Related papers: The Expected Time to End the Tug-of-War in a Wedge
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide…
Explicit numerical methods based on Lax-Friedrichs and Leap-Frog finite difference approximations are constructed to find the numerical solution of the first-order hyperbolic partial differential equation with point-wise delay or advance,…
This paper presents a wp-style calculus for obtaining bounds on the expected run-time of probabilistic programs. Its application includes determining the (possibly infinite) expected termination time of a probabilistic program and proving…
In this paper we find viscosity solutions to an elliptic system governed by two different operators (the Laplacian and the infinity Laplacian) using a probabilistic approach. We analyze a game that combines the Tug-of-War with Random Walks…
We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not…
Solutions $u(x)$ to the class of inhomogeneous nonlinear ordinary differential equations taking the form \[u'' + u^2 = \alpha f(x) \] for parameter $\alpha$ are studied. The problem is defined on the $x$ line with decay of both the solution…
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for…
We introduce a new class of strongly degenerate nonlinear parabolic PDEs $$((p-2)\Delta_{\infty,X}^N+\Delta_X)u(X,Y,t)+(m+p)(X\cdot\nabla_Yu(X,Y,t)-\partial_tu(X,Y,t))=0,$$ $(X,Y,t)\in\mathbb R^m\times \mathbb R^m\times \mathbb R$, $p\in…
We present a novel method for solving a class of time-inconsistent optimal stopping problems by reducing them to a family of standard stochastic optimal control problems. In particular, we convert an optimal stopping problem with a…
In this paper we establish that the time-harmonic elasticity problem in a half-strip with non-homogeneous Dirichlet conditions on its boundary section and traction-free conditions on its upper and lower boundaries, has a unique weak…
In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t…
We investigate the limiting behavior of solutions to the inhomogeneous $p$-Laplacian equation $-\Delta_p u = \mu_p$ subject to Neumann boundary conditions. For right hand sides which are arbitrary signed measures we show that solutions…
In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized…
In neuroscience, the time elapsed since the last discharge has been used to predict the probability of the next discharge. Such predictions can be improved taking into account the last two discharge times, and possibly more. Such multi-time…
We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincar\'e…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…
We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the $p$-Laplacian with $1<p<2$. For this version, the asymptotic H\"older continuity of solutions can be directly derived from…
Classically time is kept fixed for infinitesimal variations in problems in mechanics. Apparently, there appears to be no mathematical justification in the literature for this standard procedure. This can be explained canonically by…
We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overline{\Omega}$ in $\mathbb{R}^n$($n\geq2$). First, we establish the…
In this manuscript, we investigate a fractional stochastic neutral differential equation with time delay, which includes both deterministic and stochastic components. Our primary objective is to rigorously prove the existence of a unique…