In this lecture, we show that the diffusion semigroup that was constructed in the previous lectures appears as the solution of a parabolic Cauchy problem. Under an ellipticity and completeness assumption, it is moreover the unique square integrable solution.
Proposition: Let be an essentially self-adjoint diffusion operator and let be the corresponding diffusion semigroup. Let , and let Then is a weak solution of the Cauchy problem
Proof: For , we have
If the operator is furthermore assumed to be elliptic, then as we have seen in the previous lecture, the map is smooth and therefore, the above solution is also strong.
We now address uniqueness questions. We need further assumptions that already have been met before. We consider an elliptic diffusion operator with smooth coefficients on such that:
- There is a Borel measure , symmetric and invariant for on ;
- There exists an increasing sequence , , such that on , and , as .
Under these assumptions we already know that is essentially self-adjoint. The next proposition implies that is the unique solution of the parabolic Cauchy problem.
Proposition Let be a diffusion operator that satisfies the above assumptions. Let be a smooth solution of the Cauchy problem where . Assume that . Then .
Proof: Let be as above. On one hand, we have .
On the other hand, we have .
From Cauchy-Schwarz inequality, we now have
We deduce that . As a conclusion we obtain that . Letting , yields and thus