Interpolation Results

In order to prove convergence results in particular Sobolev spaces, we need to know the smoothing properties of the semigroup Sε(t). Estimates from analytic semigroup theory

tell us which interpolation spaces ofLε the solutions will live in. We would therefore like

to obtain some embedding result between these interpolation spaces and the usual Sobolev spaces. It would be futile to look for an embedding result uniformly inε, the best we can do is the following lemma, which, for a price, grants us the ability to switch back and forth between interpolation spaces and Sobolev spaces.

Lemma 2.3.5. One has the following two inequalities

k(1−∂_x2)γfk_.ε−2γk(1− L_ε)γfk (2.37) k(1− L_ε)−γfk_.ε−2γk(1−∂_x2)−γfk (2.38)

for anyγ∈[0,1]and anyf for which the two norms are finite.

Proof. We start by proving the first inequality, the second will follow with a simple argu- ment. To prove the first claim we apply the Cald´eron-Lions interpolation theorem [RS75] to obtain a relationship between the interpolation spaces (in the notation of [RS75]) given by

k · k(0)_X =k · k, k · k_X(1) =k(1− Lε)· k,

It guarantees that, for the identity operatorI, one has kIk_L(X(γ)_,Y(γ)₎≤ kIk1 −γ L(X(0)_,Y(0)₎kIk γ L(X(1)_,Y(1)₎, (2.39)

whereX(γ) andY(γ)are the interpolation spaces given by completingL2[0,2π]with re- spect to the normsk(1− Lε)γ· kandk(1−∂x2)γ· krespectively.

It is clear that

kIk_L(X(0)_,Y(0)₎= 1,

since this is just the norm of the identity operator inL2[0,2π]. The first claim thus follows if we can show that

kIk_L(X(1)_,Y(1)₎.ε−2,

which is equivalent to proving that

k(1−∂_x2)(1− Lε)−1fk.ε−2kfk. (2.40)

We will achieve this by simplifying the operatorLεthrough two transformations. Firstly, for

the generatorL, one can easily find a change of variablesz=φ(x)with inversex=ψ(z)

such that Lf(x) = B(ψ(z))∂z+∂z2 (f◦ψ)(z), (2.41) whereB=√2_σb − _√1 2σ

0 _{andφsolves the ordinary differential equation}

φ0(x) = √1

2σ(φ(x)), (2.42)

with boundary conditionφ(0) = 0. Given this change of variables, it is easy to find the corresponding change of variables forLε, in fact, if we setz=εφ(x/ε)we have that

L_εf(x) = 1 εB(ψ(z/ε))∂z+∂ 2 z (f◦ψε)(z), (2.43)

whereψε(·) =εψ(·/ε). Secondly, we hope to make the operator self-adjoint. To do this,

we weight our space using the invariant measure of the underlying generator. Letg(y)be the invariant density for the generator

√ 2B(y) σ(y) ∂y+∂ 2 y

. One can show that

whereu=g(ψ(·/ε))1/2f ◦ψε. The Schr¨odinger operatorAεis defined by A_εu(z) = 1 ε2W(ψ(z/ε))u(z) +∂ 2 zu(z) whereW =g1/2 √ 2B σ ∂y+∂y2

g−1/2. We then have that

k(1−∂2_x)(1− L_ε)−1f(x)k ≤ε−2k(g−1/2)00k∞k(1− Aε)−1u(εφ(x/ε))k +ε−1k(g−1/2)0k∞k∂x(1− Aε)−1u(εφ(x/ε))k +kg−1/2k∞k∂2x(1− Aε)−1u(εφ(x/ε))k.

One can easily deduce the boundedness ofg−1/2and its derivatives from Assumption 2.2.2. Moreover, we have that

k∂x(1− Aε)−1u(εφ(x/ε))k2=k (1− Aε)−1u)0(εφ(x/ε) φ0(x/ε)k2 = 1 2π Z 2π 0 | (1− Aε)−1u)0(εφ(x/ε) φ0(x/ε)|2dx = 1 2π Z εφ(2π/ε) 0 |∂z(1− Aε)−1u(z)|2|φ0(ψ(z/ε))|dz ≤ kφ0k∞k∂z(1− Aε)−1uk2φ,

wherek · kφdenotes the usualL2norm but over the interval[0, εφ(2π/ε)]as in the integral

above. We can similarly show that

k∂_x2(1− A_ε)−1u(εφ(x/ε))k ≤ k(φ0)3k1_∞/2k∂_z2(1− A_ε)−1uk_φ +ε−1k(φ 00₎2 φ k 1/2 ∞ k∂z(1− Aε)−1ukφ.

We can deduce the boundedness of the above expressions involving φ using (2.42) and Assumption 2.2.2. We therefore have the bound

k(1−∂_x2)(1− L_ε)−1fk_.ε−2k(1− A_ε)−1uk_φ+ε−1k∂z(1− Aε)−1ukφ +k∂2_z(1− A_ε)−1uk_φ.

We now claim the following bounds to hold, as operator norms fromL2_φ→L2_φin the sense of the norm defined above:

k(1− A_ε)−1k_φ≤1, (2.45)

Note that these bounds immediately implyk∂z(1− Aε)−1kφ . ε−1 which follows from

the Cauchy-Schwartz inequality. These three operator bounds are enough to prove (2.40), since by changing back to thexvariables, we have that

kuk_φ=kg(x/ε)1/2f(x)(φ0(x/ε))1/2k ≤ kgk1_∞/2kφ0k1_∞/2kfk.

Hence we need only prove the claimed bounds. To prove (2.45), we utilise the identity

spec(1− A_ε) = spec (1− L_ε) ,

which follows from the fact thatAε andLε are conjugated via a bounded operator with

bounded inverse. SinceL_εgenerates a Markov semigroup, elements in its spectrum have positive real part. Since(1− A_ε)is self-adjoint in the Hilbert space generated by the norm k · kφwith the corresponding inner product, it thus follows that

k(1− A_ε)−1k_φ≤1

using the spectral theorem [Hai09]. By writing∂_z2in terms ofAεandW, we also have that

k∂_z2(1− A_ε)−1k_φ≤1 +k 1 + 1 ε2W(ψ(·/ε)) (1− A_ε)−1k_φ .1 +ε−2(1 +kWk∞)k(1− Aε)−1kφ,

which proves (2.46) and hence (2.37). To prove the second part of the lemma, just as in (2.40) it is sufficient to show that

k(1− Lε)−1(1−∂x2)fk ≤Cε−2kfk.

But we can use the fact that the operator norm is preserved under taking the adjoint, so that

k(1− L_ε)−1(1−∂_x2)k=k(1−∂_x2)(1− L∗_ε)−1k.

It is therefore sufficient to prove (2.40) with Lε replaced with its adjoint L∗ε. An easy

calculation shows that

L∗_ε =Le_ε+ 1 ε2U(x/ε) where e L_ε= 1 ε˜b(x/ε)∂x+ 1 2σ 2_(x/ε)∂2 x .

We can reduceLe_εto a Schr¨odinger operator with potentialWˆ in the same way that we did

forL_ε, and hence reduceL∗

claim then follows similarly to the first.

Remark 2.3.6. We would like to briefly comment on the sharpness of the two estimates obtained in Lemma 2.3.5. The second estimate (2.38) is sharp. In fact, in the caseσ = 1, upon rewriting the estimate in the adjoint setting, as done in the proof, it is clear that taking

f =ρ(x/ε)will prove sharpness. Unfortunately, this argument does not work for the first estimate (2.37). This comes down to the unlucky fact that the zero eigenvector ofLε is

the constant function (and notρ(x/ε)), which of course does not yield powers of εwhen integrated. In fact, we believe that estimate (2.37) is not sharp. However, improving the estimate would not considerably improve the strength of results in the sequel, so we do not attempt to do so.