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I was wondering if the conservation of mass which is obvious for the heat equation in $\mathbb{R}^n$ holds also for the heat kernel in a general compact manifold without boundary.

I mean I want to know if the heat kernel generally satisfies

$$\int dx K(x,y,t)=1$$

If so, may I have some references to look at?

Thanks in advance

Bric
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  • Very cool question. Some related questions about the heat kernel on compact manifolds: https://math.stackexchange.com/q/2374862/532409 , https://math.stackexchange.com/q/3268595/532409 , https://math.stackexchange.com/q/3881917/532409 , https://math.stackexchange.com/q/984254/532409 , https://math.stackexchange.com/q/2700522/532409 , MathOverflow: https://mathoverflow.net/q/331870/333546 . Regarding the "the conservation of mass" (on the torus): https://math.stackexchange.com/q/3759470/532409 – Quillo Apr 20 '22 at 14:28

1 Answers1

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Yes, it's true, and is easy to prove in this setting. Differentiate under the integral sign to get $$\frac{d}{dt} \int K(x,y,t)\,dx = \int \partial_t K(x,y,t)\,dx = \int \Delta K(x,y,t)\,dx$$ the Laplacian $\Delta$ taken in the $y$ variable. Now integrating by parts, $$\int \Delta K(x,y,t)\,dx = \int K(x,y,t) \Delta 1\,dx = 0$$ so that $\int K(x,y,t)\,dx$ is constant with respect to $t$.

To show it's a constant with respect to $y$, you could again differentiate under the integral sign to show that $F(y,t) = \int K(x,y,t)\,dx$ also solves the heat equation, and since it's independent of $t$, it is harmonic. The only continuous harmonic functions on a compact manifold without boundary are constants, by the maximum principle.

More generally, any Riemannian manifold for which this holds is said to be stochastically complete. It's well known (but not quite so easy to prove) that every complete Riemannian manifold with Ricci curvature bounded below is stochastically complete; see for instance

Hsu, Pei, Heat semigroup on a complete Riemannian manifold, Ann. Probab. 17, No. 3, 1248-1254 (1989). ZBL0694.58043.

and references therein.

Nate Eldredge
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  • Thank you very much! And can I say something about other $L^p$ norms as well? – Bric Jun 25 '21 at 17:14
  • @Bric: What do you want to say? – Nate Eldredge Jun 25 '21 at 18:00
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    Sorry it's not a precise question. I was wondering for example if there's an estimate for the $L^p$ norm like in the $R^n$ case. Actually the thing I really need is if I can have an estimate on $\sup_t|u(x,t)|=\int dy K(x,y,t)u_0(y)$ only having $u_0\in L^1$ – Bric Jun 25 '21 at 18:14
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    I don't have the results off the top of my head, but a good place to look is the book Heat kernels and spectral theory by E.B. Davies. – Nate Eldredge Jun 25 '21 at 18:16
  • Thank you. I had a look to the book you suggested and it is really good but I solved my doubt in another way ;) Anyway thanks again – Bric Jun 25 '21 at 20:15