Tug-of-war games with varying probabilities and the normalized <i>p</i>(<i>x</i>)-laplacian

Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

“…As we noted at the beginning of this work, the authors in [AHP17] showed that the solutions u ε of the DPP (1.1) converge uniformly as ε → 0 to a viscosity solution of the normalized p(x)-Laplace equation ∞] is a continuous function. In this section we consider a different DPP whose solutions are asymptotically related in the same way to the normalized p(x)-Laplace equation when p(x) > 2 for all x ∈ Ω.…”

“…Proof of Lemma 2.2. case |x − z| ≤ N 10 ε In the previous section, we proved Lemma 2.2 in the case |x − z| > N 10 ε. The other case |x − z| ≤ N 10 ε is similar to [AHP17]. In Section 2.2 we briefly commented that in this case we need an annular step function f 2 ε. Recalling (3.6) and for large enough (4.1) C > 8M r + 1, we obtain the following rough estimate for f 1 ,…”

“…where the function p : Ω → (1, ∞] is continuous and bounded away from 1. Under these assumptions, Arroyo, Heino and Parviainen [AHP17] showed that for a given continuous boundary data and a suitable boundary cut-off function, it holds that u ε → u uniformly when ε → 0, where u is the viscosity solution of the normalized p( Moreover, by [AHP17, Theorem 4.1], the function u ε is asymptotically Hölder continuous.…”

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

“…This argument plays an important role in obtaining the desired estimate. 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.…”

Random walks and random tug of war in the Heisenberg group

“…We will be concerned with the so-called p-sub-Laplacian of v, with exponent p ∈ (1, ∞): 4) and with its normalized (sometimes called game-theoretical) version:…”