An L _p -theory for diffusion equations related to stochastic processes with non-stationary independent increment

Let X =(X _t ) _t≥0 be a stochastic process which has a (not necessarily stationary) independent increment on a probability space (Ω, P). In this paper, we study the following Cauchy problem related to the stochastic process X: (Formula presented) We provide a sufficient condition on X (see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in L _p ([0,T]; H _p ^φ ), where H _p ^φ is a φ-potential space on R ^d (see Definition 2.9). Furthermore we show that for this solution, (Formula presented), where N is independent of u and f.