Parabolic comparison revisited and applications

Abstract

We consider the Cauchy-Dirichlet problem ${\partial }_{t}u-F(t,x,u,Du,D^{2}u)=0on(0,T)\times {\mathbb{R}}^n$ in viscosity sense. Comparison is established for bounded semi-continuous (sub-/super-)solutions under structural assumption (3.14) of the User's Guide plus a mild condition on $F$ such as to cope with the unbounded domain. Comparison on $(0,T]$, space-time regularity and existence are also discussed. Our analysis passes through an extension of the parabolic theorem of sums which appears to be useful in its own right.

Cited by in corpus (5)

• Time-uniform log-Sobolev inequalities and applications to propagation of chaos
• Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs
• Stochastic Perron's method and elementary strategies for zero-sum differential games
• Stochastic control with rough paths
• Rough path stability of (semi-)linear SPDEs