Random walks and random tug of war in the Heisenberg group

Abstract

We study the mean value properties of p-harmonic functions on the first Heisenberg group H, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres-Sheffield [32] to provide the game-theoretical interpretation of the sub-elliptic p-Laplacian; and of Manfredi-Parviainen-Rossi [25] to characterize its viscosity solutions via the asymptotic mean value expansions.

Convergence of Natural $p$-Means for the $p$-Laplacian in the Heisenberg Group

“…For tug-of-war means the Perturbation Lemma above is valid for 1 < p < ∞, and it is due to Lewicka [Lew18,Lew20] in R n and to [LMR20] in H. The key to proving this lemma in H is a strengthening of the expansion in ε of µ H p (u, ε)(x 0 ) that we are able to prove for p > 2. Proposition 1.8.…”

“…It is known that domains with C 1,1 boundary in the Euclidean sense satisfy the exterior H-corkscrew condition (see [CG98], Theorem 14 for domains in the Heisenberg group and [MM05], Theorem 1.3 for the more general case of domains in step 2 Carnot groups). This regularity is optimal in the sense that for every α ∈ [0, 1) there exist domains with C 1,α boundary in the Euclidean sense that do not satisfy the condition in Definition 1.10 (see Example 8.2 in [LMR20]).…”

“…1}, so we can choose q 0 = 0 and v(x) = |x| s H . From the calculations in the proof of Theorem 12.1 in [LMR20] there exist s ≥ 4 and ε such that ∆ N H,p (U + εv) ≥ sε in Ω for all 0 < ε < ε. From the expansion in Proposition 1.8, there exists C > 0 such that…”

Variational p-harmonious functions: existence and convergence to p-harmonic functions

“…Classes of p-harmonious functions for a general p ∈ (1, ∞] have been introduced in [24,25], and in [11,12] also for the case p = 1 in the plane, in connection with dynamic programming principles associated with tug-of-war games (see also [28,29]). We also refer the reader to [2,7,[10][11][12][20][21][22] for other related works that motivate these issues.…”

A mean value formula for the variational p-Laplacian

“…In the last years, several other mean value formulas for the normalized p-Laplacian have been found, and the corresponding program (equivalence of solutions in the viscosity and classical sense and study of the associated dynamic programming principle) has been developed. See for instance [2,7,9,14,[16][17][18], and [22]. We also want to mention [8] and [10], where two other nonlinear mean value formulas are studied, with some similarities with ours.…”