Reports of the Academy of Sciences of the USSR
1958. Vol. 123, No. 3

MATHEMATICS

A. G. POSTNIKOV

SOLUTION OF A SYSTEM OF FINITE-DIFFERENCE EQUATIONS CORRESPONDING TO THE DIRICHLET PROBLEM BY MEANS OF A NORMAL SEQUENCE OF SIGNS

(Presented by Academician I. M. Vinogradov on 26 IX 1958)

Let \(g \geqslant 2\) be natural. Suppose there is an infinite sequence composed of the signs \(0, 1, 2, \ldots, g-1\):

\[ \varepsilon_1,\ \varepsilon_2,\ldots \tag{1} \]

Given a natural number \(s \geqslant 1\), write the first \(P+s-1\) signs of sequence (1) in the following form:

\[ (\varepsilon_1\varepsilon_2\ldots\varepsilon_s) (\varepsilon_2\varepsilon_3\ldots\varepsilon_{s+1})\ldots (\varepsilon_P\varepsilon_{P+1}\ldots\varepsilon_{P+s-1}) \tag{2} \]

and call the sequence of brackets (2) the caterpillar of length \(P\) (of rank \(s\)) of sequence (1). Let \(\Delta=(\delta_1\ldots\delta_s)\) be some group composed of \(s\) signs \(0,1,\ldots,g-1\). Denote by \(N_P(\Delta)\) the number of times \(\Delta\) occurs in the caterpillar of length \(P\).

We shall call sequence (1) a normal sequence of signs if, whatever \(s \geqslant 1\) we take and whatever group \(\Delta\) we prescribe,

\[ \lim_{P\to\infty}\frac{N_P(\Delta)}{P}=\frac{1}{g^s}. \tag{3} \]

It is easy to show that if \(\alpha\) is a number of the interval \([0,1]\) such that the fractional parts \(\{\alpha g^x\}\) are uniformly distributed, and if \(\alpha = 0,\varepsilon_1\varepsilon_2\ldots\) is the \(g\)-adic expansion of \(\alpha\), then the sequence \(\varepsilon_1,\varepsilon_2,\ldots\) is normal. Conversely, if sequence (1) is normal, then

\[ \alpha=\frac{\varepsilon_1}{g}+\frac{\varepsilon_2}{g^2}+\ldots \]

has the property that \(\{\alpha g^x\}\) are uniformly distributed.

There exist many methods of constructing normal sequences of signs \((^{1,2})\).

Denote \(\mu\Delta=1/g^s\) and call \(\mu\Delta\) the measure of the combination \(\Delta\). Consider a set \(\mathfrak{M}\) of various combinations \(\Delta\), in which there may be combinations of different lengths (different \(s\)), but combinations of infinite length do not enter into \(\mathfrak{M}\). By the measure \(\mu\mathfrak{M}\) of \(\mathfrak{M}\) we shall mean the sum of the measures of the combinations entering into it (if this sum is finite).

We shall consider sets \(\mathfrak{M}\) satisfying the following two requirements:

1. The set \(\mathfrak{M}\) is such that if a combination \((\delta_1\delta_2\ldots\delta_s)\) belongs to it, then the combinations \((\delta_1)\), \((\delta_1\delta_2)\), \(\ldots\), \((\delta_1\delta_2\ldots\delta_{s-1})\) do not belong to it.

2. Let \(k\) be an integer. In the combinations of \(\mathfrak{M}\) with a number of signs greater than or equal to \(k+1\), leave the first \(k\) signs. We obtain a set of combinations, which we denote by \(\overline{\mathfrak{M}}_{(k)}\).

Suppose that

\[ \lim_{k\to\infty}\mu\overline{\mathfrak{M}}_{(k)}=0. \]

Take in some normal sequence of signs (1) the \(k\)-th place and form the combinations \((\varepsilon_k)\), \((\varepsilon_k\varepsilon_{k+1})\), \((\varepsilon_k\varepsilon_{k+1}\varepsilon_{k+2})\ldots\) until it turns out that \((\varepsilon_k\varepsilon_{k+1}\ldots\varepsilon_l)\) belongs to \(\mathfrak M\). In that and only in that case, if this happens, we shall say that at the \(k\)-th place of the sequence (1) a combination from \(\mathfrak M\) has appeared. Denote by \(N_P(\mathfrak M)\) the number of appearances of combinations from \(\mathfrak M\) up to the \(P\)-th place of the sequence (1).

Theorem. If the sequence (1) is normal, and \(\mathfrak M\) satisfies the assumptions made above, then

\[ \lim_{P\to\infty}\frac{N_P(\mathfrak M)}{P}=\mu\mathfrak M . \]

Proof. Denote by \(\mathfrak M^{(s)}\) the set of combinations from \(\mathfrak M\) consisting of \(s\) signs. It is clear that for any \(k\)

\[ \sum_{s=1}^{k} N_P\bigl(\mathfrak M^{(s)}\bigr) \leq N_P(\mathfrak M)\leq \sum_{s=1}^{k} N_P\bigl(\mathfrak M^{(s)}\bigr)+N_P\bigl(\overline{\mathfrak M}_{(k)}\bigr). \]

Hence, by virtue of the normality of the sequence (1),

\[ \sum_{s=1}^{k}\mu\mathfrak M^{(s)} \leq \underline{\lim}_{P\to\infty}\frac{N_P(\mathfrak M)}{P} \leq \overline{\lim}_{P\to\infty}\frac{N_P(\mathfrak M)}{P} \leq \sum_{s=1}^{k}\mu\mathfrak M^{(s)}+\mu\overline{\mathfrak M}_{(k)}. \]

Since \(\mathfrak M\) contains no infinite combinations, then

\[ \lim_{k\to\infty}\sum_{s=1}^{k}\mu\mathfrak M^{(s)}=\mu\mathfrak M . \]

By the condition of the theorem,

\[ \lim_{k\to\infty}\mu\overline{\mathfrak M}_{(k)}=0 . \]

Letting \(k\) tend to infinity, we obtain

\[ \lim_{P\to\infty}\frac{N_P(\mathfrak M)}{P}=\mu\mathfrak M, \]

as was required.

In solving the Dirichlet problem in the plane \((^3)\) one has to solve the following system of linear equations. Suppose that in the plane a bounded simply connected domain \(D\) is given and that there is a square grid (see Fig. 1). We shall call a grid point a boundary point if at least one of its four neighboring points does not belong to \(D\), and an interior point if all four neighbors belong to \(D\).

Fig. 1

Suppose that at the boundary points some numbers \(\lambda_1,\lambda_2,\ldots,\lambda_s\) are prescribed. Consider the system of linear equations:

\[ u(x,y)=\frac14[u(x+1,y)+u(x-1,y)+u(x,y+1)+u(x,y-1)], \]

if \((x,y)\) is an interior point;

\[ u(x,y)=\lambda_i,\quad \text{if }(x,y)\text{ is a boundary point.} \tag{4} \]

The number of unknowns is equal to the number of interior points. We wish to determine \(u(x,y)\) at some fixed interior point. For this purpose we take a normal sequence composed of the signs \(0,1,2,3\):

\[ \varepsilon_1,\varepsilon_2,\varepsilon_3,\ldots \]

Suppose a particle moves along the grid points; let the sign \(0\) mean an order to the particle to move one unit to the left, the sign \(1\) an order to move one unit upward, the sign \(2\)—to the right, the sign \(3\)—downward; let the boundary points be absorbing points, i.e., once the particle reaches a boundary point, it remains there regardless of any orders. Fix some boundary-

boundary point \(B_i\) and denote by \(\mathfrak M_{B_i}\) the set of such combinations \((\delta_1\delta_2\ldots)\) that the particle, which was initially at the node \((x,y)\), following these orders, after the last order will be absorbed at the point \(B_i\). It is clear that the first requirement on \(\mathfrak M\) is satisfied. Obviously, \(\mu\mathfrak M\) is equal to the probability that a particle which begins random wandering at the point \((x,y)\) and can move each time in all four directions with probability \(1/4\) will eventually be absorbed at the point \(B_i\). Denote
\[ u_i(x,y)=\mu\mathfrak M_{B_i}. \]

In Feller’s book ((\(^{4}\), p. 307)) it is shown that \(u_i(x,y)\) satisfies the system of linear equations
\[ \begin{aligned} u_i(x,y)&=\frac14\bigl(u_i(x+1,y)+u_i(x-1,y)+u_i(x,y+1)+u_i(x,y-1)\bigr),\\ &\hspace{2.7em}\text{if }(x,y)\text{ is an interior point;}\\ u_i(x,y)&=1,\quad \text{if }(x,y)=B_i;\\ u_i(x,y)&=0,\quad \text{if }(x,y)\text{ is a boundary point distinct from }B_i. \end{aligned} \]

Further, in Feller’s book it is shown that the probability of “eternal wandering” is equal to zero, i.e.
\[ \lim_{k\to\infty}\overline{\mathfrak M}(k)=0 \]
(\(\overline{\mathfrak M}_{B_i(k)}\) does not depend on \(B_i\)). Applying the theorem, we obtain
\[ \lim_{P\to\infty}\frac{N_P(\mathfrak M_{B_i})}{P}=\mu\mathfrak M_{B_i}. \]

Further, it is obvious that
\[ u(x,y)=\sum_{i=1}^{s}\lambda_i\mu\mathfrak M_{B_i}, \]
i.e.
\[ u(x,y)=\lim_{P\to\infty}\sum_{i=1}^{s}\lambda_i\frac{N_P(\mathfrak M_{B_i})}{P}. \tag{5} \]

We carry out motions from the point \((x,y)\) according to the orders
\[ (\varepsilon_1\varepsilon_2\ldots)(\varepsilon_2\ldots)(\varepsilon_3\ldots)\ldots(\varepsilon_P\ldots)\ldots \]
and count how many times absorptions take place at the points \(B_1,B_2,\ldots,B_s\), i.e. determine the quantities \(N_P(\mathfrak M_{B_i})\), \(i=1,2,\ldots,s\). Formula (5) gives the solution of system (4).

REFERENCES

\(^{1}\) D. G. Champernowne, J. London Math. Soc., 8, 254 (1933).
\(^{2}\) N. M. Korobov, Izv. AN SSSR, matem. ser., 14, No. 3, 215 (1950).
\(^{3}\) R. Courant, K. Friedrichs, H. Lewy, Uspekhi matem. nauk, vol. 8, 125 (1941).
\(^{4}\) W. Feller, Introduction to Probability Theory and Its Applications, IL, 1952.