Fall workshop on nonlinear and nonlocal PDEs

Plenary talk

Regularity estimates for a class of nonlocal equations arising from discrete stochastic processes

Ángel Arroyo

 Tu, 11:20in  Room 520for  50min

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).

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