All functions are locally ss-harmonic up to a small error

We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.Comment: To appear in J. Eur. Math. Soc. (JEMS

An obstacle problem for Tug-of-War games

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war

Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

Approximate solution for Fokker-Planck equation

In this paper, an approximate solution to a specific class of the Fokker-Planck equation is proposed. The solution is based on the relationship between the Schr\"{o}dinger type equation with a partially confining and symmetrical potential. To estimate the accuracy of the solution, a function error obtained from the original Fokker-Planck equation is suggested. Two examples, a truncated harmonic potential and non-harmonic polynomial, are analyzed using the proposed method. For the truncated harmonic potential, the system behavior as a function of temperature is also discussed.Comment: 12 pages, 8 figure

Subquadratic harmonic functions on Calabi-Yau manifolds with Euclidean volume growth

We prove that on a complete Calabi-Yau manifold $M$ with Euclidean volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville type theorem for harmonic $1$-forms, which follows from a new local $L^2$ estimate of the differential. We also give another proof based on the construction of harmonic functions with polynomial growth in Ding, and the algebraicity of tangent cones in Liu-Sz\'ekelyhidi.Comment: 30 pages. Comments are welcom