Second-order parabolic viscosity subsolutions and supersolutions

Let $u$ be upper semicontinuous and $v$ lower semicontinuous on the periodic space-time cylinder. A second-order parabolic viscosity subsolution of Periodic diagonal controlled-diffusion HJB Cauchy problem is a function $u$ such that every $C^{1,2}$ test function $\varphi$ for which $u-\varphi$ has a local maximum at $(x_0,t_0)$, $0<t_0<T$, satisfies ${\varphi }_t(x_0,t_0)-{\max }_{a\in A}\{{\sum }_j({\alpha }_j^a{\varphi }_{x_j{x}_j}+b_j^a{\varphi }_{x_j})(x_0,t_0)+f^a(x_0,t_0)\}\le 0.$ Here $C^{1,2}$ means that ${\varphi }_t$, all first spatial derivatives, and all second spatial derivatives exist and are continuous near the contact point. A supersolution is lower semicontinuous and satisfies the reverse inequality at every local minimum. A viscosity solution is both. For the Cauchy problem, the relaxed initial conditions are ${\mathrm{lim\hspace{0.17em}sup}}_{(y,s)\to (x,0),s>0}u(y,s)\le u_0(x),{\mathrm{lim\hspace{0.17em}inf}}_{(y,s)\to (x,0),s>0}v(y,s)\ge u_0(x)\mathrm{.}$

This extends the contact convention in Viscosity subsolutions and supersolutions of a first-order equation and of the Cauchy problem to tests with the spatial Hessian required by the diffusion operator. The definition does not assert a comparison principle, existence, or uniqueness; those are separate theorems.