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In this paper, we study the following Cauchy problem related to the stochastic process $X$:\n  $\\label{main eqn} \\frac{\\partial u}{\\partial t}(t,x) = \\cA(t)u(t,x) +f(t,x), \\quad u(0,\\cdot)=0, \\quad (t,x) \\in (0,T) \\times \\mathbf{R}^d, \\end{align} where $f \\in L_p( (0,T) ; L_p(\\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \\begin{align*} \\cA(t)u(t,x) = \\lim_{h \\downarrow 0}\\frac{\\mathbb{E}\\left[u(t,x+X_{t+h}-X_t)-u(t,x)\\right]}{h}$. 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