Applications

• Optimal control problems with Levy processes give rise to the Bellman equation, or in general any equation derived from jump processes will be some fully nonlinear integro-differential equation.
• In financial mathematics it is particularly important to study models involving jump processes. This can be considered a particular case of the item above (stochastic control), but it is a very relevant one. The pricing model for American options involves the obstacle problem.
• Nonlocal electrostatics is a very promising tool for drug design which could potentially have a strong impact in medicine in the future.
• The denoising algorithms in nonlocal image processing are able to detect patterns in a better way than the PDE based models. A simple model for denoising is the nonlocal mean curvature flow.
• The Boltzmann equation models the evolution of dilute gases and it is intrinsically an integral equation. In fact, simplified kinetic models can be used to derive the fractional heat equation without resorting to stochastic processes.
• In conformal geometry, the Paneitz operators encode information about the manifold, they include fractional powers of the Laplacian, which are nonlocal operators.
• In oceanography, the temperature on the surface may diffuse though the atmosphere giving rise to the surface quasi-geostrophic equation.
• Models for dislocation dynamics in crystals.
• Several stochastic models, in particular particle systems, can be used to derive nonlocal equations like the Nonlocal porous medium equation, the Hamilton-Jacobi equation with fractional diffusion, conservation laws with fractional diffusion, etc...
• In hydrology, the so called Fractional Advection Dispersion Equation involves fractional operators similar to sums of one-dimensional fractional Laplacians to describe super-diffusive spreading rates of tracer particles flowing in underground aquifers. The more typically used second order (local) models cannot correctly capture this spreading phenomenon in a suitable way.

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