Regularity estimates for a class of nonlocal equations arising from discrete stochastic processes
In the recent years, the connection between nonlocal equations and discrete stochastic processes has received an increasing attention. This interplay can be described by a \textit{dynamic programming equation}, which is stated as the nonlocal equation
$$ \int_{B_1}\frac{u(x+\varepsilon y)-u(x)}{\varepsilon^2}\ d\mu_x(y) = f(x), \qquad x\in\Omega, $$
where $\varepsilon>0$ and $x\mapsto\mu_x$ is any (non necessarily continuous) choice of symmetric probability measures on $B_1$ satisfying certain uniform ellipticity condition.
The importance, among others, of this equation lies in the fact that its solutions approximate a viscosity solution of a PDE as $\varepsilon\to 0$. This allows to explore new regularity techniques for solutions of PDEs by obtaining regularity estimates that hold uniformly for every sufficiently small $\varepsilon>0$.
We present a proof of an asymptotic H"older estimate and Harnack inequality for solutions of dynamic programming equations with bounded and measurable increments, an analogous result to the celebrated Krylov-Safonov regularity method for non-divergence form elliptic equations. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators, which allow to extend the result to a wider class of equations.
(Joint work with Pablo Blanc and Mikko Parviainen, University of Jyv"askyl"a).