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Register a new userNon-local tug-of-war with noise for the geometric fractional $p$-Laplacian
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Marta Lewicka
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This paper concerns the fractional $p$-Laplace operator $Delta_p^s$ in non-divergence form, which has been introduced in [Bjorland, Caffarelli, Figalli (2012)]. For any $pin [2,infty)$ and $sin (frac{1}{2},1)$ we first define two families of non-local, non-linear averaging operators, parametrised by $epsilon$ and defined for all bounded, Borel functions $u:mathbb{R}^Nto mathbb{R}$. We prove that $Delta_p^s u(x)$ emerges as the $epsilon^{2s}$-order coefficient in the expansion of the deviation of each $epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $mathbb{R}^N$ at the rate $epsilonto 0$. Second, we consider the $epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $epsilonto 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $Delta_p^s$, when posed in a domain $mathcal{D}$ that satisfies the external cone condition and subject to bounded, uniformly continuous data on $mathbb{R}^Nsetminus mathcal{D}$. Finally, we interpret such $epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on $p$ and $N$, and whose $epsilon$-tip has been removed.
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This is a preprint of Chapter 2 in the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, reproduced with permission of Springer Nature Switzerland AG. We present the basic relation between the linear potential theory and random walks. This fundamental connection, developed by Ito, Doob, Levy and others, relies on the observation that harmonic functions and martingales share a common cancellation property, expressed via mean value properties. It turns out that, with appropriate modifications, a similar observation and approach can be applied also in the nonlinear case, which is of main interest in our Course Notes. Thus, the present Chapter serves as a stepping stone towards gaining familiarity with more complex nonlinear constructions. After recalling the equivalent defining properties of harmonic functions, we introduce the ball walk. This is an infinite horizon discrete process, in which at each step the particle, initially placed at some point $x_0$ in the open, bounded domain $mathcal{D}subsetmathbb{R}^N$, is randomly advanced to a new position, uniformly distributed within the following open ball: centered at the current placement, and with radius equal to the minimum of the parameter $epsilon$ and the distance from the boundary $partialmathcal{D}$. With probability one, such process accumulates on $partialmathcal{D}$ and $u^epsilon(x_0)$ is then defined as the expected value of the given boundary data $F$ at the process limiting position. Each function $u^epsilon$ is harmonic, and if $partialmathcal{D}$ is regular, then each $u^epsilon$ coincides with the unique harmonic extension of $F$ in $mathcal{D}$. One sufficient condition for regularity is the exterior cone condition.
We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.
We study the mean value properties of $mathbf{p}$-harmonic functions on the first Heisenberg group $mathbb{H}$, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres-Scheffield to provide the game-theoretical interpretation of the sub-elliptic $mathbf{p}$-Laplacian; and of Manfredi-Parviainen-Rossi to characterize its viscosity solutions via the asymptotic mean value expansions.
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Delta)^s_p$ with $sin(0,1)$ and $pge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Omega)$ and in the weighted Holder space $C^0_s(overlineOmega)$, respectively, do coincide.
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