Type: Article
Publication Date: 1986-01-01
Citations: 56
DOI: https://doi.org/10.1090/s0002-9947-1986-0825735-5
The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals left-brace upper G left-parenthesis t right-parenthesis right-brace"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G = \{ G(t)\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="GL left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>GL</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {GL}}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induces processes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals upper G upper G Superscript upper T"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X = G{G^T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals upper G Superscript upper T Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y = {G^T}G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is analysed. It is further shown that the orthonormal frame of eigenvectors of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ultimately behaves like Brownian motion on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>O</mml:mtext> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\text {O}}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, while that of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> converges to a limiting value. The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> process is that studied by Dynkin and Orihara. From a naive standpoint, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> process would seem to provide a more natural model.