Local regularity for time-dependent tug-of-war games with varying probabilities

Abstract

We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain Hölder and Harnack estimates. The games have a connection to the normalized p(x, t)parabolic equation (n + p(x, t))ut = ∆u + (p(x, t) − 2)∆ N ∞ u.

Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

“…Several regularity results for functions satisfying various time-independent DPPs were proved by calculations based on this argument (see [LP18,AHP17,ALPR]). It was proved in [PR16] that functions satisfying another time-dependent DPP have Hölder regularity. Our proof differs from that in [PR16] due to the difference of the setting of DPP.…”

“…Meanwhile, for the Lipschitz regularity when p is constant, a core approach in the proof is based on game theory. The aim of this paper is to extend regularity results in [PR16] from the case 2 < p < ∞ to the case 1 < p < ∞. It is hard to apply the game theoretic argument in that paper to our DPP.…”

The tusk condition and Petrovskiĭ criterion for the normalized p‐parabolic equation

“…These are easily obtained from the cylinders used in the weak Harnack inequality in [, Theorem 2.1], by the solution‐preserving dilation . When , the weak Harnack inequality also follows by a game‐theoretic argument (see Parviainen–Ruosteenoja [, Theorem 4.7]). Here and below, denotes the integral average, that is, .…”

“…used in the weak Harnack inequality in [17, Theorem 2.1], by the solution-preserving dilation (x, t) → (2rx, (2r) 2 t). When p 2, the weak Harnack inequality also follows by a gametheoretic argument (see Parviainen-Ruosteenoja [24,Theorem 4.7]). Here and below, ffl denotes the integral average, that is,…”

“…Boundary regularity for the normalized ‐parabolic equation was first studied by Banerjee–Garofalo (see also their earlier paper ). More recently, Jin–Silvestre established the interior ‐regularity for solutions of (see also Imbert–Jin–Silvestre , Attouchi–Parviainen and Parviainen–Ruosteenoja for related regularity results).…”