Following the embedding idea of generalized functions into the Colombeau algebra of generalized functions, we construct a new algebra of generalized stochastic processes and denote it (W ^2,2;(S)₋₁). This is done by using the chaos expansion form of generalized stochastic processes regarded as linear continuous mappings from the Sobolev space into the Kondratiev space (S)₋₁. As an application, we prove existence and uniqueness of the solution of the equation Lu = h with given stochastic boundary condition. The operator L is assumed to be strictly elliptic in divergence form Lu = ∇·(A·∇ u) + c·∇ u + du. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be deterministic Colombeau generalized functions.

Stochastic evolution equations with multiplicative noise

Since the surgical treatment of hemorrhoidal disease has been characterized by intense postoperative pain, recent studies have tried to modify the standard Milligan-Morgans technique. The up-to-date literature, in the experience of authors, has confirmed that the new method of Harmonic Scalpel? hemorrhoidectomy reduces postoperative pain. The aim of our study was to statistically evaluate, based on our experience, the efficacy of this surgical approach in terms of reducing postoperative pain and establishing a stable hemostasis. Seventy-seven (77) patients suffering from hemorrhoid disease, stage III and IV, underwent surgery in our clinic during the last five years. The postoperative pain was determined using the visual analog scale on the 1st, 2nd and 7th postoperative days. Patients were divided into two groups in regard to the surgical procedure applied. The data were statistically processed using the Statistica 7.0 software. We concluded that Harmonic Scalpel? hemorrhoidectomy, due to less thermal damage, statistically significantly reduced postoperative pain with better hemostasis, compared with Milligan-Morgan's method of treating hemorrhoidal disease. .