scholarly journals Equivariant discretizations of diffusions, random walks, and harmonic functions

2021 ◽  
Vol 67 (3) ◽  
pp. 331-367
Author(s):  
Werner Ballmann ◽  
Panagiotis Polymerakis
Keyword(s):  
Random Walks ◽  
Harmonic Functions

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilitiesp0,1= 1,pk,k–1=gk,pk,k+1= 1–gk(0 &lt;gk&lt; 1,k= 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

scholarly journals Discrete Harmonic Functions in the Three-Quarter Plane

2021 ◽  
Author(s):  
Amélie Trotignon
Keyword(s):  
Functional Equation ◽  
Conformal Mapping ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Harmonic Functions ◽  
Symmetric Convex ◽  
Planar Lattice ◽  
Quarter Plane ◽  
Lattice Random Walks ◽  
Discrete Harmonic Functions

AbstractIn this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane—resolution of a functional equation via boundary value problem using a conformal mapping—to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.

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