 e [t-s, x(s) + ξ] ds} o [x(s) + ξ] dwx.In addition an inequality which gives bounds for Wiener integrals of the type∫C0 [s,t] exp {-∫t/s F[x( r )] dr} dwx has been deduced.Further, certain parabolic partial differential equations have been solved by building suitable Green’s functions through Gaussian-Markov stochastic processes. Two stochastic processes which exhibit certain interesting features have been obtained and briefly discussed.Ball State UniversityMuncie, IN 47306