# Constructing discrete harmonic functions in wedges

@article{Hoang2020ConstructingDH, title={Constructing discrete harmonic functions in wedges}, author={Viet Hung Hoang and Kilian Raschel and Pierre Tarrago}, journal={arXiv: Spectral Theory}, year={2020} }

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

## 5 Citations

### Discrete harmonic functions for non-symmetric Laplace operators in the quarter plane

- Mathematics
- 2022

. We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coeﬃcients…

### Polyharmonic Functions in the Quarter Plane

- Mathematics, Computer ScienceAofA
- 2022

A novel method to compute all discrete polyharmonic functions in the quarter plane for models with small steps, zero drift and a finite group is proposed, and convergence between the discrete and continuous cases is shown.

### Harmonic functions for singular quadrant walks

- Mathematics
- 2022

. We consider discrete (time and space) random walks conﬁned to the quarter plane, with jumps only in directions ( i, j ) with i + j (cid:62) 0 and small negative jumps, i.e., i, j (cid:62) − 1.…

### Marches al\'eatoires dans un c\^one et fonctions discr\`etes harmoniques

- Mathematics
- 2022

Les marches aléatoires dans un cône présentent le double attrait de se trouver au cœur de nombreux problèmes probabilistes et d’être liées à de multiples domaines mathématiques, comme la théorie…

## References

SHOWING 1-10 OF 49 REFERENCES

### On discrete harmonic functions

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1949

A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equation This equation can be…

### Random walks in the quarter plane, discrete harmonic functions and conformal mappings

- Mathematics
- 2014

### Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

- Mathematics
- 2015

We give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive…

### Analytic functions of several complex variables

- Mathematics
- 1965

In the previous chapters we have repeatedly made use of holomorphic functions of a complex vector. Now we discuss their properties in more detail.

### Martin boundary of random walks in convex cones

- MathematicsAnnales Henri Lebesgue
- 2022

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete…

### About a possible analytic approach for walks in the quarter plane with arbitrary big jumps

- Mathematics
- 2014

### Polyharmonic Functions And Random Processes in Cones

- MathematicsAofA
- 2020

It is shown that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems.

### t -Martin Boundary of Killed Random Walks in the Quadrant

- Mathematics
- 2016

We compute the t-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the)…

### Boundary Behaviour of Conformal Maps

- Mathematics
- 1992

1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. Curve…

### On the nature of the generating series of walks in the quarter plane

- MathematicsInventiones mathematicae
- 2018

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this…