Brownian motion, de Rham complex and index formulas (Q2752169)

The author gives a lapidary description of his results on the Atiyah-Singer index theorem for a relative de Rham complex and its interpretation in terms of a generalization of Brownian motion. A full description of the ideas of the paper can be found in the book [`Brownian motion and index formulas for the de Rham complex' (1998; Zbl 0917.58042)] of the author.NEWLINENEWLINENEWLINEIn the first two sections the author recalls classical results concerning the de Rham complex for both cases of manifolds with vacuous and non-vacuous boundary, namely a relation between homology of the de Rham complex and singular homology, the Hodge decomposition theorem, and expression of the Euler characteristic in terms of the index of a suitable elliptic operator or elliptic boundary value problem. The author concludes the second section with an interpretation of the above-mentioned boundary value problems in terms of Brownian motion. As this interpretation seems to lie at the very heart of the ideas presented in the paper we describe it in more detail. Equip a smooth manifold \(M^n\) with non-vacuous boundary \(\partial M^n\) with a Riemannian metric, which is of product nature near the boundary. In a tubular neighborhood \(B\) of the boundary any differential form \(\alpha\in \Omega M^n\) can be decomposed as \(\alpha= \alpha_1+ \alpha_2\wedge d\alpha\), where \(\alpha_1, \alpha_2 \in\Omega (B)\), and \(\alpha\) is the ``normal'' coordinate in \(B\). Then the Euler characteristic \(\chi(M, \partial M)\) (resp. \(\chi(M))\) equals the index of the ``Euler characteristic'' operator acting on differential forms on \(M\) subject to boundary condition \(B_r(\alpha)=0\), where \(B_r(\alpha)= \alpha_1 \mid \partial M\) (resp. \(B_a(\alpha) =0\), where \(B_a(\alpha)= \alpha_2 \mid \partial M)\). The author links \(B_a\) to the reflection phenomenon at the boundary \(\partial M\), i.e. a Brownian particle reflects at the boundary \(\partial M\). Similarly, \(B_r\) is linked to the absorption phenomenon, i.e. a Brownian particle is absorbed whenever it hits \(\partial M\). There are no further explanations of these concepts, which therefore remain a little mysterious.NEWLINENEWLINENEWLINEIn the third, and final, section the author generalizes these concepts to the case of a closed manifold \(X^n\) with a submanifold \(Y^m\). The generalization consists in defining an operator \(D\), the index of which is the Euler characteristic \(\chi(X,Y)\) of the singular cohomology complex of the pair \((X,Y)\). The appropriate operator turns out to be a (non-trivial) variation of the usual ``Euler characteristic'' operator \((d+\delta)_e\) acting on a Sobolev space of currents on \(X\) subject to certain ``boundary like'' condition on \(Y\). Then we get the following formula: NEWLINE\[NEWLINE\text{Index} (D)=\chi (X,Y)=\chi (X)-\chi (Y).NEWLINE\]NEWLINE This formula generalizes the formulas for the Euler characteristic for both the cases of closed manifold \(X\) (if we take empty \(Y)\) and a manifold \(M\) with boundary (if we take \(X\) to be the double of \(M\) and \(Y=\partial M\subseteq X)\).NEWLINENEWLINENEWLINEThe paper is concluded with a (suggestion of) Brownian-like motion interpretation of the formula above, namely a motion on \(X\setminus Y\) governed by the equation \({\partial\over \partial t}+\Delta (1+\Delta)^a\), where \(a\) is the integral part of \(1/2(n-m)\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].