Aleksandrov–Bakelman–Pucci–Krylov–Tso maximum principle for $$L^p$$-viscosity solutions of parabolic equations with drift term in $$L^q$$

The main result of the paper is the Aleksandrov–Bakelman–Pucci–Krylov–Tso (ABPKT) maximum principle for $$L^{n+1}$$ -viscosity sub/super-solutions of fully nonlinear uniformly parabolic equations $$u_t+F(t,x,u,Du,D^2u)=f(t,x)$$ in $$(0,T]\times \Omega ,$$ where $$\Omega \subset {\mathbb {R}}^n.$$ In this version of the maximum principle, the $$L^{n+1}$$ norm of f is taken over the so-called contact set. Equations have measurable and unbounded terms and we assume that the “drift” term which governs the dependence of F on the gradient variable is unbounded and is a function in $$L^{n+2}(Q).$$ Other versions of the ABPKT maximum principle for $$L^p$$ -viscosity solutions and its pointwise version are also obtained. We use the maximum principles to prove various properties of $$L^p$$ -viscosity solutions and build basic theory of $$L^p$$ -viscosity solutions for uniformly parabolic equations with the unbounded drift term.