Reports of the Academy of Sciences of the USSR

1. Volume 141, No. 2

MATHEMATICS

E. B. Dynkin

NONNEGATIVE EIGENFUNCTIONS OF THE LAPLACE–BELTRAMI OPERATOR AND BROWNIAN MOTION IN SOME SYMMETRIC SPACES

(Presented by Academician A. N. Kolmogorov on 5 VI 1961)

1. Our aim is to describe all nonnegative solutions of the differential equation

\[ Af - cf = 0, \tag{1} \]

where \(c\) is a constant and \(A\) is the Laplace–Beltrami operator in a certain symmetric space \(E\) having negative curvature. We shall give a solution of this problem for the case when the motion group of the space \(E\) is isomorphic to the complex unimodular group of order \(l\). An analogous solution can also be given for other complex semisimple Lie groups.

The basis of the present investigation is Martin’s method \((^1)\), used by him to describe all nonnegative solutions of the Laplace equation in an arbitrary domain of Euclidean space.

1. Let \(L\) be an \(l\)-dimensional complex Euclidean space; \(G\) the group of all linear transformations of the space \(L\) with determinant 1; \(E\) the set of all transformations \(x \in G\) to which there corresponds a positive definite Hermitian form \((x\xi,\eta)\) \((\xi,\eta \in E)\). To each \(g \in G\) there corresponds a transformation \(S_g\) of the space \(E\), defined by the formula \(S_g x = g^* xg\). The characteristic roots of any operator \(x \in E\) are positive and may be written in the form \(e^{\rho_1}, e^{\rho_2}, \ldots, e^{\rho_l}\), where \(\rho_1 \geq \rho_2 \geq \cdots \geq \rho_l\) and \(\rho_1 + \cdots + \rho_l = 0\). We shall agree to denote the set \((\rho_1,\ldots,\rho_l)\) by \(\rho(x)\). In the space \(E\) there exists a Riemannian metric \(d(x,y)\), invariant with respect to all transformations \(S_g\) \((g \in G)\). This metric is determined uniquely up to a constant factor. It is completely determined if one requires that

\[ \frac{d(e,x)}{|\rho(x)|} \to 1 \]

as \(|\rho(x)| \to 0\).* We denote by \(A\) the Laplace–Beltrami operator corresponding to this metric.

Put \(\delta = (\delta_1,\ldots,\delta_l)\), where \(\delta_j = \frac12(l+1-2j)\). It is proved that for \(c < -\delta^2\) every nonnegative solution of equation (1) is equal to zero. Therefore in what follows one may assume that \(c \geq -\delta^2\).

1. A function \(h(x,y)\) will be constructed satisfying the following conditions: a) for each \(y \in E\), \(h(x,y)\), as a function of \(x\), satisfies equation (1) in the domain \(E \setminus y\); b)

\[ \lim_{y \to x}\frac{h(x,y)}{d(x,y)^{N-2}} = 1, \]

where \(N = l^2 - 1\) is the dimension of the space \(E\); c) \(h(x,y) \to 0\) as \(d(x,y) \to \infty\). We shall call such a function the Green’s function for equation (1).

Let \(I_\nu(z)\) and \(K_\nu(z)\) be Bessel functions of imaginary argument \((^2,\ \text{No. }3.7)\). For even \(l\) put

\[ \Phi_l(z)=\pi^{1/2}2^{2q-1/2}(2q)!\,[(4q)!]^{-1} z^{2q+1/2}\left(I_{-2q-1/2}(z)+I_{2q+1/2}(z)\right)= \]

\[ = e^{-z}\sum_{k=0}^{2q}\frac{1}{k!}\binom{2q}{k}\binom{4q}{k}^{-1}(2z)^k, \]

* By \(e\) is denoted the identity transformation. For any set of real numbers \(\rho=(\rho_1,\ldots,\rho_l)\) we put \(\rho^2=\rho_1^2+\cdots+\rho_l^2\), \(|\rho|=(\rho^2)^{1/2}\).

where \(q=\frac14 l^2-1\). For odd \(l\) put

\[ \Phi_l(z)=[(\nu-1)!]^{-1}2^{1-\nu}z^\nu K_\nu(z) =\sum_{m=0}^{\nu-1}(-1)^m(m!)^{-2} \binom{\nu-1}{m}^{-1}2^{-2m}z^{2m}+ \]

\[ +\sum_{m=0}^{\infty}[m!(\nu+m)!(\nu-1)!]^{-1} \left[2\ln \frac12 z-\psi(m+1)-\psi(\nu+m+1)\right], \]

where \(\nu=\frac12(l^2-3)\), \(\psi(m+1)=1+\frac12+\cdots+\frac1m-C\) (\(C\) is Euler’s constant).

Theorem 1. For each \(c\geq -\delta^2\), equation (1) has a Green’s function, which is given by the formula

\[ h(x,y)=\Phi_l(a|\rho|)|\rho|^{3-l^2} \prod_{j<k}\frac{\rho_j-\rho_k}{\operatorname{sh}^{1/2}(\rho_j-\rho_k)}, \]

where \(\rho=\rho(x^{-1/2}yx^{-1/2})\) and \(a=(\delta^2+c)^{1/2}\). This function is everywhere positive, and as \(d(x,y)\to\infty\)

\[ h(x,y)\sim a_l e^{-a|\rho|}|\rho|^{\frac12(3-l^2)} \prod_{j<k}\frac{\rho_j-\rho_k}{\operatorname{sh}^{1/2}(\rho_j-\rho_k)}, \]

where \(a_l\) is a certain constant.

1. We shall call a solution \(f\) of equation (1) minimal if \(f\geq 0\) and if every nonnegative solution \(\tilde f\) subject to the inequality \(\tilde f\leq f\) differs from \(f\) only by a constant factor. Let \(B\) denote the set of all bases of the space \(L\), and let \(R\) denote the set of all sets \(\rho=(\rho_1,\ldots,\rho_l)\) of real numbers satisfying the conditions \(\rho_1\geq\cdots\geq\rho_l\), \(\rho_1+\cdots+\rho_l=0\). For each \(b=(e_1,\ldots,e_l)\in B\), \(\rho\in R\), put

\[ f_{b,\rho}(x)=\prod_{k=1}^{l}[d_{b,k}(x)]^{-1-\rho_k+\rho_{k+1}}, \]

where

\[ d_{b,k}(x)= \left| \begin{array}{ccc} (xe_1,e_1)&\ldots&(xe_1,e_k)\\ \ldots&\ldots&\ldots\\ (xe_k,e_1)&\ldots&(xe_k,e_k) \end{array} \right|, \qquad \rho_{l+1}=1-\delta_l . \]

Put \(\rho\in R_c\) if \(\rho\in R\) and \(\rho^2=\delta^2+c\).

Theorem 2. The set of minimal solutions of equation (1) coincides with the set of functions \(f_{b,\rho}(x)\) \((b\in B,\rho\in R_c)\).

The group \(G\) acts in the space \(B\) by the formula \(g(e_1,\ldots,e_l)=(ge_1,\ldots,ge_l)\). It is not difficult to see that \(f_{b,\rho}(gx)=f_{gb,\rho}(x)\).

Consider the totality of all orthonormal bases of the space \(E\), and denote by \(V\) the set obtained from this totality by identifying proportional bases. (We call the bases \(\{e_j\}\) and \(\{e'_j\}\) proportional if \(e'_j=\lambda_j e_j\), where \(\lambda_j\) are certain complex numbers.) Since the functions \(d_{b,k}\) and \(f_{b,\rho}\) are the same for all proportional bases \(b\), it makes sense to speak of the functions \(d_{v,k}\), \(f_{v,\rho}\), corresponding to \(v\in V\).

Theorem 3. Every minimal solution of equation (1) can be represented, and moreover uniquely, in the form \(a f_{v,\rho}\) \((a>0,\ v\in V,\ \rho\in R_c)\). The formula

\[ f(x)=\int_{V\times R_c} f_{v,\rho}(x)\,d\mu \]

establishes a one-to-one correspondence between all nonnegative solutions of equation (1) and all finite measures* on \(V \times R_c\).

1. As is known \({}^{(3)}\), to each symmetric polynomial \(Q\) in the \(l\) variables \(\rho_1,\ldots,\rho_l\) there corresponds a certain differential operator \(A(Q)\) in the space \(E\), commuting with all shifts \(f(x)\to f(S_g x)\) \((g\in G)\). (In particular, to the polynomial \(Q=\rho_1^2+\cdots+\rho_l^2-\delta^2\) there corresponds the Laplace–Beltrami operator.) Functions that are eigenfunctions for all the operators \(A(Q)\) are called spherical functions.

Theorem 4. The set of all nonnegative spherical functions is given by the formula

\[ f(x)=\int_V f_{v,\rho}(x)\,d\mu, \tag{2} \]

where \(\rho\in R\); \(\mu\) is an arbitrary finite measure on \(V\). The pair \(\rho,\mu\) is uniquely determined by the function \(f\). If \(f\) is given by formula (2), then for every \(Q\)

\[ A(Q)f=Q(\rho)f. \]

The unitary operator \(g\) leaves invariant the set of all orthonormal bases of the space \(L\), and therefore induces a certain transformation of the set \(V\). Denote by \(\mu_0\) the probability measure on \(V\) invariant with respect to all such transformations.

A spherical function \(f\) is called zonal if \(f(S_g x)=f(x)\) for all unitary \(g\in G\). It follows from Theorem 4 that every nonnegative zonal spherical function can be written in the form

\[ f(x)=\int_V f_{v,\rho}(x)\,d\mu_0 \]

(cf. \({}^{(6)}\)); in particular,

\[ 1=\int_V \pi(x,v)\,d\mu_0, \]

where

\[ \pi(x,v)=f_{v,\delta}(x)=\prod_{k=1}^{l-1} d_{v,k}(x)^{-2}. \]

Theorem 5. For \(c\ne 0\), all nonzero nonnegative solutions of equation (1) are unbounded. The set of all bounded solutions of the equation \(Af=0\) is given by the formula

\[ f(x)=\int_V \pi(x,v)F(v)\,d\mu_0, \]

where \(F\) is an arbitrary bounded Borel function on \(V\). The function \(F\) is determined by \(f\) uniquely up to a set of measure zero.

The main theorem of the paper \({}^{(4)}\) follows at once from Theorem 5.

1. For each operator \(x\in E\) there exists an orthonormal eigenbasis. Denote by \(v(x)\) the corresponding element of the space \(V\). The function \(v(x)\) is, generally speaking, multivalued, but if all characteristic roots of \(x\) are pairwise distinct, then \(v(x)\) is uniquely determined by \(x\).

To the differential operator \(A\) there corresponds a certain continuous Markov process \(x_t\), which is called Brownian motion in the space \(E\) \({}^{(5)}\).

* All measures under consideration are assumed to be defined on the Borel subsets of the corresponding topological space.

Theorem 5. For any initial state \(x\), almost surely the limits

\[ \lim_{t\to\infty}\frac{\rho(x_t)}{|\rho(x_t)|}=\frac{\delta}{|\delta|}, \qquad \lim_{t\to\infty} v(x_t^*)=\eta . \]

exist. Here \(\delta\) is the vector defined in item 2, and the probability distribution of \(\eta\) is given by the formula

\[ P_x\{\eta\in\Gamma\}=\int_{\Gamma}\pi(x,v)\,d\mu_0 . \]

Moscow State University
named after M. V. Lomonosov

Received
5 VI 1961

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