A Parabolic Transform and Some Stochastic Ill-Posed Problems

Abstract

It is well known that the Cauchy problem for elliptic partial dierential equations is ill-posed. The question, which arises, how a priori knowledge about solutions can bring about stability? A parabolic transform is dened to discuss the stability of the Cauchy problem for some stochastic partial dierential equations under a priori knowledge about solutions. With the help of the parabolic transform, existence results are established for general linear and nonlinear stochastic partial dierential equations, without any restrictions on the characteristic forms. Many physical and engineering problems in areas like seismology, geophysics and biology require the solutions of ill-posed problems. The Cauchy problem for general stochastic dierential equations has many dierent important applications with amazing range.