Hand in y our solutions un til Monda y , July 16, 8.30 am (PO b o x 183 in V3-128)
total points: 20
Prof. Dr. Moritz Kaßmann Fakultät für Mathematik
Sommersemester 2018 Universität Bielefeld
Partial Differential Equations III Exercise sheet XIII, July 9
Exercise XIII.1 (3 points)
Let (X,B(X)) be a nice measure space, e.g., X a Polish space and B(X) the Borel σ-algebra. Let P :B(X)→B(X)be a linear operator1 satisfying P1 = 1and P f ≥0 if f ≥0. We say that a measureµis invariant for P if for every f ∈L1(dµ)∩L∞(dµ)
ˆ
P fdµ= ˆ
fdµ .
Letµ be invariant forP. Show thatP extends to a bounded linear operator on Lp(dµ) for 1≤p <∞.
Exercise XIII.2 (4 points)
(i) Fort >0, x∈R, definept(x) = (4πt)11/2 e−x
2
4t. Consider the semigroup (Tt)t>0 on B([0,1]), given by
Ttf(x) = ˆ1
0
X
k∈Z
f(y)pt(x−(k+y))dy .
Compute the infinitesimal generator. Which measure µis invariant for Tt? (ii) Consider the semigroup(Pt)t>0 onCb(Rd)given by
Ptf(x) = ˆ
f xe−t+yp
1−e−2t
νd(dy),
whereνd(dx) = (2π)1d/2e−|x|
2
2 dx. Compute the infinitesimal generator. It is sufficient if you provide the proof in the case d= 1 and guess the result for generald. Which measure µis invariant for Pt?
Given a semigroup and its infinitesimal generatorL, one can define related special bilinear forms. The carré du champ operator Γ and its higher iterationΓ2 are defined by
2Γ(f, g) =L(f g)−f L(g)−gL(f), 2Γ2(f, g) =LΓ(f, g)−Γ(f, Lg)−Γ(g, Lf),
1B(X)denotes the space of real-valued bounded measurable functions.
wheref andgare taken from some subspaceA ⊂D(L)that is closed under multiplication.
Givenρ∈Randn∈N∪ {∞}, we say that a semigroup satisfies thecurvature dimension inequality CD(ρ,n) if for every f ∈ A
Γ2(f, f)≥ρΓ(f, f) + 1 n(Lf)2
µ-almost everywhere, where all objects correspond to the semigroup under consideration.
Exercise XIII.3 (6 points)
(i) Show that in each of the cases (a) the heat semigroup on Rd, (b) the semigroup (Tt)t>0 from above, and (c) the semigroup(Pt)t>0 from above, the carré du champ
satisfies
Γ(f, g) =h∇f,∇gi
if one uses the appropriate invariant measure and the corresponding scalar product.
(ii) Show that the heat semigroup onRd satisfies CD(0,d).
(iii) Show that the semigroup(Pt)t>0 on Cb(Rd) from above satisfies CD(1,∞). Show that it does not satisfy CD(ρ,n) for any finiten.
Exercise XIII.4 (7 points)
Let (Pt)t>0 be any of the semigroups mentioned on this exercise sheet. Show that the following statements are equivalent:
(i) CD(ρ,∞) holds for someρ∈R. (ii) Γ(Ptf, Ptf)≤e−2ρtPt Γ(f, f)
for f ∈ A, t≥0.
(iii) Pt(f2)−(Ptf)2 ≤ 1−eρ−2ρtPt Γ(f, f)
for f ∈ A, t≥0.
(iv) Pt(f2)−(Ptf)2 ≥ e2ρtρ−1Γ Ptf, Ptf)
for f ∈ A, t≥0.
Hint: The usage of the auxiliary functionss7→e−2ρsPsΓ Pt−sf, Pt−sf and s→Ps(Pt−sf)2 might be helpful.
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