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Results tagged with laplacian
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user 16702
The Laplacian matrix is the representation of a graph in matrix form.
5
votes
2
answers
664
views
Closed formula for heat kernel
Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum_j \frac{d^2}{dx_j^2} + 2\sqrt{-1} \sum_j \lambda_j \frac{d}{dx_j} …
- 9,853
5
votes
Accepted
Stochastic interpretation of heat kernel on fiber bundle
Let $P\longrightarrow M$ be a $G$ principal bundle endowed with a connection $1$-form $\omega$ (which has values in the Lie algebra $\mathfrak{g}$). If $X_t^x$ denotes Brownian motion on $M$ starting …
- 9,853
4
votes
Accepted
heat kernel on closed manifolds - error in Chavel's book?
Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that
$$L(H_k * F) = (LH_k)*F - F,$$
so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then
$$ L p = LH_ …
- 9,853
3
votes
The complex heat kernel on a Riemann manifold
As far as I know, the term "Mehler Kernel" is used for the integral kernel of the heat equation corresponding the the harmonic oscillator,
$$ \partial_tu + \Delta u + x^2 u = 0.$$
The equation you are …
- 9,853