Journal of Lie Theory Volume 11 (2001) 469–481 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Brownian Motion and the Heat Kernels of Iwasawa NA-Type Groups

M. J. Cornwall

Communicated by M. Cowling

Abstract. Let G be a semisimple Lie group with a finite centre and Iwasawa decomposition NAK. We shall consider the heat equation on NA and prove a structure theorem for a diffusion on the group which is typically called a Brownian motion. This theorem then shows how we may calculate the heat kernel of real hyperbolic space in a relatively easy fashion.

1. Introduction

Throughout this paper we shall consider a semisimple Lie group G with finite centre and Iwasawa decomposition NAK. The Lie algebra 𝔤 of G then has the decomposition 𝔫 ⊕ 𝔞 ⊕ 𝔨. We recall that the tangent vectors at the identity correspond to unique left invariant vector fields. Using the Killing form and a Cartan involution we may construct a Riemannian metric on G and hence on the component NA. We may then consider the heat equation on NA:

Δu = \frac{∂}{∂t}u,  (1)

where Δ is the Laplace–Beltrami operator associated with the metric. Using the root space decomposition of 𝔫 we may form a distinguished Laplacian L on NA,

L = \frac{1}{2} (∑_{k=1}^{n} Y_{k}^{2} + ∑_{k=1}^{m} H_{k}^{2}) ,

where {Y_{1}, …, Y_{n}} is an orthonormal basis of 𝔫 compatible with its root space decomposition and {H_{1}, …, H_{m}} is an orthonormal basis of 𝔞 (see [2]). To solve the heat equation on NA, it is then enough to solve the equation

Lu = \frac{∂}{∂t}u.  (2)

That we may consider this equation rather than (1) follows from the geometric relationship between them. The following discussion about this relationship was communicated to the author by Professor J. C. Taylor.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

For the solvable group NA that occurs in the Iwasawa decomposition of a semisimple Lie group we always have Δ = L + Y where Y = −2H_{ρ}, ρ is the half sum of the positive roots and H_{ρ} is the unique element of 𝔞 associated with ρ (see [13, Cor. A.1.2]). Now for any Riemannian manifold M there is an intrinsic number determined by the spectrum of −Δ in L^{2}. This number λ_{0}(M) is the minimum of the L^{2}-spectrum. If λ ≥ λ_{0}(M), there are positive global solutions of the equation Δu + λu = 0 (see [11, 14]). When M = G/K, a symmetric space of noncompact type, this constant λ_{0}(M) is ∥H_{ρ}∥ = ∥ρ∥. Consequently on NA we have (see [1])

e^{−ρ(log(a))} \frac{1}{2}Δ(fe^{ρ(log(a))}) + ∥ρ∥^{2}f = Lf.

Thus with u = fe^{ρ(log(a))},

\frac{1}{2}Δu + ∥ρ∥u = \frac{∂u}{∂t} if and only if Lf = \frac{∂f}{∂t} .

Hence if v(x, t) = e^{ct}u(x, t) and g(x, t) = e^{ct}f(x, t) we have

\frac{1}{2}Δv + (∥ρ∥ − c)v = \frac{∂v}{∂t} if and only if Lg − cg = \frac{∂g}{∂t} ,

and we may use the heat kernel for one value of c to determines all the others.

In this paper we shall consider the stochastic process (ξ_{t})_{t≥0} which has L as its generator and satisfies the stochastic differential equation

dξ_{t} = ∑_{k=1}^{n} Y_{k}(ξ_{t}) ∘ dB_{t}^{(k)} + ∑_{k=1}^{m} H_{k}(ξ_{t}) ∘ dB_{t}^{(n+k)},  (3)

where (B_{t})_{t≥0} is a standard (n+m)-dimensional Brownian motion and ∘ indicates a Stratonovich stochastic integral. This process is often called a Brownian motion on NA or an L-diffusion (see [5]). Using the structure of the Lie algebra 𝔫 ⊕ 𝔞 we show it is possible to write down the solution of (3) in a way which is compatible with the root space decomposition. Consequently since the density of ξ_{t} is the kernel p_{t} of (2) we may write p_{t} compatibly with this decomposition.

2. Review of Results

In this section we shall present some results for stochastic differential equations on manifolds that we shall require later. These results have been proved in [7, 8] and are collected here for convenience.

A key result in stochastic calculus is Itô’s lemma. In [8], Kunita extends the Itô formula and gives a general discussion about its formulation for processes on manifolds. We will now summarise this discussion by stating a proposition found in [8].

Proposition 2.1. Extended Itô Formula (see [8]) Suppose that M is a smooth n-dimensional manifold and (F_{t}(p))_{t≥0} is an ℝ-valued process which is continuous in (t, p) almost surely and satisfies

1. for each t > 0, p ↦ F_{t}(p) is a smooth map from M into ℝ almost surely 2. for each p, F_{t}(p) is a continuous semimartingale with the representation

F_{t}(p) = F_{0}(p) + ∑_{j=1}^{m} ∫_{0}^{t} f_{s}^{(j)}(p) ∘ dN_{s}^{(j)},

where (N_{s})_{s≥0} = ((N_{s}^{(1)}, …, N_{s}^{(m)}))_{s≥0} is a continuous semimartingale and

for each j, (f_{s}^{(j)}(p))_{s≥0} is a process, continuous in (s, p), such that
(a) for each s, p ↦ f_{s}^{(j)}(p) is a smooth map of M into ℝ

(b) for each p, (f_{s}^{(j)}(p))_{s≥0} the stochastic integral with respect to N^{(j)} is well defined.

Then if (ξ_{t})_{t≥0} satisfies the stochastic differential equation

dξ_{t} = ∑_{i=1}^{d} X_{i}(ξ_{t}) ∘ dM_{t}^{(i)},

where (M_{t})_{t≥0} is a continuous semimartingale, we have

F_{t}(ξ_{t}) = F_{0}(ξ_{0}) + ∑_{j=1}^{m} ∫_{0}^{t} f_{s}^{(j)}(ξ_{s}) ∘ dN_{s}^{(j)} + ∑_{i=1}^{d} ∫_{0}^{t}(X_{i}F_{s})(ξ_{s}) ∘ dM_{s}^{(i)}.

This result underlies the main result of [7] on the decomposition of solutions. Its proof is an easy application of Proposition 2.1, once we know that the stochastic process generates a stochastic flow. For then we may consider the notion of a “pullback” of a stochastic process and its inverse. Kunita shows in [7] that the processes we are considering do indeed generate stochastic flows. For a process (η_{t})_{t≥0} on the manifold satisfying the stochastic differential equation

dη_{t} = ∑_{i=1}^{m} X_{i}(η_{t}) ∘ dW_{t}^{(i)},

we define the “pullback” to be the process (η_{t⋆})_{t≥0}, where for each vector field X and smooth function f on the manifold we have

(η_{t∗}(X))f(p) = X(f ∘ η_{t})(η_{t}^{−1}(p))

for each p ∈ M. The inverse of the pullback (η_{t⋆}^{−1})_{t≥0} satisfies the stochastic differential equation

(η_{t∗})^{−1} = Id + ∑_{i=1}^{m} ∫_{0}^{t}(η_{s∗})^{−1} ad(X_{i}) ∘ dW_{s}^{(i)}.

Theorem 2.2. (see [7, Theorem 4.3]) Suppose that (η_{t})_{t≥0} and (ζ_{t})_{t≥0} are processes on M which satisfy the stochastic differential equations

dη_{t} = ∑_{i=1}^{m} X_{i}(η_{t}) ∘ dM_{t}^{(i)}

and

dζ_{t} = ∑_{j=1}^{d}(η_{t∗})^{−1}Y_{j}(ζ_{t}) ∘ dN_{t}^{(j)}.

Then the process (η_{t} ∘ ζ_{t})_{t≥0} satisfies the stochastic differential equation

d(η_{t} ∘ ζ_{t}) = ∑_{i=1}^{m} X_{i}(η_{t} ∘ ζ_{t}) ∘ dM_{t}^{(i)} + ∑_{j=1}^{d} Y_{j}(η_{t} ∘ ζ_{t}) ∘ dN_{t}^{(j)}.

We may use this theorem to reflect the structure of the Lie algebra generated by the vector fields of the manifold in the solution of the stochastic differential equation.

3. Solutions of Stochastic Differential Equations on Symmetric Spaces.

By repeatedly applying the theorem on the decomposition of solutions and the Dambis, Dubins–Schwarz theorem (see [6, Theorem 3.4.6]) we may write down the solution of a stochastic differential equation on a symmetric space as the combination of a solution of an stochastic differential equation on N and a solution of an stochastic differential equation on A.

Firstly let G be a real semisimple Lie group with a finite centre K and an Iwasawa decomposition NAK. We shall denote the Lie algebras by 𝔤, 𝔨 and 𝔫 ⊕ 𝔞 ⊕ 𝔨. Let Σ^{+} denote a set of positive roots.

Theorem 3.1. Suppose that {Y_{1}, …, Y_{n}, H_{1}, …, H_{r}} is an orthonormal basis of 𝔫 ⊕ 𝔞 such that {H_{1}, …, H_{r}} is an orthonormal basis of 𝔞 and {Y_{1}, …, Y_{n}} is an orthonormal basis of 𝔫 which is compatible with its root space decomposition. Consider the stochastic differential equation

dξ_{t} = ∑_{i=1}^{r} H_{i}(ξ_{t}) ∘ dB_{t}^{(i)} + ∑_{j=1}^{n} Y_{j}(ξ_{t}) ∘ dB_{t}^{(r+j)}
ξ_{0} = e,

and suppose that Σ^{+} = {λ_{1}, …, λ_{p}}, where the multiplicity of λ_{i} is m_{i}. Then (ξ_{t})_{t≥0} may be decomposed as ξ_{t} = η_{t} · α_{t}, where α_{t} = exp(H_{t}) with

H_{t} = ∑_{i=1}^{r} B_{t}^{(i)}H_{i},

and there is an n-dimensional Brownian motion (W_{t})_{t≥0} such that (η_{t})_{t≥0} may be decomposed as

η_{t} = ζ̃_{A_{t}^{λ_{1}}}^{1} ∘ ⋯ ∘ ζ̃_{A_{t}^{λp}}^{p},

where

dζ̃_{t}^{1} = ∑_{j=1}^{m_{1}} Y_{i}(ζ̃_{t}^{1}) ∘ dW_{t}^{(j)},

dζ̃_{t}^{i} = ∑_{j=k_{i}+1}^{k_{i}+m_{i}}(ζ̃^{i−1}(T_{t}^{i,i−1})_{∗})^{−1} ⋯ (ζ̃^{1}(T_{t}^{i,1})_{∗})^{−1}Y_{j}(ζ̃_{t}^{i}) ∘ dW_{t}^{(j)}

Cornwall 473

for i = 2, …, p,

A_{t}^{λ} = ∫_{0}^{t} exp(2λ(H_{s})) ds,
T_{t}^{λ} = inf{s : A_{s}^{λ} > t}

for a root λ, and for q = 1, …, i − 1,

T_{t}^{i,q} = t + ∫_{T_{t}^{λq}}^{T_{t}^{λ_{i}}} exp(2λ_{q}(H_{s})) ds.

Proof. By Theorem 2.2, we may decompose (ξ_{t})_{t≥0} as ξ_{t} = α_{t} ∘ η_{t}(e), where

dα_{t} = ∑_{i=1}^{r} H_{i}(α_{t}) ∘ dB_{t}^{(i)}
dη_{t} = ∑_{j=1}^{n}(α_{t∗})^{−1}Y_{i}(η_{t}) ∘ dB_{t}^{(r+j)}.  (4)

Since H_{1}, …, H_{r} commute, α_{t} = exp(H_{t}) where

H_{t} = ∑_{i=1}^{r} B_{t}^{(i)}H_{i}.

Note that then

(α_{t∗})^{−1} = e^{ad(H_{t})}.

Thus we may write (4) as

dη_{t} = ∑_{i=1}^{p} ∑_{j=k_{i}+1}^{k_{i}+m_{i}} e^{ad(H_{t})}Y_{j}(η_{t}) ∘ dB_{t}^{(r+j)},  (5)

where k_{1} = 0 and k_{i} = m_{1} + ⋯ + m_{i−1} for i = 2, …, p. Since each Y_{j} is an eigenvector we may rewrite (5) as

dη_{t} = ∑_{i=1}^{p} ∑_{j=k_{i}+1}^{k_{i}+m_{i}} Y_{j}(η_{t})e^{λ_{i}(H_{t})} ∘ dB_{t}^{(r+j)}.

Now for each j we may apply the Dambis, Dubins–Schwarz theorem to yield a standard Brownian motion (W_{t}^{(j)})_{t≥0} such that

dη_{t} = ∑_{i=1}^{p} ∑_{j=k_{i}+1}^{k_{i}+m_{i}} Y_{j}(η_{t}) ∘ dW_{A_{t}^{λ_{i}}}^{(j)},

where

A_{t}^{λ_{i}} = ∫_{0}^{t} exp(2λ_{i}(H_{s})) ds.

For notational convenience we shall now restrict our attention to the case where p = 2. By Theorem 2.2, we may write (η_{t})_{t≥0} as the decomposition of the solutions to the stochastic differential equations

dζ_{t}^{1} = ∑_{j=1}^{m_{1}} Y_{j}(ζ_{t}^{1}) ∘ dW_{A_{t}^{λ_{1}}}^{(j)}

dζ_{t}^{2} = ∑_{j=m_{1}+1}^{m_{1}+m_{2}}(ζ_{t∗}^{1})^{−1}Y_{j}(ζ_{t}^{2}) ∘ dW_{A_{t}^{λ_{2}}}^{(j)}.

By a corollary to the Dambis, Dubins–Schwarz Theorem (see [6, Proposition 3.4.8]) we have

ζ_{t}^{1} = ζ_{0}^{1} + ∑_{j=1}^{m_{1}} ∫_{0}^{A_{t}^{λ_{1}}} Y_{j}(ζ_{T_{s}^{λ_{1}}}^{1}) ∘ dW_{t}^{(j)},

where T_{s}^{λ_{1}} = inf{u : A_{u}^{λ_{1}} > s}. Therefore

ζ_{T_{t}^{λ_{1}}}^{1} = ζ_{0}^{1} + ∑_{j=1}^{m_{1}} ∫_{0}^{t} Y_{j}(ζ_{T_{s}^{λ_{1}}}^{1}) ∘ dW_{t}^{(j)},

and hence if (ζ̃_{t}^{1})_{t≥0} is the unique solution to the stochastic differential equation

dζ̃_{t}^{1} = ∑_{j=1}^{m_{1}} Y_{j}(ζ̃_{t}^{1}) ∘ dW_{t}^{(j)},

then ζ_{T_{s}^{λ_{1}}}^{1} = ζ̃_{t}^{1}, or in other words, ζ̃_{A_{t}^{λ_{1}}} = ζ_{t}^{1}. Applying the same corollary to (ζ_{t}^{2})_{t≥0}, we get

ζ_{t}^{2} = ζ_{0}^{2} + ∑_{j=k_{2}+1}^{k_{2}+m_{2}} ∫_{0}^{A_{t}^{λ_{2}}}(ζ_{T_{s}^{λ_{2}}∗}^{1})^{−1}Y_{j}(ζ_{T_{s}^{λ_{2}}}^{2}) ∘ dW_{t}^{(j)}.

Now ζ^{1}(T_{s}^{λ_{2}}) = ζ̃^{1}(A_{T_{s}^{λ_{2}}}^{λ_{1}}), and

A_{T_{s}^{λ_{2}}}^{λ_{1}} = ∫_{0}^{T_{s}^{λ_{1}}} e^{2λ_{1}(H_{u})} du + ∫_{T_{s}^{λ_{1}}}^{T_{s}^{λ_{2}}} e^{2λ_{1}(H_{u})} du

= s + ∫_{T_{s}^{λ_{1}}}^{T_{s}^{λ_{2}}} e^{2λ_{1}(H_{u})} du
= T_{s}^{2,1}

(say). Thus if we consider the unique solution to the stochastic differential equation

dζ̃_{t}^{2} = ∑_{j=k_{2}+1}^{k_{2}+m_{2}}(ζ̃^{1}(T_{t}^{2,1})_{∗})^{−1}Y_{j}(ζ̃_{t}^{2})dW_{t}^{(j)},

we have ζ̃_{A_{t}^{λ_{2}}}^{2} = ζ_{t}^{2}. For p > 2, a similar argument gives the other ζ̃_{t}^{i} and we are done.

Corollary 3.2. If (ξ_{t}^{i})_{t≥0} is the solution to the stochastic differential equation

dξ_{t}^{i} = ∑_{j=1}^{r} H_{j}(ξ_{t}^{i}) ∘ dB_{t}^{(j)} + ∑_{j=k_{i}+1}^{k_{i}+m_{i}} Y_{j}(ξ_{t}^{i}) ∘ dB_{t}^{(j)},

then there is an m_{i}-dimensional Brownian motion (W_{t})_{t≥0} such that (ξ_{t}^{i})_{t≥0} has the decomposition ξ_{t}^{i} = η_{A_{t}^{λ_{i}}} · α_{t} where α_{t} = exp(B_{t}^{(1)}H_{1} + ⋯ + B_{t}^{(r)}H_{r}) and

dη_{t} = ∑_{j=k_{i}+1}^{k_{i}+m_{i}} Y_{j}(η_{t}) ∘ dW_{t}^{(j)}.

It follows from the above corollary that we may immediately write down the density of the random variable η_{t} · α_{t} once we have the densities of η_{t} and α_{t}. For the density of α_{t} we have the following result from [15].

Proposition 3.3. (see [15, Section 6, Proposition 2]) Suppose (B_{t})_{t≥0} is a real Brownian motion starting from 0 and

A_{t} = ∫_{0}^{t} e^{2B_{s}} ds.

If we let P(A_{t} ∈ du | B_{t} = x) = a_{t}(x, u) dx, then we have

\frac{1}{\sqrt{2πt}}e^{−x^{2}/2t}a_{t}(x, u) = \frac{1}{u} exp(−(1 + e^{2x})/2u)Θ_{e^{x}/u}(t),

where

Θ_{e^{x}/u}(t) = \frac{e^{x}}{u\sqrt{2π^{3}t}} ∫_{0}^{∞} e^{−(y^{2}−π^{2})/2t} exp(−e^{x} cosh(y)/u) sinh(y) sin(πy/t) dy.

Thus from Corollary 3.2 and Proposition 3.3, we get the following result.

Proposition 3.4. Under the assumptions of Theorem 3.1, suppose dim 𝔞 = 1 and (ξ_{t})_{t≥0} is the solution of the stochastic differential equation

dξ_{t} = H_{1}(ξ_{t}) ∘ dB_{t}^{(1)} + ∑_{j=1}^{n} Y_{j}(ξ_{t}) ∘ dB_{t}^{j+1}
ξ_{0} = e,

with λ_{1}(H_{1}) = 1. Then there is an n-dimensional Brownian motion (W_{t})_{t≥0} such that ξ_{t} = η_{A_{t}} · α_{t}, where α_{t} = exp(B_{t}^{(1)}H_{1}) and

dη_{t} = ∑_{j=1}^{n} Y_{j}(η_{t}) ∘ dW_{t}^{j}.

Moreover ξ_{t} has the density

p_{t}(n, e^{x}) = \frac{1}{\sqrt{2πt}} ∫_{0}^{∞} q_{u}(n)a_{t}(x, u)e^{−x^{2}/2t} du,

where q_{t}(n) is the density of η_{t}.

We shall now use this proposition to calculate heat kernels of the real hyperbolic spaces.

4. The Heat Kernel of the NA-component of SO_{0}(n, 1)

The NA-component of SO_{0}(n, 1) may be identified with ℝ^{n−1} ⋊ℝ (see [1]). We may choose H ∈ 𝔞 such that, in local coordinates, we have

(Hf)(x, a) = a\frac{∂f}{∂a} (x, a).

Thus the A-component, ((0, α_{t}))_{t≥0} satisfies the stochastic differential equation

d(0, α_{t}) = H(0, α_{t}) ∘ dB_{t},

where (B_{t})_{t≥0} is a standard one-dimensional Brownian motion. Therefore α_{t} = exp(B_{t}).

If we consider N in isolation it is just ℝ^{n−1} and so the left Brownian motion on N is ((W_{t}, 1))_{t≥0}, where (W_{t})_{t≥0} is a standard (n − 1)-dimensional Brownian motion. Thus the N -component (η_{t})_{t≥0} of the Brownian motion on ℝ^{n−1} ⋊ ℝ^{+} is

η_{t} = (W_{A_{t}}, 1),

where

A_{t} = ∫_{0}^{t} e^{2B_{s}} ds,

and hence

η_{t} · α_{t} = (W_{A_{t}}, e^{B_{t}})

is the Brownian motion on ℝ^{n−1} ⋊ ℝ^{+}. This stochastic process has also been considered in [1]. The expression given by Bougerol for the Brownian motion (ζ_{t})_{t≥0} is

ζ_{t} = (∫_{0}^{t} e^{B_{s}} dW_{s}, e^{B_{t}}) ,

where (B_{t})_{t≥0} is a standard one-dimensional Brownian motion and (W_{t})_{t≥0} is a standard (n−1)-dimensional Brownian motion. However by the Dambis, Dubins– Schwarz theorem we have equality of the random variables in his expression and our own. In [1, Proposition 2.3.2] the expression for the density of η_{t} · α_{t} is also given. The proof given uses an expression derived by Lohoué and Rychener [9], which is ultimately based on pages 329 and 330 of the thesis of Takahashi [12]. Bougerol notes that the expression in [9] has some minor mistakes in the constants and corrects these.

Rather than follow the route of [9], we shall now give our own derivation of the heat kernel. The density of W_{t} is

P(W_{t} ∈ dx) = \frac{e^{−||x||/2t}}{(2πt)^{(n−1)/2}} dx.

Consequently, by Proposition 3.4, the density of (W_{A_{t}}, e^{B_{t}}) is given by

p_{t}(z, e^{x}) = \frac{1}{\sqrt{2πt}} ∫_{0}^{∞} \frac{e^{−||z||/2u}}{(2πu)^{(n−1)/2}} a_{t}(x, u) e^{−x^{2}/2t} du.

This expression for p_{t}(z, e^{x}) differs from that which appears in [1]. We shall now show how they are in fact the same by transforming our expression into the one given in [1].

Substituting in the expression for a_{t}(x, u) given in Proposition 3.3 we get

p_{t}(z, e^{x}) = \frac{e^{x}}{\sqrt{2π^{3}t}} ∫_{0}^{∞} \frac{e^{−(y^{2}−π^{2})/2t}}{(2π)^{(n−1)/2}} sinh(y) sin(πy/t) F (y; z, e^{x}) dy,

where

F (y; z, e^{x}) = ∫_{0}^{∞} \frac{e^{−(||z||+1+e^{2x})/2u}}{u^{(n−1)/2}} \frac{e^{−e^{x} cosh(y)/u}}{u^{2}} du

= ∫_{0}^{∞} u^{(n−1)/2}e^{−ue^{x}( \frac{1}{2} e^{−x}||z||+cosh(x)+cosh(y))} du

= \frac{Γ ((n + 1)/2)}{(e^{x}(\frac{1}{2}e^{−x}||z|| + cosh(x) + cosh(y)))^{(n+1)/2}} .

Hence

p_{t}(z, e^{x}) = κ(x, t) ∫_{0}^{∞} e^{−(y^{2}−π^{2})/2t} \frac{sinh(y) sin(πy/t)}{(cosh(r) + cosh(y))^{(n+1)/2}} dy,

where

κ(x, t) = (2π)^{−(n−1)/2} \frac{e^{−(n+1)x/2}}{\sqrt{2π^{3}t}} Γ((n + 1)/2)
cosh(r) = \frac{1}{2}e^{−x}∥z∥^{2} + cosh(x).

We shall now consider

I = ∫_{0}^{∞} e^{−(y^{2}−π^{2})/2t} \frac{sinh(y) sin(πy/t)}{(cosh(r) + cosh(y))^{(n+1)/2}} dy

= \frac{1}{2}Im(∫_{−∞}^{∞} e^{(y−iπ)^{2}/2t} \frac{sinh(y)}{(cosh(r) + cosh(y))^{(n+1)/2}} dy),

and suppose that n − 1 is of the form 2k + 1. On integrating by parts, we see that I is equal to

\frac{1}{2Γ ((2k + 1)/2)}Im(∫_{−∞}^{∞} (\frac{1}{sinh(y)} \frac{d}{dy} )^{k+1} (e^{(y−iπ)^{2}/2t})\frac{sinh(y)}{\sqrt{cosh(r) + cosh(y)}} dy).

To find the imaginary part of the integral we let R be large and ϵ be small and consider

∫_{γ_{1}} f(ξ) dξ − ∑_{i=2}^{8} ∫_{γ_{i}} f(ξ) dξ,

where

f : ℂ ∋ ξ ↦ (\frac{1}{sinh(ξ)} \frac{d}{dξ} )^{k+1} (e^{(ξ−iπ)^{2}/2t})\frac{sinh(ξ)}{\sqrt{cosh(r) + cosh(ξ)}}

[figure: Re; Im; γ_{1}; −R; γ_{2}; γ_{3}; ϵ; −r + iπ; γ_{4}; iπ; γ_{5}; ϵ; r + iπ; γ_{6}; γ_{7}; R; γ_{8}]

Figure 1: Odd case Contour

and each γ_{i} is as in Figure 1.

It is straightforward to show that the integrals along γ_{2} and γ_{8} tend to zero as R tends to infinity, and that

∫_{γ_{3}} f(ξ) dξ = ∫_{γ_{7}} f(ξ) dξ

= i ∫_{r+ϵ}^{R}(−1)^{k} (\frac{1}{sinh(y)} \frac{d}{dy} )^{k+1} (e^{−y^{2}/2t})\frac{sinh(y)}{\sqrt{cosh(y) − cosh(r)}} dy.

It is also easy to show that the integrals along γ_{4} and γ_{6} tend to zero as ϵ tends to zero. Finally the integral along γ_{5} is real, and hence we need not consider it further.

By letting R tend to ∞ and ϵ tend to zero the above considerations allow us to conclude that

I = \frac{1}{Γ ((2k + 1)/2)} ∫_{r}^{∞}(−1)^{k} (\frac{1}{sinh(y)} \frac{d}{dy} )^{k+1} (e^{−y^{2}/2t})\frac{sinh(y)}{\sqrt{cosh(y) − cosh(r)}} dy,

and hence

p(z, e^{x}) = κ_{2}(x, t) ∫_{r}^{∞} (\frac{1}{sinh(y)} \frac{d}{dy} )^{k+1} (e^{−y^{2}/2t})\frac{sinh(y)}{\sqrt{cosh(y) − cosh(r)}} dy,

where

κ_{2}(x, t) = (−1)^{k} \frac{(\sqrt{2π})^{−2k−1}}{\sqrt{π}} \frac{e^{−(2k+1)x/2}}{\sqrt{2πt}} .

The above expression is that given by [1] for the heat kernel of ℝ^{2k+1} ⋊ ℝ^{+}. When n − 1 is of the form 2k, we may follow a similar procedure. After integrating by parts we consider the contour integral

∫_{γ} f(ξ) dξ,

where

f : ℂ ∋ ξ ↦ (\frac{1}{sinh(ξ)} \frac{d}{dξ} )^{k} (e^{−(ξ−iπ)^{2}/2t})\frac{sinh(ξ)}{cosh(r) + cosh(ξ)}

[figure: Re; Im; γ_{1}; −R; γ_{2}; γ_{3}; 2πi; R; γ_{4}; ·; −r + iπ; ·; r + iπ]

Figure 2: Even case Contour

and γ is the contour in Figure 2.

After finding the imaginary part, we can show that

p_{t}(z, e^{x}) = (−2π)^{−k} \frac{e^{−kx}}{\sqrt{2πt}} (\frac{1}{sinh(r)} \frac{d}{dr})^{k} (e^{−r^{2}/2t}),

which is the expression given in [1] for the heat kernel of ℝ^{2k} ⋊ ℝ.

5. Some sub-Laplacians and heat kernels from H -type groups

For the NA-component of SO_{0}(n, 1) the Laplacian and the sub-Laplacian coincide. However we may also apply Proposition 3.4 for groups where N is an H - type group and consider the sub-Laplacian. Initially let us consider the Heisenberg group H_{2n+1} of real dimension 2n + 1. Accordingly we may realise H_{2n+1} as the group ℝ^{n} × ℝ^{n} × ℝ with

(x, y, t) · (u, v, s) = (x + u, y + v, t + s − ∑_{i=1}^{n}(y_{i}u_{i} − v_{i}x_{i})).

The process (η_{t})_{t≥0} on H_{2n+1} associated with the sub-Laplacian adapted to the root space decomposition is well known (see [3, 4]), namely

η_{t} = (X_{t}, Y_{t}, ∑_{i=1}^{n} ∫_{0}^{t} X_{s}^{(i)}dY_{s}^{(i)} − ∫_{0}^{t} Y_{s}^{(i)}dX_{s}^{(i)}),

where (X_{t})_{t≥0} and (Y_{t})_{t≥0} are independent n-dimensional Brownian motions. In [3, 4], we also find the density q_{t}(x, y, τ) of η_{t},

q_{t}(x, y, τ) = ∫_{−∞}^{∞} (\frac{2s}{sinh(2s)})^{n} exp (\frac{2isτ}{t} − \frac{∥x + y∥^{2}}{2t} \frac{2s}{tanh(2s)}) ds.

If we now consider H_{2n+1} ⋊ℝ^{+}, which is the NA-component of SU(n, 1), then the process on ℝ^{+} is again ((0, α_{t}))_{t≥0}, where α_{t} = exp(B_{t}). As before the density of the process on H_{2n+1} ⋊ ℝ^{+} is

p_{t}(x, y, τ, e^{v}) = \frac{1}{\sqrt{2πt}} ∫_{−∞}^{∞} q_{u}(x, y, τ)a_{t}(v, u)e^{−v^{2}/2t} du.

We may write this explicitly as

\frac{2}{(2π)^{n+3/2}} ∫_{−∞}^{∞} (\frac{2s}{sinh(2s)})^{n} \frac{e^{v}}{t^{1/2}} ∫_{0}^{∞} e^{−(ỹ^{2}−π^{2})/2t} sinh(ỹ) sin(πỹ/t) F (ỹ, s) dỹds,

where

F (y′, s; x, y, τ, e^{v}) = \frac{Γ(n + 2)}{(s∥x + y∥^{2}/2 tanh(2s) + e^{v}(cosh(v) + cosh(y′)) − isτ)^{n+2}} ,

by taking the required Laplace transform.

Suppose now that N is a generalized Heisenberg or H -type group. Then the heat kernel h_{t}(ν, ζ) of N is known (see [10]) to be

\frac{1}{2^{d}} \frac{1}{(2πt)^{k/2+d}} ∫_{0}^{∞} \frac{σ^{k/2}}{|ζ|^{k/2−1}} J_{k/2−1}(σ|ζ|/t) (\frac{σ}{sinh(σ)})^{d} exp (\frac{−|ν|σ}{4t tanh(σ)}) dσ,

where J_{k/2−1} is a Bessel function. Again by applying Proposition 3.4, we may write down the heat kernel p_{t}(ν, ζ, e^{x}) of N ⋊ ℝ^{+}, namely,

p_{t}(ν, ζ, e^{x}) = \frac{1}{\sqrt{2πt}} ∫_{−∞}^{∞} h_{u}(ν, ζ)a_{t}(x, u)e^{−x^{2}/2t} du.

This may be written more explicitly as

\frac{e^{x}}{\sqrt{2π^{3}t}} ∫_{0}^{∞} \frac{σ^{k/2}}{|ζ|^{k/2−1}} (\frac{σ}{2 sinh(σ)})^{d} ∫_{0}^{∞} \frac{e^{−(y^{2}−π^{2})/2t}}{(2π)^{k/2+d}} sinh(y) sin(πy/t)F (y, σ) dy dσ,

where

F (y, σ; ν, ζ, e^{x}) = \frac{Γ(k + d)}{(\sqrt{p^{2} + σ^{2}|ζ|^{2}})^{k/2+d+1}} P_{k/2+d}^{1−k/2} (\frac{p}{\sqrt{p^{2} + σ^{2}|ζ|^{2}}} ) ,

where P_{k/2+d}^{1−k/2} is a Legendre function and

p = \frac{|ν|^{2}σ}{4 tanh(σ)} + e^{x}(cosh(y) + cosh(x)).

The function F is a Laplace transform.

References

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[9] Lohoué, N., and T. Rychener, Die resolvente von Δ auf symmetrischen Räumen vom nichtkompakten typ, Comment. Math. Helv. 57 (1982), 445– 468.

[10] Randall, J., The heat kernel for generalized Heisenberg groups, J. Geom. Anal. 6 (1996), 287–316.

[11] Sullivan, D., Related aspects of positivity in Riemannian geometry, J. Diff. Geom. 25 (1987), 327–351.

[12] Takahashi, R., Sur les représentations unitaires des groupes de Lorentz généralisés, Bull. Soc. Math. France 91 (1963), 289–433.

[13] Taylor, J. C., The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of non-compact type, Contemp. Math. 73 (1988), 303–332.

[14] —, The Martin Compactification of a Symmetric Space of Non-Compact Type: An Introduction in: “Potential Theory”, ed. by M. Kishi., Walter de Gruyter, Berlin, New York, 1992.

[15] Yor, M., On some exponential functionals of Brownian motion, Adv. Appl. Prob. 24 (1992), 509–531.

M. J. Cornwall 1 Martin Place Sydney N.S.W. 2000 Australia JCornwal@macquarie.com.au

Received June 22, 2000 and in final form March 6, 2001