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8URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 177-186
DOI: 10.15826/umj.2022.2.015
BIHARMONIC GREEN FUNCTION AND BISUPERMEDIAN
ON INFINITE NETWORKS
Manivannan Varadha Raj
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632014, India varadharaj [email protected]
Venkataraman Madhu
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632014, India [email protected]
Abstract: In this article, we have discussed Biharmonic Green function on an infinite network and bimedian functions. We have proved some standard results in terms of supermedian and bimedian. Also, we have proved the Discrete Riquier problem in the setting of bimedian functions.
Keywords: Biharmonic Green function, Bimedian function, Dirichlet problem, Discrete Riquier problem, Hyperbolic networks.
1. Introduction
An electrical network X is a finite graph consisting of a finite number of nodes and branches, each branch connecting some two nodes. There is a certain resistance r(a, b) on each branch [a, b] connecting the nodes a and b in X; the reciprocal of the resistance is called the conductance c(a, b). Thus, an electrical network X can be considered as a graph {X, c(a, b)} with finite number of vertices (nodes) and a finite number of edges (branches); the non-negative conductance c(a, b) is positive if and only if a and b are neighbors, that is [a, b] is an edge in X. A vertex e in X is called a terminal vertex if e has only one neighbor in X. If a and b are neighbors, we write b ~ a. We assume also that if a ~ b, then there is only one branch [a, b] connecting a and b and there is no self-loops in X; there is no edge of the form [a, a] so that c(a, b) = 0 for all a in X. We also assume that there is always a path {a = a0, a1, a2,..., an = b} connecting two vertices a and b in X where ai ~ ai+1 for 0 < i < n — 1.
An electrical current regime voltage is considered on the finite network {X, c(a, b)} assuming the Ohm-Kirchhoff laws: if ^ is the potential function on X, when extremal currents are applied on X, then the voltage on the branch [a, b] is [^(b) — ^(a)] and the current is c(a, b)[^(b) — ^(a)] so that the total current at the node a is ^ c(a, b)[^(a) — ^(b)]. Based on these basic notions, the
condenser principle, the equilibrium principle, the minimum principle, etc. are proved on X in [4].
In an abstract sense, can we consider these principles on an infinite network in a meaningful manner? Is it possible to think of an infinite electrical network with Ohm-Kirchhoff laws suitably modified by Nash-Williams [10] in his remarkable paper on random walks and electrical currents
in networks, where it is shown that the random walk in probability theory has features analogous to electrical networks. A random walk considered as an irreducible, reversible Markov chain will serve as a model to develop a function theory on infinite graphs analogous to that of electrical networks (Abodayeh and Anandam [1, 2], Woess [12] and Zemanian [13]). Let {X,p(a,b)} stand for a countably infinite state X with the transition probabilities p(a, b). Assume that {X,p(a, b)} is irreducible, i.e., is it possible to move along a path from any state a to any other state b in X; it is also reversible, i.e., there is a function <^(a) > 0 on X such that ^(a)p(a, b) = ^>(b)p(b, a) for any two states a and b in X. Then, as an example of the analogy between random walks and electrical currents, consider two disjoint subsets A and B. Denote by 0(a) the probability that the walker starting at the state a reaches A before meeting any state in B. Then 0(a) = 1 for a € A and ■0(a) = 0 for a € B; if a € A U B, then <^(x) = ^ p(a, b)0(a) and, since ^ p(a, b) = 1, we have
Y p(a, b)[0(b) — 0(a)] = 0, which is equality analogous to the situation where the total current at
the node a is 0.
In a random walk, the Green function G(a, b) represents the expected number of visits that the walker starting at a makes to reach b. The function G(a, b) takes the value to if {X,p(a, b)} is recurrent, i.e., the walker starting at any state a comes back to a infinitely often; G(a, b) < to for all pairs a, b if {X,p(a, b)} is transient, i.e., the walker starting at a vertex a definitely wanders off from a. A situation similar to this occurs in the study of Riemann surfaces. If a Riemann surface R is parabolic, there is no Green potential on R. If R is hyperbolic, then there is a Green kernel on R.
When this analogy between random walks and functions on Riemann surfaces is properly developed, a successful application of function-theoretic methods on Riemann surfaces to solve problems in random walks on an irreducible, reversible {X,p(a, b)} is possible. For this case, we can define the Dirichlet norm on a, and then the functional analysis methods enable us to establish a correspondence between some function-theoretic problems on a Riemann surface and problems connected with a random walk on X. For example, Lyons [9], modifying a Royden criterion on Riemann surfaces, gives a necessary and sufficient condition for a reversible Markov chain to be transient. However, these arguments establishing relations between random walks and Riemann surfaces are valid only when it is assumed that the random walk is reversible. Intending to develop a function theory on infinite networks that will be applicable even in the case of non-reversible Markov chains, we adopt here potential theoretic methods on locally compact spaces. The basic result is the solution to a generalized Dirichlet problem in infinite networks; using which we introduce the analogous of balayage, maximum principle, equilibrium principle, condenser principle, the classifications based on the notions of transient and recurrent random walks, etc. in infinite networks.
2. Preliminaries
Let N be an infinite graph that is connected and locally finite but without self-loops [4, 7]. Let ^>(a, b) > 0 be a nonnegative number associated with each pair of vertices a and b in N such that ^>(a, b) > 0 iff b ~ a. Then {N, <^(a, b)} is called an infinite network. We do not assume that <^(a, b) is symmetric. Given a set B in N, say that a vertex a is an interior vertex of B if a and all its
o o
neighbors are in B; denote by B the set of all interior vertices of B, and let dB = B \ B. If f (a) is a real-valued function on B, write
Af (a) = £ p(a,b)[f (b) - f (a)]
o o
for any a € B. Say that f (a) is superharmonic on B if Af(a) < 0 for any a € B; and f (a) is said
o
to be harmonic on B if Af (a) = 0 for any x € B. A function f (a) on B is subharmonic if —f (a) is superharmonic on B. The following statements are valid.
(1) If {fn(a)} is a sequence of superharmonic functions on B and f(a) = limnfn(a) is real-valued on B, then f(a) is superharmonic on B; consequently, if {gn(a)} is a sequence of superharmonic functions on B such that g(a) = ^ngn(a) is finite for each a in B, then g(a) is superharmonic on B.
(2) Minimum Principle: If s(a) > 0 is superharmonic on N and s(a0) = 0 for some vertex ao, then s = 0.
(3) Greatest harmonic minorant: Let f be superharmonic on B and g be subharmonic on B such that f > g on B. Let u(a) be the sequence F of all subharmonic functions on B such that u < f. Let
A(a) = sup u(a) ueF
for x € B. Then A(a) is a harmonic function on B such that if A'(a) is another harmonic function and A' < f on B, then A' < A. We call A(a) the greatest harmonic minorant of f on B. Similarly, we define the least harmonic majorant of g on B.
o
(4) Generalised Dirichlet Solution: Let F be an arbitrary set in N and B C F. Suppose that u(a) is a real-valued function on F \ B such that there exist a superharonic function f and a subharmonic function g on F such that f > g on F and f > u > g on F \ B. Then there exists a function A on F such that A = u on F \ B and AA(a) = 0 for a € B. This generalised Dirichlet solution A on F is uniquely determined if F is a finite set.
3. Biharmonic Green Function
Definition 1 (Potential). A nonnegative superharmonic function p defined on a subset B is said to be potential if and only if the greatest harmonic minorant of p on B is 0.
Definition 2 (Bipotential). A potential u in (N,p) is said to be a bipotential if and only if (—A)u = p, where p is a potential in N. We say that N is a bipotential network if there exists a positive bipotential on N.
Definition 3 (Biharmonic Green function). For a fixed vertex z in N, a potential uz(a) in (N,p) is said to be the biharmonic Green function with biharmonic support {z} if and only if (—A)uz(a) = Gz(a), where Gz(a) is the harmonic Green function with harmonic support z.
Proposition 1. The biharmonic Green function exists on (N,p) if and only if there is a positive bipotential on (N, p).
Proof. Clearly, the binarmonic Green function is a bipotential. Conversely, let u be a positive bipotential, (—A)u = p. Then,
u(a) = ^ G(a, b)p(b). b
Let z be a fixed vertex. Then, for some A > 0, Gz(b) < Ap(b) for any b € N (Domination Principle). Hence,
Qz (a) = £ G(a,b)Gz (b) b
is a well-defined potential such that (—A)uz(a) = Gz(a). □
Theorem 1. Let (N,p) be a bipotential infinite network, and let uy (a) be the biharmonic Green potential on (N,p). IfYb f (b)ub(a) is finite at some vertex a0 for some f > 0, then
u(a) = ^ f (b)ub(a) b
is a bipotential on (N,p). Conversely, every bipotential u(a) can be represented as
u(a) = ^ f (b)ub(a), b
where f (a) = (—A)2u(a).
Proof. Let (—A)u = p on (N,p). For a finite set E in (N,p), write
s(a) = u(a) — ^ (—A)p(b)ub (a). bee
Then,
(—A)s(a) = p(a) — ^(—Aq)p(b)Gb(a) = A)p(b)Gb(a) > 0. bee b/e
Hence, s is superharmonic on (N,p), and since
—s(a) < ^(—A)p(b)ub(a), bee
we conclude that —s < 0. Hence,
u(a) > ^(—A)p(b)ub(a). bee
Allow E to grow into (N,p), to conclude that
u(a) > ^(—A)p(b)ub(a). bee
Write
<p(a) = q(a) — ^ (—A)p(b)ub(a).
be(N,p)
Then,
(—A)p(a) = p(a) — ^ (—A)p(b)Gb(a) = 0.
be(N,p)
Hence, ^>(a) is a nonnegative harmonic function majorized by the potential u(a). Hence, ^ = 0 so that
u(a) = ^ (—A)p(b)ub(a)
be(N,p)
for any a € (N,p). If f(a) = (—A)p(a), then f > 0 and f(a) = (—A)2u(a). Conversely, suppose that
u(a) = ^ f (b)ub(a),
which is a convergent sum of potentials if u(a0) is finite at some vertex a0. Then, u(a) is a potential in (N,p) and
(-A)u(a) = ^ f (b)(-A)ub(a) = ^ f (b)Gb(a), b b
being finite at each a, defines a potential p(a). Thus, (—A)u(a) = p(a). □
Proposition 2. Let (N,p) be a bipotential infinite network. For z € (N,p), if uz(a) and Gz(a) are the biharmonic and harmonic Green potentials, then uz (a) > Gz (a) for any a € (N,p).
Proof. Since (—A)uz(a) = Gz(a), (—A)Gz(a) = ¿z(a), and Gz(z) > Gz(a) for all a (Domination principle), we have
(-A)
Hence,
Mz(a)
> ¿z(a) = (—A)Gz(a).
u(a) = ^^y -Gz(a)
is a superharmonic function such that —u(a) < Gz(a); hence, — u < 0 on (N,p). Consequently,
uz (a) > Gz (z)Gz (a) > Gz (a)
since Gz (z) > 1. □
Proposition 3. Let u be a potential in (N,p), (—A)u = p. Suppose that 0 < f < p. Then, there exists a potential v, v < u, such that (—A)v = f on (N,p).
Proof. Let
u(a) = ^G(a,b)p(b) > ^G(a,b)f(b) = v(a), b b
then v(a) is a potential, v < u and (—A)v(a) = f (a) for all a € (N,p). □
Corollary 1. Let (N,p) be a bipotential infinite network. If u is a potentials with finite harmonic support in (N,p), then there exist is a bipotential v on (N,p) such that (—A)v = u on (N,p).
Proof. By hypothesis, there are postive potentials p and q such that (—A)u = p on (N,p). Since u has finite harmonic support, u < Ap on (N,p) for some A > 0 (Domination Principle). Hence use the above Proposition , there is a potential v < Aq such that (—A)v = u on (N,p). □
o
Lemma 1. Let F be a finite subset in any infinite network (N,p). Let E c F and f > 0 be a real-valued function on E. Then there exist a potential u on F such that (—A)u(a) = f (a) for any a € E.
Proof. Assume that f is defined on F by giving it values 0 in F \ E. Let G^(a) be the
o
Green function in F with point harmonic support b € F such that G^ (a) = 0 if a € dF. Let
u(a) = £ f (b)Gf (a).
beF
Then, u is a potential in F such that (—A)u(a) = f (a) if a € E. □
4. Transient and hyperbolic networks
Let {N,p(a, b)} be a countable set of state space N with transition probabilities {p(a, b)}. Let E be a fixed vertex in N. If a walker starting at e comes back to the state e infinitely often, i.e., with probability 1, then {N,p(a, b)} is said to be recurrent; otherwise, it is transient (see [5, 6]).
In classical potential theory (Brelot [8] and Al-Gwaiz M.A., Anandam V. [3]), a superharmonic function s(a) > 0 is called a potential if its greatest harmonic minorant is 0. In the discrete case, we can show that the superharmoinc function s(a) > 0 on an infinite network {N, <^(a, b)} is a potential if its greatest harmonic minorant is 0. If there exists a potential p(a) > 0 on N, then we say that {N, <^(a, b)} is a hyperbolic network; otherwise, it is called a parabolic network. Let {N, <^(a, b)} be an infinite network. Write
^(a) = Y1 ^(a,b).
b~a
Then 0 < <^(a) < to. Write
< ^(a, b)
p(a'6) = w
Then {N,p(a, b)} becomes a probability space, which need not be reversible. Therefore, we can say that {N, <^(a, b)} is transient when the associated probability space {N,p(a, b)} is transient.
Theorem 2. The infinite network {N, <^(a, b)} is transient if and only if it is hyperbolic.
Proof. Let e be a fixed vertex in N. Consider a sequence of finite subsets {Fn} such that
o o
e € Fi, Fn c Fn+i, and N = |JFn. For a vertex a in N, let 0n(a) denote the probability that
n
the walker starting at a reaches the vertex e before contacting any vertex in F^. Then 0n(e) = 1, 0n(a) = 0 for a € Fn, and
for a € {e} U {FjC }. Since for all a, we have
0n(a) = ^ p(a,b)0n(b) b
p(a, b) = 1
b
J>(a,b)[0n (b) — 0n(a)] =0
that is A0n(a) = 0 if a € {e} U {F^7}. Since {0n(a)} is an increasing sequence, 0(a) = lim0n(a)
nn
exists and 0 < 0(a) < 1 for all a in N. Clearly, 0(a) denotes the probability that the walker starting at {e} returns to {e}. Consequently, 0 = 1 if and only if N is recurrent. Hence, {N,0(a, b)} is transient if and only if 0 is not the constant 1.
Now, another interpretation of 0n(a) is that it is the Dirichlet solution with boundary values
o
0n(e) = 1 and 0n(a) = 0 if a € Fn. Hence, if we extend 0n to the whole space N assuming it equal to 0 on Fn, then 0n(a) becomes subharmonic at each vertex other than e, harmonic at each
o
vertex in Fn \ {e}, and superharmonic at e. Hence, in the limit, we find that 0(a) is a nonnegative superharmonic function on N that is harmonic outside the vertex e. Consequently, if 0 is not the constant 1, then 0 is a positive superharmonic function that is not harmonic on N. Let h(a) be the greatest harmonic minorant of 0(a) on N. Then, p(a) = 0(a) — h(a) is a positive superharmonic function that is a potential on N. That is, {N, 0(a, b)} is hyperbolic. Then, the following statements are equivalent:
(1) the function is not the constant 1;
(2) the probability space {N,p(a, b)} is transient;
(3) the infinite network {N, ^>(a, b)} is hyperbolic.
□
5. Bimedian functions on infinite networks
In this section, we assume that {N, ^>(a, b)} is an infinite network that is a tree without terminal vertices. Write
Au(a) = u(a) — ^ ^>(a, b)u(b) for a real-valued function u(a) on N. Note that A is the Lapalcian operator —A if
^(a) = Y1 ^(a,b) =1
for all a in N.
Definition 4. A real-valued function u(a) on N is said to be supermedian if
u(a) > ^ ^>(a, b)u(b)
for all a in N; u(a) is said to be median if
u(a) = ^ ^>(a, b)u(b)
for all a in N. Remark 1.
(1) A supermedian function is the same as superharmonic if and only if <^(a) = ^ ^(a, b) = 1 for all a in N.
(2) A solution to the Schrodinger equation corresponds to a median function if and only if ^>(a) < 1 for all a in N and ^>(a0) < 1 for at least one vertex ao in N.
We can develop a theory of supermedian functions exactly in the same way as the theory of discrete superharmonic functions. For example, we have the following.
(a) If u(a) is supermedian and v(a) is submedian such that u(a) > v(a) on N, then there exists a median function h(a) on N such that u(a) > h(a) > v(a); and if h (a) is another median function such that u(a) > h > v(a), then h (a) > h(a).
(b) If u(a) > 0 is supermedian, then there exists a unique decomposition u(a) = p(a) + h(a), where p(a) is a superpotential (i.e., a nonnegative supermedian function whose greatest median minorant is 0) and h(a) > 0 is a median function. Recall that a finite or infinite graph is known as a tree if there is no closed path of the form {a0, ai,..., an = a0} with more than 2 distinct vertices.
Corollary 2. Let {N, <^(a, b)} be an infinite tree without terminal vertex. Then, for any vertex e in N, there exists a supermedian function <^e(a) on N such that (a) is a median function at each vertex in N \ {e}, i.e., (a) is not median at e.
Proof. Let F be the set consisting of {e} and all its neighbors. Define a function u(a) on F such that u(a) = 1 and u(a) = 0 at each neighbor of e. Then extend v(a) to N as in the above theorem to get the function v(a) which equal to u(a) on F and is a median function at each vertex a = e. Note that u(a), and hence v(a), is superharmonic at e but not median. Denote the function v(x) by (a) to prove the statement in the corollary. □
Remark 2. This function (a) is an analog of the Newtonian potential function 1/|x| in R3 if there is a positive superpotential on N; otherwise, <^e(a) is an analog of the logarithmic function log(1/|x|) in R2.
Theorem 3. Let {N, <^(a, b)} be an infinite tree without terminal vertices. Let F be a connected subset of N, and let u(a) be a real-valued function on F. Then, there exists a real-valued function
o
v(a) on N such that v(a) = u(a) if a € F and v(a) is a median function at each vertex not in F.
Proof. Let ao € dF. Let {ai, a2,..., ak, bi, b2,..., bm} be the neighbors of ao, where {ai,..., ak} are in F and {bi,..., bm} are outside F. Note that the latter subset {bi, b2,..., bm} is non-empty since a0 € dF. Choose a constant A and define a function v(a) on F U {all neighbors of ao} such that
u(a) for a € F,
v(a) = | A for a € F. Now, if the constant A is chosen so that
k m
v(ao) = ^(ao,ai)u(aj) + A^i(ao,bj), i=i j=i
then v(a) is a median function at the vertex ao.
This procedure can be adopted with respect to each vertex on dF. Denoting this extended function also by v(a), we get a function v(a) defined on Nbr(F), which consists of F and all neighbors of each vertex in F such that v(x) = u(x) if a € F and v(a) is a median function at each vertex in dF.
Repeat this procedure with respect to v(x) as Nbr(f). Since N is a connected network,
N = ... Nbr[Nbr[Nbr(f)]] so that v(a) is a function defined on N such that v(a) = u(a) if a € F and v(a) is a median function
o
at each vertex not in F.
Theorem 4. Let f (a) be a real-valued function on N. Then there exists a function u(a) on N such that Au(a) = f (a) for every a in N.
Proof. From Theorem 3, we have a function <^e(a) such that (a) = A^e(a), where A > 0 is a constant and ¿e(a) is the Dirac function. Write qe(a) = 1/A ■ <^e(a). Thus, we conclude that given any vertex e in N, there exists a real-valued function qe(a) on N such that Aqe(a) = (a).
Take a finite exhaustion {En} of N, i.e., En is a non-empty finite set, En c En+1, and N = UEn. For n > 1, let
un(a) = qe(a)f (e).
eeE„+i\En
Then, Aun(a) = 0 for x € En+1 \ En and Aun(a) = f (a) for x € En+1 \ En. Define
u(a) = un(a),
n=1
where
U1(a) = ^ qe(a)f (e)
eeEx
is such that Au1(a) = 0 for x € E1 and Au1(a) = f (a) for a € E1. Note that the infinite sum
o
is well-defined. For, if a0 is any vertex in N, then a0 € Em for some m and ^un(a) is a convergent series consisting of functions that are median at the vertex a0. Consequently, u(a) is a well-defined function on N such that Au(a) = f(a) for all a € N.
Definition 5 (Bimedian) [11]. A real-valued function v(a) on N is said to be bimedian if there exists a median function u(a) on N such that Av(a) = u(a) for all a € N. If A is the Laplacian operator, then v(a) is called a biharmonic function on N.
Theorem 5 (Discrete Riquier problem). Let E be a finite subset of N. Let f and g be two real-valued functions on dE. Then, there exists a unique bimedian function v on E such that Av(a) = f (a) and v(a) = g(a) for a € dE.
o
Proof. Let h1 (a) be the unique Dirichlet solution on E such that Ah1(a) = 0 for a € E and h1(a) = f (a) for a € dE. By Theorem 4, we can choose a function s(a) on E such that As(a) = h1(a) on E.
o
Let h2(a) be the unique Dirichlete solution on E such that Ah2(a) = 0 for a € E and h2(a) = g(a) — s(a) on dE. Take v(a) = s(a) + h2(a). Then, v(a) = g(a) on dE and Av(a) = As(a) = h1(a)
oo
for a € E, so that A[Av(a)] = Ah1(a) = 0 for a € E; further, Av(a) = f (a) for a € dE. Thus, v(a) is the unique bimedian function on E such that Av(a) = f (a) and v(a) = g(a) for a € dE. □
oo
Acknowledgement
We thank the referee for very useful comments. The first author acknowledges the support given by the Vellore Institute of Technology through the Teaching cum Research Associate fellowship (VIT/HR/2019/5944 dated 18th September 2019).
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