*Published Paper*

**Inserted:** 18 jul 2017

**Last Updated:** 2 jul 2019

**Journal:** SIAM Journal on Mathematical Analysis

**Volume:** 50

**Number:** 5

**Pages:** 5198-5242

**Year:** 2018

**Abstract:**

We study the stochastic homogenization for a Cauchy problem for a first-order Hamilton-Jacobi equation whose operator is not coercive w.r.t. the gradient variable. We look at Hamiltonians like $H(x,\sigma(x)p,\omega)$ where $\sigma(x)$ is a matrix associated to a Carnot group. The rescaling considered is consistent with the underlying Carnot group structure, thus anisotropic. We will prove that under suitable assumptions for the Hamiltonian, the solutions of the $\varepsilon$-problem converge to a deterministic function which can be characterized as the unique (viscosity) solution of a suitable deterministic Hamilton-Jacobi problem.

**Keywords:**
Stochastic homogenisation, Carnot groups, non-coercive Hamiltonian

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