I am confused about equivalence of (i) and (i.a) in Theorem 10.19. I don't see how it follows from Theorem 10.11. Shouldn't one use a Laplace transform instead?
Best,
Anna
Christian Seifert, 2023/01/31 12:51
Dear Anna,
Many thanks; this is indeed a mistake. The equivalence follows from the Laplace transform.
Best,
Christian
Sascha Trostorff, 2023/02/02 10:25
Dear Christian,
I am still confused. I agree that you can use Laplace transform for the implication (i) ⇒ (ia) with the for all statements, but I do not see how to come back. I thought, one needs some exponential formula, which we not have at hand, or do I overlook something?
Best regards
Sascha
Marcel Schmidt, 2023/02/02 11:04
Dear Sasha,
first of all the formula in Theorem 10.19 (i.a) is not correct. It should read α(L+α)−11=1.
As discussed in the lecture (Theorem~10.9) the following formula holds for all α>0 and all x∈X:
(L+α)−11(x)=∫∞0e−tαe−tL1(x)dt
This shows two things:
- If e−tL1(x)=1 for all t>0, then (L+α)−11(x)=1/α.
- If e−sL1(x)<1 for some s>0, then by continuity this holds on a whole neighborhood of s. Hence, we obtain
∫∞0e−tαe−tL1(x)dt<∫∞0e−tαdt=1/α.
Best,
Marcel
Sascha Trostorff, 2023/02/03 09:36
Dear Marcel,
thanks a lot. I simply overlooked the pointwise continuity of the semigroup.
Best regards
Sascha
Ines Joel Tatang Demano, 2023/01/30 12:08
Good morning and thanks for the lecture on Stochastic complete graph with link to solution of Heat equation. Please, in practice, what represent the function Af(t) of the lemma 10.18 ?
Christian Seifert, 2023/01/30 16:44
Dear Ines,
The quantity Af(t) describes the heat content of the solution of the heat equation with initial condition f at time t. Think of f as being positive. Then f describes the heat density at time t=0, and the solution ut=e−tLf of the heat equation the heat density at time t. Now, we take the sum (or put differently the ℓ1-norm as our densities are positive) which gives Af(t), the total amount of heat at time t.
Best,
Christian
discussion/lecture12.txt · Last modified: 2022/11/15 18:12 by matcs
Discussion on Lecture 12
Dear lecturers,
thank you for the lecture!
Dear all,
I am confused about equivalence of (i) and (i.a) in Theorem 10.19. I don't see how it follows from Theorem 10.11. Shouldn't one use a Laplace transform instead?
Best, Anna
Dear Anna,
Many thanks; this is indeed a mistake. The equivalence follows from the Laplace transform.
Best, Christian
Dear Christian,
I am still confused. I agree that you can use Laplace transform for the implication (i) ⇒ (ia) with the for all statements, but I do not see how to come back. I thought, one needs some exponential formula, which we not have at hand, or do I overlook something?
Best regards Sascha
Dear Sasha,
first of all the formula in Theorem 10.19 (i.a) is not correct. It should read α(L+α)−11=1.
As discussed in the lecture (Theorem~10.9) the following formula holds for all α>0 and all x∈X:
(L+α)−11(x)=∫∞0e−tαe−tL1(x)dt
This shows two things:
- If e−tL1(x)=1 for all t>0, then (L+α)−11(x)=1/α.
- If e−sL1(x)<1 for some s>0, then by continuity this holds on a whole neighborhood of s. Hence, we obtain
∫∞0e−tαe−tL1(x)dt<∫∞0e−tαdt=1/α.
Best, Marcel
Dear Marcel,
thanks a lot. I simply overlooked the pointwise continuity of the semigroup.
Best regards
Sascha
Good morning and thanks for the lecture on Stochastic complete graph with link to solution of Heat equation. Please, in practice, what represent the function Af(t) of the lemma 10.18 ?
Dear Ines,
The quantity Af(t) describes the heat content of the solution of the heat equation with initial condition f at time t. Think of f as being positive. Then f describes the heat density at time t=0, and the solution ut=e−tLf of the heat equation the heat density at time t. Now, we take the sum (or put differently the ℓ1-norm as our densities are positive) which gives Af(t), the total amount of heat at time t.
Best, Christian