Hand in y our solutions un til F rida y , June 22, 8.30 am (PO b o x 183 in V3-128)

Hand in y our solutions un til F rida y , June 22, 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 X, June 16

The aim of the first two exercises is to review the Markov property resp. its meaning for the resolvent, the generator and the corresponding bilinear form.

Exercise X.1 (6 points)

Let(Gλ)λ>0 be a strongly continuous contraction resolvent onL²(X, m), where(X, m)is a measure space. Let(T_t)_t>0 be the corresponding semigroup andA the corresponding generator, cf. Exercise VII.1. Show that the following are equivalent:

(i) (G_λ) is Markov.

(ii) (Tt)is Markov.

(iii) (Au,(u−1)⁺)≤0for every u∈D(A).

Exercise X.2 (6 points)

LetE :D(E)×D(E)→Rbe a symmetric closed form, where D(E) is a dense subspace ofL²(X, m). Let (G_λ)_λ>0 be the corresponding strongly continuous contraction resolvent.

Show that the following statements sare equivalent:

(i) u∈D(E), `≥0 =⇒ u∧`∈D(E),E(u∧`, u−u∧`)≥0.

(ii) u∈D(E) =⇒ u⁺∧1∈D(E),E(u⁺∧1, u−u⁺∧1)≥0.

(iii) u∈D(E) =⇒ u⁺∧1∈D(E),E(u+u⁺∧1, u−u⁺∧1)≥0.

(iv) (G_λ) is Markov.

The aim of the next exercise is to provide examples of continuous convolution semigroups of measures. One should think of νt as the distribution of a random variable Xt that describes the position of the corresponding stochastic process at timet >0assuming that it started at0∈Rat timet= 0.

Exercise X.3 (8 points)

(i) Prove that the family (ν_t)_t>0 of measures on B(R) defined respectively by a) ν_t=δ_at(dx) (a∈Rfixed) ,

b) ν_t=e^−t

∞

P

k=0 t^k

k! δ_ak(dx) (a≥0 fixed) , c) ν_t= ^√1

4πte^−x

2 4t dx d) νt= _π¹_t2+x^{t 2} dx

defines a continuous convolution semigroup. (ii) Computeν_t(R^d) is each case. (iii) It is a wonderful fact that continuous convolution semigroups are in a one-to-one correspondence with functions ψthrough the relation

νb_t(ξ)

1

= ˆ

e^i(x,ξ)ν_t(dx) =e^−tψ(ξ). Compute the functionψ in the examples above.

1The definition of the Fourier transform is slightly different from the one that we used in the study of Sobolev spaces. The definition used here is the standard one on the area of Probability Theory.

2