×

Remark about heat diffusion on periodic spaces. (English) Zbl 0949.58027

Let \(M\) be a complete connected oriented \(n\)-dimensional Riemannian manifold. Let \(k(t,m_1,m_2)\) be the time-\(t\) heat kernel on \(M\). The usual ansatz to approximate \(k(t,m_1,m_2)\) is to say that \[ k(t,m_1,m_2)\sim P(t,m_1,m_2) e^{-{d(m_1, m_2)^2\over 4t}}, \] where \(e^{-{d(m_1, m_2)^2\over 4t}}\) is considered to be the leading term and \(P(t,m_1,m_2)\) is a correction term which can be computed iteratively. The main interesting question is the asymptotic behaviour for \(t\to 0\), \(t\to\infty\) and certain conditions for \(d(m_1, m_2)\). The author looks at the case when \(M\) has a periodic metric, meaning that \(\mathbb{Z}^k\) acts freely by orientation-preserving isometries on \(M\), with \(X= M/\mathbb{Z}^k\) compact and considers the asymptotic regime in which \(t\to\infty\) and \(d(m_1,m_2)\sim \sqrt t\). As the typical time-\(t\) Brownian path will travel a distance comparable to \(\sqrt t\), this is the regime which contains the bulk of the diffusing heat.
Let \({\mathcal F}\) be a fundamental domain in \(M\) for the \(\mathbb{Z}^k\)-action. Given \({\mathbf v}\in\mathbb{Z}^k\), put \[ k(t,{\mathbf v})= \int_{\mathcal F} k(t,m,{\mathbf v}\cdot m) d\text{ vol}(m). \] This is independent of the choice of fundamental domain \({\mathcal F}\). The covering \(M\to X\) is classified by a map \(\nu:X\to B\mathbb{Z}^k\), defined up to homotopy, which is \(\pi_1\)-surjective. It induces a surjection \(\nu_*: H_1(X;\mathbb{R})\to \mathbb{R}^k\) and an injection \(\nu^*:(\mathbb{R}^k)^*\to H^1(X; \mathbb{R})\). Let \(\langle\cdot,\cdot\rangle_{H^1(X;\mathbb{R})}\) be the Hodge inner product on \(H^1(X;\mathbb{R})\).
Definition. The inner product \(\langle\cdot, \cdot\rangle_{(\mathbb{R}^k)^*}\) on \((\mathbb{R}^k)^*\) is given by \[ \langle\cdot, \cdot\rangle_{(\mathbb{R}^k)^*}= {(\nu^*)^*\langle\cdot, \cdot\rangle_{H^1(X; \mathbb{R})}\over \text{vol}(X)}. \] The inner product \(\langle\cdot, \cdot\rangle_{\mathbb{R}^k}\) is the dual inner product on \(\mathbb{R}^k\).
Let \(\text{vol}(\mathbb{R}^k/ \mathbb{Z}^k)\) be the volume of a lattice cell in \(\mathbb{R}^k\), measured with \(\langle\cdot, \cdot\rangle_{\mathbb{R}^k}\). Then the author proves the following
Theorem. Fix \(C>0\). Then in the region \(\{(t,{\mathbf v})\in \mathbb{R}^+\times \mathbb{Z}^k: \langle{\mathbf v},{\mathbf v}\rangle_{\mathbb{R}^k}\leq Ct\}\), as \(t\to\infty\) we have \[ k(t,{\mathbf v})= {\text{vol}(\mathbb{R}^k/\mathbb{Z}^k)\over (4\pi t)^{k/2}} e^{-\langle{\mathbf v},{\mathbf v}\rangle_{\mathbb{R}^k}/(4t)}+ O(t^{-{k+ 1\over 2}}) \] uniformly in \({\mathbf v}\).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J65 Diffusion processes and stochastic analysis on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv