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By Cedric Villani
A memoir that makes an attempt at a scientific research of convergence to desk bound kingdom for yes sessions of degenerate diffusive equations, taking the final shape ${\frac{\partial f}{\partial t}}+ L f =0
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Nc + 1}, Ck h = 0. This will be achieved by finite induction on k. 4) is true for k = 0. Assume now that Cj h = 0 for j ≤ k; it is obvious that also Cj Ah = 0 for j ≤ k; then our assumption on Rk+1 implies that Rk+1 h = 0. So Ck+1 h = Ck Bh − BCk h − Rk+1 h = 0. 4). 5) ((h, Lh)) ≥ 1 Ah 2 Nc 2 + k=0 ak Ck Ah 2 2 + bk Ck+1 h 2 2 . 5. 6) ((h, Lh)) = Ah 2 k k k k ak [(I)A + (I)B ] + 2bk [(II)A + (II)B ] . + k=0 To alleviate the notation, assume for a moment that we are working in a real Hilbert space, so there is no need to take real parts (otherwise, just put real parts everywhere).
18 and deduce the finiteness of the Fisher information Iµ (f ) for positive time. Then the rest of the argument is based on Theorem 28 as in the first proof above. 8. The method of multipliers A crucial ingredient in the L2 treatment of the Fokker–Planck equation was the use of the mixed second derivative CAh = ∇v ∇x h to control the error term [B, C]h = (∇2 V )·∇v h. 2) thanks to well-chosen auxiliary operators, typically multipliers. In the case of the Fokker–Planck equation, this method leads to less general results; it is however of independent interest, and can certainly be applied to many equations.
This concludes the proof of Theorem 18. 10) a, Remark 23. There are other possible ways to conduct these calculations. In an early version of this work, the last term (III)A was rewritten in three different forms to create helpful terms in AC ∗ h 2 and A∗Ch 2 , at the cost of requiring additional assumptions on [C, C ∗ ]. 5. Generalization Now I shall present a variant of Theorem 18 which covers more general situations. Theorem 24. Let H be a Hilbert space, let A : H → Hn and B : H → H be unbounded operators, B ∗ = −B, let L := A∗A + B and K := Ker L.