Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.Comment: Published in at http://dx.doi.org/10.1214/13-AOP833 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions $\Phi$ and $\tilde \Phi$ of the PDE $$F(D^2\Phi) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \}$$ for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $\alpha^*, \tilde\alpha^* > -1$ which describe the homogeneity of $\Phi$ and $\tilde \Phi$. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of $F(D^2u) = 0$ in $\mathbb{R}^n \setminus \{0 \}$ which are bounded on one side in a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.Comment: 35 pages, typos and minor mistakes correcte

Discovering Employment Listings from Imagery

Generally, the present disclosure is directed to discovering employment listings from imagery. In particular, in some implementations, the systems and methods of the present disclosure can include or otherwise leverage one or more machine-learned models to identify employment listings based on image data

Using Imagery of Vegetation and Rooftops to Predict Solar Roof Candidacy

Generally, the present disclosure is directed to predicting whether a property can be conducive to solar energy system installation with minimal adjustment to peripheral vegetation. In particular, in some implementations, the systems and methods of the present disclosure can include or otherwise leverage one or more machine-learned models to predict whether a property can be conducive to solar energy installation with minimal adjustment to peripheral vegetation based on imagery and/or publicly available or user-submitted information about the property or area surrounding the property

Using Imagery of Lawns to Estimate Lawn Care Need

Generally, the present disclosure is directed to identifying properties whose owners are likely to be receptive to lawn care services. In particular, in some implementations, the systems and methods of the present disclosure can include or otherwise leverage one or more machine-learned models to predict that a property exhibits a need for lawn care disproportional to surrounding properties and/or has an owner or owners with the financial capability to afford lawn care services based on imagery and/or real estate information for the property and/or surrounding properties

Using Imagery of Property Improvements to Direct Marketing

Generally, the present disclosure is directed to using imagery of properties containing property improvements to direct advertising for such improvements. In particular, in some implementations, the systems and methods of the present disclosure can include or otherwise leverage one or more machine-learned models to predict that a property has a property improvement based on imagery of the property