Proof of the main result

Z

O

ψ((ρδ∗µext)|^O) dx

= Z

R^d

ψ(ρδ∗µext)1_Odx

≤ Z

R^d

ψ(ρδ∗µext) dx≤ Z

R^d

ψ(µext) = Z

O

ψ(µ), which finishes the proof.

Corollary 3.3.25. Together with Remark 3.3.4, Proposition 3.3.24 immediately implies lim sup

ε&0

Z

O

ψ(µε,δ_ε)≤ Z

O

ψ(µ),

whereµ∈ M(O)and0< δε≤ ^w(ε)2 for eachε >0.

Proof of Theorem 3.3.8. Foruas in Theorem 3.3.8, we show that the sequence (un)_n∈N:=

u1 n,¹₂w(¹_n)

n∈N

, wherewwas defined in Remark 3.3.12, meets all requirements.

By construction,u_n ∈ M(O)∩H⁻¹ for alln∈N, and by Lemma 3.3.19, the density of u_n is bounded and thus in L²(O). Property (3.3.8) is proved in the first part of Proposition 3.3.20. For Property (3.3.9), note that Corollary 3.3.25 especially shows that (ψ(u_n))_n∈N is uniformly bounded in the TV norm, which means that it contains a subsequence that converges weakly* toψ(u) by Proposition 3.3.20, Corollary 3.3.25 and Lemma 2.1 in [45]. Since this argument can be carried out for any subsequence, we get weak* convergence for the whole sequence and, also by Lemma 2.1 in [45],

ψ

u1 n,¹₂w(_n¹)

_{T V} =

Z

O

ψ u1

n,¹₂w(¹_n)

→ Z

O

ψ(u) =kψ(u)kT V as n→ ∞. This yields (3.3.9) and thereby concludes the proof.

3.4 Proof of the main result

Throughout this section, we work under Assumptions 3.2.1.

We first solve a modified SPDE by the variational approach, which will yieldε-approximate solutions.

Moreover, we show improved regularity for those approximations, which is used later to prove their convergence to a limit inL²(Ω;C([0, T];H⁻¹)) forε→0.

We consider the SPDE

dX_t^ε=ε∆X_t^εdt+ ∆φ^ε(X_t^ε)dt+B(t, X_t^ε)dWt,

X₀^ε=x0, (3.4.1)

where we use the notation for the Yosida approximation of Appendix 3.D and assumex0∈L²(Ω,F⁰;L²).

Now and in the following we omit the domain O when using Lebesgue and Sobolev spaces as well as spaces of continuous or continuously differentiable functions, as introduced in Section 3.1.2.

Lemma 3.4.1. For allT >0, Problem (3.4.1) gives rise to a solution in sense of Definition 3.B.1 with respect to the Gelfand tripleV :=L²,→H⁻¹,→(L²)⁰=V⁰.

Proof. We prove that (3.4.1) fits into the framework of Appendix 3.B with the operator A(u) = ∆(εu+φ^ε(u)) foru∈L².

In [112, Example 4.1.11], it is shown that an operator A of the form u 7→ ∆(Ψ(u)) satisfies the four properties of Appendix 3.B with respect to the Gelfand triple L^p ,→ H⁻¹ ,→ (L^p)⁰, if the following conditions are satisfied.

(Ψ1) Ψ is continuous.

(Ψ2) For alls, t∈Rwe have

(t−s)(Ψ(t)−Ψ(s))≥0.

(Ψ3) There existp∈[2,∞), a∈(0,∞), c∈[0,∞) such that for alls∈Rwe have sΨ(s)≥a|s|^p−c.

(Ψ4) There existc3, c4∈(0,∞) such that for alls∈R

|Ψ(s)| ≤c₄+c₃|s|^p−1, where pis as in (Ψ3).

We briefly check (Ψ1) – (Ψ4) for Ψ :=εId_R+φ^ε. The first condition is satisfied by Lemma 3.D.2, the second one by the maximal monotonicity of φ^ε, together with [7, Corollary 2.1]. Using φ^ε(0) = 0 and again the monotonicity ofφ^ε, we obtainsφ^ε(s)≥0 and thereby

sΨ(s)≥sεId_R(s) =ε|s|².

Thus, (Ψ3) is satisfied for p= 2, a=εandc= 0. (Ψ4) is then clear by Lemma 3.D.2. Thus, Theorem 3.B.2 is applicable as required.

The following lemma provides an important estimate on the regularity of these approximate solutions and corresponds to [74, Lemma B.1]:

Lemma 3.4.2. Let ε >0, x₀ ∈L²(Ω,F⁰;L²) andT >0. Then for the solution (X_t^ε)_t∈[0,T] to (3.4.1) we have

E sup

t∈[0,T]kX_t^εk²2+εE Z T

0 kX_r^εk²H₀¹dr≤C(Ekx₀k²2+ 1) with a constantC >0 independent ofε.

Proof. Let (e_i)_i∈N ⊂ C₀² be a sequence of smooth eigenvectors to −∆, i. e. −∆e_i = λ_ie_i for some (λ_i)_i∈N⊂(0,∞), such that (e_i)_i∈Nis an orthonormal basis inH⁻¹. Such a sequence can be obtained by first choosing anL²-orthonormal basis of (−∆)-eigenvectors (˜e_i)_i∈N⊂C₀²⊂L², where

−∆˜ei =λie˜i for someλi>0. (3.4.2) Then, setting

e_i=p

λ_ie˜_i fori∈N

keeps (3.4.2) true for ˜ei replaced byei and makes (ei)i∈Nan orthonormal basis inH⁻¹ as required. The latter can be seen by computing fori, j∈N

hei, ejiH⁻¹ =p λiλj

−∆⁻¹e˜i,˜ej

L² = pλiλj

λ_i he˜i,˜ejiL² =δij.

We further letPⁿ :H⁻¹ →Hn := span{e1, . . . , en}be the H⁻¹-orthogonal projection onto the span of the firstneigenvectors, i. e.

Pⁿ(y) =

n

X

i=1

hy, eiiH⁻¹ei.

Recall that the unique variational solutionX^εto (3.4.1) is constructed in [112, Section 4.2] as a (weak) limit inL²([0, T]×Ω;L²) of the solutions to the Galerkin approximation

dX_tⁿ = εPⁿ∆X_tⁿdt+Pⁿ∆φ^ε(X_tⁿ)dt+PⁿB(t, X_tⁿ)dW_tⁿ X₀ⁿ = Pⁿx₀,

inHn, where for simplicity we omit theε-dependence ofXⁿ, and for an orthonormal basis (gi)_i∈NofU (as defined in Assumption 3.2.1 (A1)) we let

W_tⁿ =

n

X

i=1

J⁻¹(Wt), gi

Ugi.

We first note that forx∈Hn we have ∆x∈Hn ⊂L² and thusPn(−∆x) =−∆x. UsingX_tⁿ ∈Hn for allt∈[0, T], we have

hX_tⁿ, Pⁿ(−∆X_tⁿ)iL²=kXⁿk²H¹₀. We note by Lemma 3.D.3 and (3.2.5) that

|φ^ε(Xⁿ)|²≤C(1 + (Xⁿ)²),

so φ^ε(Xⁿ) ∈ L² since Xⁿ ∈ Hn ⊆ L². Thus, φ^ε(Xⁿ) ∈ H₀¹ by [126, Theorem 2.1.11], and we can compute

hXⁿ, Pⁿ(∆φ^ε(Xⁿ))i^L2 =

* Xⁿ,

n

X

i=1

h∆φ^ε(Xⁿ), eiiH⁻¹ei

+

L²

=

*

∆φ^ε(Xⁿ),

n

X

i=1

hXⁿ, eiiL²ei

+

H⁻¹

=

*

∆φ^ε(Xⁿ),

n

X

i=1

h−∆Xⁿ, eiiH⁻¹ei

+

H⁻¹

= h∆φ^ε(Xⁿ),−∆XⁿiH⁻¹

= h∆φ^ε(Xⁿ), XⁿiH⁻¹×H₀¹. Again by [126, Theorem 2.1.11], we obtain for allr∈[0, T]

h∆φ^ε(X_rⁿ), X_rⁿiH⁻¹×H₀¹ =− h∇X_rⁿ,∇φ^ε(X_rⁿ)iL² =−(φ^ε)⁰(X_rⁿ)kX_rⁿk²H₀¹ ≤0,

where we used that (φ^ε)⁰(X_rⁿ) ≥ 0 almost everywhere by the monotonicity of φ^ε. Along with the

finite-dimensional Ito formula, this can be used to estimate

Using lemma 3.E.1, the Burkholder-Davis-Gundy inequality (see e. g. [112, Appendix D]) and (3.2.3), we get for the stochastic integral term in (3.4.3)

E sup

We can now estimate from (3.4.3) and the previous calculation that

E sup

where we absorbed the second-to-last term in (3.4.3) into the terms with the constantsCand ˜C. Thus,

we get

E sup

r∈[0,T]

e^−KrkX_rⁿk²L²

+ε

Z T 0

e^−KrkX_rⁿk²H₀¹dr

≤4Ekx0k²L²+ 4(C−K)E Z T

0

e^−KrkX_rⁿk²L²dr+ ˜C,

and by choosingKlarge enough and multiplying bye^KT, which we absorb in the constant, we obtain E sup

r∈[0,T]kX_rⁿk²L²+εE Z T

0 kX_rⁿk²H₀¹dr≤C(Ekx₀k²L²+ 1).

Thus, (Xⁿ)n∈N is bounded in L²(Ω;L^∞([0, T];L²)) and in L²(Ω×[0, T];H₀¹). The latter is a Hilbert space, thus we can extract a weakly converging subsequence whose limit can be identified with the unique weakL²(Ω×[0, T];L²) limitX^ε. Furthermore, we can interpret the former as the dual space of L²(Ω;L¹([0, T];L²)) which is separable. Thus, we can extract a weak* converging subsequence whose limit can again be identified withX^ε. By weak (respectively weak*) lower-semicontinuity of the norms, we can thus pass to the limitn→ ∞to obtain the required inequality.

Proof of Theorem 3.2.6. The proof will be carried out in three steps. We first construct a solution candidate as a limit of solutions to (3.4.1). Then we show that this limit indeed is an SVI solution and we conclude by showing uniqueness, which relies on the same construction which was already used to show the existence of a solution.

Step 1: We begin by showing that the solutions (X^ε)_ε>0 to (3.4.1) forε→0 form a Cauchy sequence in L²(Ω;C([0, T];H⁻¹)). To this end, we first consider two of those solutions X^ε1, X^ε2 with respective initial conditionx¹₀, x²₀∈L²(Ω,F⁰;L²). By subsequently applying the Ito formula for the squared norm in Hilbert spaces (see e. g. [112, Theorem 4.2.5]) and the finite-dimensional Ito formula (see e. g. [114, IV.§3]), we have forK >0

e^−KtkX_t^ε1−X_t^ε2k²H⁻¹ =

x¹₀−x²₀

2 H⁻¹

+ 2 Z t

0

e^−Krhε1∆X_r^ε1−ε2∆X_r^ε2, X_r^ε1−X_r^ε2iH⁻¹dr + 2

Z t 0

e^−Krh∆φ^ε1(X_r^ε1)−∆φ^ε2(X_r^ε2), X_r^ε1−X_r^ε2iH⁻¹dr + 2

Z t 0

e^−KrhX_r^ε1−X_r^ε2, B(r, X_r^ε1)−B(r, X_r^ε2)dW_riH⁻¹

+ Z t

0

e^−KrkB(r, X_r^ε1)−B(r, X_r^ε2)k²L2(U,H⁻¹)dr

−K Z t

0

e^−KrkX_r^ε1−X_r^ε2k²H⁻¹dr.

(3.4.5)

We note that

hε1∆X_r^ε1−ε2∆X_r^ε2, X_r^ε1−X_r^ε2iH⁻¹=− Z

O

(ε1X_r^ε1−ε2X_r^ε2)(X_r^ε1−X_r^ε2)dx

≤C(ε₁+ε₂)

kX_r^ε1k²L²+kX_r^ε2k²L²

and, using Corollary 3.D.9 for the second step,

h∆φ^ε1(X_r^ε1)−∆φ^ε2(X_r^ε2), X_r^ε1−X_r^ε2i^H−1=

=− Z

O

(φ^ε1(X_r^ε1)−φ^ε2(X_r^ε2)) (X_r^ε1−X_r^ε2) dx

≤C(ε1+ε2)

1 +kX_r^ε1k²L²+kX_r^ε2k²L²

dt⊗dP-almost everywhere. Using this and the Lipschitz property (3.2.2) ofB, we continue (3.4.5) by

e^−KtkX_t^ε1−X_t^ε2k²H⁻¹ ≤

x¹₀−x²₀

2 H⁻¹

+C(ε1+ε2) Z t

0

e^−Kr

1 +kX_r^ε1k²L²+kX_r^ε2k²L²

dr + 2

Z t 0

e^−KrhX_r^ε1−X_r^ε2, B(r, X_r^ε1)−B(r, X_r^ε2)dW_riH⁻¹dr +C

Z t 0

e^−KrkX_r^ε1−X_r^ε2k²H⁻¹dr

−K Z t

0

e^−KrkX_r^ε1−X_r^ε2k²H⁻¹dr.

With (3.2.2), Lemma 3.4.2 and, as in (3.4.4), the Burkholder-Davis-Gundy inequality, we obtain

E sup

t∈[0,T]

e^−KtkX_t^ε1−X_t^ε2k²H⁻¹

≤CE

x¹₀−x²₀

2

H⁻¹ (3.4.6)

+C(ε₁+ε₂) E

x¹₀

2 L²+E

x²₀

2 L²+ 1

forK large enough, where we use the assumption thatx¹₀, x²₀ ∈L². If we assume thatx¹₀ =x²₀ =:x₀, (3.4.6) implies

E sup

t∈[0,T]

e^−KtkX_t^ε1−X_t^ε2k²H⁻¹

≤C(ε1+ε2)(Ekx0k²L²+ 1),

and thus, by completeness there exists a processX∈L²(Ω;C([0, T];H⁻¹)) satisfying (

Esup_t∈[0,T]kX_t^ε−X_tk²H⁻¹ →0 forε→0 X0=x0.

In particular, we have for each t ∈ [0, T] that X_t^ε → Xt for ε → 0 in L²(Ω;H⁻¹). Since X_t^ε is F^t -measurable by construction (see Theorem 3.B.2), so isX_t, which makes X an adapted process. If the initial condition is indeed inL², this will be the candidate for an SVI solution.

It remains to construct a solution candidate if the initial state is not inL²but a generalH⁻¹functional.

To this end, we first notice that for two different initial conditions x¹₀, x²₀ ∈ L²(Ω,F⁰;L²), we can construct the limit of the approximate solutions inL²(Ω;C([0, T];H⁻¹)) as before, call themX¹andX², respectively, and take the limitε1, ε2→0 in (3.4.6) to obtain

E sup

t∈[0,T]

e^−Kt

X_t¹−X_t²

2 H⁻¹

≤2E

x¹₀−x²₀

2

H⁻¹. (3.4.7)

Let now x0 ∈ L²(Ω,F⁰;H⁻¹) and select a sequence (xⁿ₀)n∈N ⊂ L²(Ω,F⁰;L²) such that xⁿ₀ → x0 in L²(Ω;H⁻¹) forn→ ∞. Let (X^ε,n)_ε>0,n∈Nbe the unique variational solutions to (3.4.1) with respective initial conditions (x₀)_n∈N, for which Lemma 3.4.2 applies. We first construct the sequence (Xⁿ)_n∈N as the unique limits in L²(Ω;C([0, T];H⁻¹)) obtained as in the argument above, and notice that it is a Cauchy sequence by (3.4.7). Thus, we obtain another limitX ∈L²(Ω;C([0, T];H⁻¹)) which we identify as a solution to (3.1.1) in the sense of Definition 3.2.4 in the following step.

Step 2: We show that the limit process satisfies the properties of Definition 3.2.4. Let ε > 0, x0 ∈ L²(Ω,F⁰;H⁻¹) and (xⁿ₀)n∈N ⊂ L²(Ω,F⁰;L²) such that xn → x ∈ L²(Ω;H⁻¹) for n → ∞. Let (X^ε,n)_ε>0,n∈N be the solutions to (3.4.1) with initial values xⁿ₀. For part (i) of Definition 3.2.4, we

apply Ito’s formula as in (3.4.5) to obtain fort∈[0, T] e^−KtkX_t^ε,nk²H⁻¹ =kx₀k²H⁻¹+ 2

Z t 0

e^−Krhε∆X_r^ε,n, X_r^ε,niH⁻¹dr + 2

Z t 0

e^−Krh∆φ^ε(X_r^ε,n), X_r^ε,niH⁻¹dr + 2

Z t 0

e^−KrhX_r^ε,n, B(r, X_r^ε,n)dWriH⁻¹dr +

Z t 0

e^−KrkB(r, X_r^ε,n)k²L2(U,H⁻¹)dr

−K Z t

0

e^−KrkX_r^ε,nk²H⁻¹dr.

(3.4.8)

Note that we have

hε∆X_r^ε,n, X_r^ε,niH⁻¹=−εkX_r^ε,nkL² ≤0.

With the notation of Appendix 3.D and setting ϕ^ε(v) =

(R

Oψ^ε(v)dx, v∈L^m+1,

+∞, otherwise, (3.4.9)

forv∈H⁻¹, m∈(0,1] as in Assumption 3.2.1 (A5) in the superlinear case, i. e. if (3.2.6) is satisfied, and m= 0 in the sublinear case, i. e. if (3.2.7) is satisfied. We can useφ^ε=∂ψ^ε, the fact thatφ^ε(X^ε,n)∈H₀¹ dt⊗P-almost everywhere by Lemma 3.4.2 and Lemma 3.D.2, and the chain rule for Sobolev functions (see e. g. [126, Theorem 2.1.11]), to obtain

h∆φ^ε(X_r^ε,n), X_r^ε,niH⁻¹ =h−∆φ^ε(X_r^ε,n),0−X_r^ε,niH⁻¹

≤ϕ^ε(0)−ϕ^ε(X_r^ε,n) =−ϕ^ε(X_r^ε,n). (3.4.10) Furthermore, we can use (3.2.2) and (3.2.4) to obtain

kB(t, X^ε,n)k²L2(U,H⁻¹)≤2

kB(t, X^ε,n)−B(t,0)k²L2(U,H⁻¹)+kB(t,0)k²L2(U,H⁻¹)

≤C(1 +kX^ε,nk²H⁻¹).

Thus, (3.4.8) implies E

e^−KtkX_t^ε,nk²H⁻¹

≤Ekxⁿ₀k²H⁻¹−2E Z t

0

e^−Krϕ^ε(X_r^ε,n) dr + (C−K)E

Z t 0

e^−KrkX_r^ε,nk²H⁻¹dr+ Z t

0

Ce^−Krds.

ChoosingK large enough, we get E

e^−KtkX_t^ε,nk²H⁻¹

≤Ekxⁿ₀k²H⁻¹+C−2e^−KtE Z t

0

ϕ^ε(X_r^ε,n) dr (3.4.11) and thus, by choosingt=T and multiplying with ¹₂e^KT,

E Z T

0

ϕ^ε(X_r^ε,n) dr≤C(1 +Ekxⁿ₀k²H⁻¹)≤C <˜ ∞, (3.4.12) for someC,C >˜ 0. Note that ˜Ccan be chosen independent ofεandndue to the convergence of (xⁿ₀)_n∈N tox₀. By Assumption 3.2.1 (A4) we can use Corollary 3.D.7 to obtain forv∈L²

|ϕ^ε(v)−ϕ(v)| ≤ Z

O|ψ^ε(v)−ψ(v)|dx

≤ Z

O

Cε(1 +v²) dx

=Cε(1 +kvk²L²).

(3.4.13)

SinceX^ε,n∈L² dt⊗P-almost everywhere by Lemma 3.4.2, this leads to E

Z T 0

ϕ^ε(X_r^ε,n) dr≥E Z T

0

ϕ(X_r^ε,n) dr−CεE Z T

0

1 +kX_r^ε,nk²L² dr. (3.4.14) With these statements about fixed values of ε, we can now consider the limit ε → 0. Taking into account that ϕ is convex and lower-semicontinuous as shown in Section 3.3 and that X^ε,n → Xⁿ in L²(Ω;C([0, T];H⁻¹)) and thus inL²(Ω×[0, T];H⁻¹), we can use [19, Proposition 16.50] and (3.4.14) to obtain

E Z T

0

ϕ(X_rⁿ) dr≤lim inf

ε→0 E Z T

0

ϕ(X_r^ε,n) dr

≤lim inf

ε→0 E

Z T 0

ϕ^ε(X_r^ε,n) dr+CεE Z T

0

1 +kX_r^ε,nk²L² dr

! .

(3.4.15)

Since, by Lemma 3.4.2, the last term converges to 0 forε→0 andn∈Nfixed, we deduce that E

Z T 0

ϕ(X_rⁿ) dr≤lim inf

ε→0 E Z T

0

ϕ^ε(X_r^ε,n) dr. (3.4.16) Thus, taking lim infε→0 in (3.4.12) and then lim infn→∞, using lower-semicontinuity ofϕas in (3.4.15), we obtain

E Z T

0

ϕ(Xr) dr≤C(1 +kx0k²H⁻¹)<∞, as required.

For the variational inequality part, let G, Z, t be as in Definition 3.2.4 (ii). Ito’s formula (e. g. [112, Theorem 4.2.5]) then implies for allt∈[0, T]

EkX_t^ε,n−Ztk²H⁻¹=Ekxⁿ₀−Z0k²H⁻¹

+ 2E Z t

0 hε∆X_r^ε,n+ ∆φ^ε(X_r^ε,n)−Gr, X_r^ε,n−ZriH⁻¹dr +E

Z t

0 kB(r, X_r^ε,n)−B(r, Zr)k²L₂(U,H⁻¹)dr.

Analogous to (3.4.10), we have

h∆φ^ε(X_r^ε,n), X_r^ε,n−ZriH⁻¹+ϕ^ε(X_r^ε,n)≤ϕ^ε(Zr) (3.4.17) dt⊗P-almost everywhere, where we recall that both X^ε,n and Z are in L² dt⊗P-almost everywhere.

Moreover, using the weighted Young inequality,

hε∆X_r^ε,n, X_r^ε,n−ZriH⁻¹ ≤εk∆X_r^ε,nkH⁻¹kX_r^ε,n−ZrkH⁻¹

≤1

2ε⁴³k∆X_r^ε,nk²H⁻¹+1

2ε²³kX_r^ε,n−Z_rk²H⁻¹

(3.4.18)

dt⊗P-almost everywhere. Hence, by (3.2.2), (3.4.17) and (3.4.18), EkX_t^ε,n−ZtkH⁻¹+ 2E

Z t 0

ϕ^ε(X_r^ε,n) dr

≤Ekxⁿ₀−Z₀k²H⁻¹+ 2E Z t

0

ϕ^ε(Z_r) dr (3.4.19)

−2E Z t

0 hGr, X_r^ε,n−ZriH⁻¹dr+CE Z t

0 kX_r^ε,n−Zrk²H⁻¹dr + 2E

Z t 0

1

2ε⁴³k∆X_r^ε,nk²H⁻¹+1

2ε²³kX_r^ε,n−Zrk²H⁻¹dr.

As for (3.4.15), we have

E Z t

0

ϕ(X_rⁿ) dr≤lim inf

ε→0 E Z t

0

ϕ^ε(X_r^ε,n) dr. (3.4.20)

We notice that by Z ∈L² dt⊗P-almost everywhere, we have ϕ^ε(Zr)≤ϕ(Zr) due to Corollary 3.D.5.

Moreover, any other term in (3.4.19) converges becauseX^ε,n→XⁿinL²(Ω;C([0, T];H⁻¹)), the require-ment ofGbelonging toL²(Ω×[0, T];H⁻¹) and Lemma 3.4.2. Thus, we can take lim inf_ε→0 in (3.4.19) to obtain

E Z t

0

ϕ(X_rⁿ) dr≤ −1

2EkX_tⁿ−Z_tkH⁻¹+1

2Ekxⁿ₀ −Z₀k²H⁻¹+E Z t

0

ϕ(Z_r) dr

−E Z t

0 hGr, X_rⁿ−ZriH⁻¹dr+1 2CE

Z t

0 kX_rⁿ−Zrk²H⁻¹dr.

Now taking lim inf_n→∞, using the lower-semicontinuity of ϕ and convergence of all the other terms, yields (3.2.12), as required.

Step 3: It remains to show that the solution constructed in the previous step is unique. To this end, let x₀, y₀ ∈L²(Ω,F⁰;H⁻¹), (y₀ⁿ)_n∈N⊂L²(Ω,F⁰;L²) satisfyingyⁿ₀ →y₀ in L²(Ω;H⁻¹) for n→ ∞. LetX be an arbitrary SVI solution to (3.1.1) with initial conditionx₀ and let (Y^ε,n)_ε>0,n∈N be the solutions to (3.4.1) with respective initial conditions (yⁿ₀)_n∈N. We first check that

Z=Y^ε,n and G=ε∆Y^ε,n+ ∆φ^ε(Y^ε,n) (3.4.21) are admissible choices for (3.2.12). First,

Y^ε,n∈L²(Ω;C([0, T];H⁻¹)) by construction and

Y^ε,n∈L²(Ω×[0, T];H₀¹)⊂L²(Ω×[0, T];L²)

by Lemma 3.4.2 with norm bounded uniformly inε. Also by Lemma 3.4.2, we have E

Z T

0 kε∆Y_t^ε,nk²H⁻¹dt=ε²E Z T

0 kY_t^ε,nk²H₀¹dt <∞.

Finally, for the nonlinear term, we have by the chain rule for the composition of Lipschitz functions with H₀¹functions (e. g. [126, Theorem 2.1.11]) that almost everywhere inO

∇φ^ε(Y_t^ε,n) = (φ^ε)⁰(Y_t^ε,n)∇Y_t^ε,n, such that we can compute using Lemma 3.D.2

Z

O|∇φ^ε(Y^ε,n)|²dx= Z

O

(φ^ε)⁰(Y_t^ε,n)∇Y_t^ε,n

2dx≤ 1

ε²k∇Y_t^ε,nk²L²

dt⊗P-almost everywhere. Consequently,

kφ^ε(Y_t^ε,n)k²H₀¹≤C(ε)kY_t^ε,nk²H₀¹, such that we can conclude by Lemma 3.4.2

E Z T

0 k∆φ^ε(Y_t^ε,n)k²H⁻¹dt=E Z T

0 kφ^ε(Y_t^ε,n)k²H₀¹dt

≤C(ε)E Z T

0 kY_t^ε,nk²H₀¹dt

≤C(ε)˜ <∞, which yields that the choices in (3.4.21) were admissible.

As a consequence, (3.2.12) yields fort∈[0, T] EkXt−Y_t^ε,nk²H⁻¹+ 2E

Z t 0

ϕ(Xr) dr

≤Ekx0−yⁿ₀k²H⁻¹+ 2E Z t

0

ϕ(Y_r^ε,n) dr (3.4.22)

−2E Z t

0 hε∆Y_r^ε,n+ ∆φ^ε(Y_r^ε,n), X_r−Y_r^ε,niH⁻¹dr +CE

Z t

0 kXr−Y_r^ε,nk²H⁻¹dr.

Foru∈L²and ϕ^ε as in (3.4.9), we have, as in (3.4.10) and (3.4.17),

h−∆φ^ε(Y^ε,n), u−Y^ε,niH⁻¹+ϕ^ε(Y^ε,n)≤ϕ^ε(u) dt⊗P-a. e. (3.4.23) Since Y^ε,n ∈ H₀¹ ⊂ L² dt⊗P-a. e. we can use Corollary 3.D.7 as in (3.4.13) to obtain dt⊗P-almost everywhere

|ϕ^ε(Y^ε,n)−ϕ(Y^ε,n)| ≤Cε

1 +kY^ε,nk²L²

. Thus, we can modify (3.4.23) and get

h−∆φ^ε(Y^ε,n), u−Y^ε,niH⁻¹+ϕ(Y^ε,n)≤ϕ(u) +Cε

1 +kY^ε,nk²L²

dt⊗P-a. e.. (3.4.24) Note that (3.4.24) is trivial if ϕ(u) = ∞. Furthermore, (3.4.24) can be deduced analogously for u ∈ ιm(L^m+1∩H⁻¹) in the superlinear setting, i. e. when ϕ is given by (3.2.8), with m as in Assumption 3.2.1 (A5). In the sublinear setting, i. e. ϕis given by (3.2.9), and u∈ M(O)∩H⁻¹, we consider an approximating sequence (µj)j∈N⊂ M ∩H⁻¹ with densities (uj)j∈N⊂L² given by Theorem 3.3.8, such that (3.4.24) is satisfied for allu_j, j∈N. We then pass to the limitj→ ∞ and notice that (µ_j)_j∈Nhas been constructed in such a way that bothϕ(u_j)→ϕ(u) and

h−∆φ^ε(Y^ε,n), u_j−Y^ε,niH⁻¹

=_H1

0hφ^ε(Y^ε,n), uj−Y^ε,niH⁻¹ −→ ^H1₀hφ^ε(Y^ε,n), u−Y^ε,niH⁻¹

=h−∆φ^ε(Y^ε,n), u−Y^ε,niH⁻¹. Consequently, replacingubyX in (3.4.24), we have in any case

h−∆φ^ε(Y^ε,n), X−Y^ε,niH⁻¹+ϕ(Y^ε,n)≤ϕ(X) +Cε

1 +kY^ε,nk²L²

dt⊗P-a. e.. (3.4.25) Using (3.4.25) and the same estimate as in (3.4.18), we can modify (3.4.22) to obtain fort∈[0, T]

EkXt−Y_t^ε,nk²H⁻¹ ≤Ekx0−y₀ⁿk²H⁻¹

+ 2E Z t

0

1

2ε⁴³k∆Y_r^ε,nk²H⁻¹dr+1

2ε²³kX_r−Y_r^ε,nk²H⁻¹dr +CE

Z t

0 kXr−Y_r^ε,nk²H⁻¹dr+CεE Z t

0

1 +kY_r^ε,nk²L²

dr.

Takingε→0 and thenn→ ∞yields

EkX_t−Y_tk²H⁻¹ ≤Ekx₀−y₀k²H⁻¹+CE Z t

0 kX_r−Y_rk²H⁻¹dr fort∈[0, T], (3.4.26) whereY is the SVI solution which has been constructed from (Y^ε,n) in the limiting procedure of the first two steps of this proof. Gronwall’s inequality then yieldsX=Y ifx0=y0, and thus uniqueness of SVI solutions. Then, estimate (3.2.13) follows by applying Gronwall’s inequality to (3.4.26) with different initial values, which concludes the proof.

3.A Generalities on convex functions

We collect and prove some statements on convex functions defined onR.

Lemma 3.A.1. Let f :R→[0,∞)be convex withf(0) = 0 andx, y∈R\ {0} with x < y. Then f(x)

x ≤f(y)

y . (3.A.1)

In particular, forx >0 this impliesf(x)≤f(y).

Proof. Note that by convexity, we have forλ∈(0,1),x∈R

f(λx) =f(λx+ (1−λ)0)≤λf(x) + (1−λ)f(0) =λf(x). (3.A.2) If x < 0 < y, the statement is obvious by the nonnegativity of f. If 0 < x < y, we use (3.A.2) with λ=^x_y to get

f(x)

x = f(λy)

λy ≤λf(y)

λy =f(y) y , while forx < y <0 we use (3.A.2) with λ:= ^y_x to get

f(y)

y =f(λx)

λx ≥ λf(x)

λx =f(x) x , as required.

Lemma 3.A.2. Let ψsatisfy Assumptions 3.2.1 and y >0. Then, ifψ(y)>0, we have ψ^∗(−x) =ψ^∗(x)≤ψ(y) forx∈

0,ψ(y)

y

, whereψ^∗ is defined as in Definition 3.3.1.

Proof. By Remark 3.3.2, the last part of Lemma 3.A.1 and the nonnegativity ofψ^∗, it is enough to show ψ^∗

ψ(y) y

≤ψ(y). (3.A.3)

To verify (3.A.3), we distinguish three cases fory⁰ ∈R. Fory⁰≥y we have by Lemma 3.A.1 ψ(y)

y y⁰−ψ(y⁰) =y⁰ ψ(y)

y −ψ(y⁰) y⁰

≤0, fory⁰≤0 we have by the nonnegativity ofψ

ψ(y)

y y⁰−ψ(y⁰)≤0, and fory⁰∈(0, y) we have

ψ(y)

y y⁰−ψ(y⁰)≤ψ(y)

y y=ψ(y), which yields the claim.

Lemma 3.A.3. Let ψsatisfy Assumptions 3.2.1. For K=dom(ψ^∗) :={x∈R:ψ^∗(x)<∞} we have supK= lim

t→∞

ψ(t)

t and sup(−K) = lim

t→∞

ψ(−t) t .

Proof. We only prove the first statement, the second one then becomes clear by symmetry. To this end, note first that the limit is actually a supremum, as ^ψ(t)_t is increasing (by (3.A.1)). Let nowx∈K, which means thatxt−ψ(t)≤cx<∞and thus ^ψ(t)_t ≥x−^ct^x for all t ∈[0,∞), which yields “≤” by letting t→ ∞.

Conversely, we have ^ψ(t)_t ∈K fort >0, ψ(t)>0 by by Lemma 3.A.2. As ψ^∗(0) = 0, this is true also if ψ(t) = 0, thereby proving “≥”.

Corollary 3.A.4. Letψ satisfy Assumptions 3.2.1. By Lemma 3.A.2 and Lemma 3.A.3, we have that ψ_∞(1) =ψ_∞(−1)≥ ψ(y)

y fory >0with ψ(y)>0, whereψ_∞ is defined as in Definition 3.3.1.

Lemma 3.A.5. Let ψ satisfy Assumptions 3.2.1. For the convex conjugate of the recession function, we have

ψ_∞^∗ (x) := (ψ_∞)^∗(x) =χ_{[−ψ∞(1),ψ∞(1)]}(x) forx∈R, where for an Interval I we have written

χ_I(x) =

(0, ifx∈I +∞, else.

Proof. In the superlinear case, i. e. (3.2.6) is satisfied, we haveψ_∞=χ_{0} and thusψ_∞^∗ ≡0, as required.

In the sublinear case, we first note thatψ∞is, by definition, positively homogeneous, which by symmetry amounts to absolute homogeneity. Thus

ψ_∞(x) =ψ_∞(1)|x|,

whereψ_∞(1)>0 by Corollary 3.A.4, which allows to conclude by the definition of the convex conjugate.

3.B Variational solutions to nonlinear SPDE

Let (Ω,F,P) a complete probability space, V ⊂ H ⊂ V⁰ a Gelfand triple, (W_t)_t∈[0,T] a cylindrical Id-Wiener process taking values in another separable Hilbert space (U,h,iU) with normal filtration (F^t)_t∈[0,T]. Let

A: [0, T]×V ×Ω→V⁰, B: [0, T]×V ×Ω→L2(U, H), be progressively measurable and satisfy the following conditions:

(H1) (Hemicontinuity) For all u, v, x∈V, ω∈Ω andt∈[0, T], the map R3λ7→^V0hA(t, u+λv, ω), xiV

is continuous.

(H2) (Weak monotonicity) There existsc∈R, such that for allu, v∈V

2_V⁰hA(·, u)−A(·, v), u−viV +kB(·, u)−B(·, v)k²L2(U,H)≤cku−vk²H

on [0, T]×Ω.

(H3) (Coercivity) There existα∈(1,∞), c1∈R, c2∈(0,∞) and an (F^t)-adapted processf ∈L¹([0, T]× Ω,dt⊗P), such that for allv∈V, t∈[0, T]

2V⁰hA(t, v), viV +kB(t, v)k²L₂(U,H)≤c1kvk²H−c2kvk^αV +f(t) on Ω. (3.B.1) (H4) (Boundedness) There existc₃∈[0,∞) and an (F^t)-adapted process

g∈L^α−1α ([0, T]×Ω,dt⊗P), such that for allv∈V, t∈[0, T]

kA(t, v)kV⁰ ≤g(t) +c3kvk^α−1V

on Ω, whereαis as in (H3).

We then consider the stochastic partial differential equation

dX_t=A(t, X_t)dt+B(t, X_t) dW_t, (3.B.2) for which we establish the following notion of solution:

Definition 3.B.1. A continuous H-valued (F^t)-adapted process (Xt)_t∈[0,T] is called a (variational) solution of (3.B.2), if for its dt⊗P-equivalence class ˆX we have

Xˆ ∈L^α([0, T]×Ω,dt⊗P;V)∩L²([0, T]×Ω,dt⊗P;H), withαas in (3.B.1), andP-a. s.

Xt=X0+ Z t

0

A(s,X¯s)ds+ Z t

0

B(s,X¯s) dWs, t∈[0, T], where ¯X is anyV-valued progressively measurable dt⊗P-version ofX.

We then have the following well-posedness result (see [112, Theorem 4.4], relying on [92]).

Theorem 3.B.2. Let X0 ∈ L²(Ω,F⁰,P;H). Then there exists a unique solution X to (3.B.2) in the sense of Definition 3.B.1.

3.C Strong solutions to gradient-type SPDE

Letϕ:H →Rbe a proper, lower-semicontinuous, convex function on a separable real Hilbert spaceH. We consider an SPDE of the type

dXt∈ −∂ϕ(Xt)dt+B(t, Xt)dWt,

X0=x0, (3.C.1)

where W is a cylindrical Wiener process in a separable Hilbert space U defined on a probability space (Ω,F,P) with normal filtration (F^t)_t≥0 andB: [0, T]×H×Ω→L₂(U, H) is Lipschitz continuous, i. e.

for allv, w∈H

kB(t, v)−B(t, w)k²L₂(U,H)≤Ckv−wk²H, and for all (t, ω)∈[0, T]×Ω. Furthermore, we assume that

kB(·,0)kL₂(U,H)∈L²([0, T]×Ω).

Definition 3.C.1. Letx₀∈L²(Ω,F⁰;H). AnH-continuous,F^t-adapted processX ∈L²(Ω;C([0, T];H)) for which there exists a selectionη∈ −∂ϕ(X), dt⊗P-a. e., is said to be astrong solution to(3.C.1) if

η∈L²([0, T]×Ω;H) andP-a. s.

Xt=x0+ Z t

0

ηr dr+ Z t

0

B(r, Xr) dWr for allt∈[0, T].

3.D Yosida approximation of multivalued operators

The theory of Yosida approximations can be applied to general maximal monotone operators from Banach spaces to their dual, see e. g. [7, Section 2]. However, we constrain ourselves to the case of the Hilbert spaceR.

Fix ε > 0. For a convex, lower-semicontinuous proper functionψ : R →[0,∞) we define its Moreau-Yosida approximation ψ^ε:R→[0,∞) by

ψ^ε(r) = inf

s∈R

|r−s|² 2ε +ψ(s)

!

. (3.D.1)

Letφ=∂ψ:R→2^Rbe the subdifferential of ψ. For each r∈R, we define the resolventJ^ε(r) as the unique solutionsto

s+εφ(s)3r.

Hereby the resolvent is well-defined, since φ is maximal monotone as a subdifferential (see e. g. [7, Theorem 2.8]), which implies that Id_R+εφ is bijective. We then define the Yosida approximation φ^ε:R→Rofφby

φ^ε(r) =1

ε(r−J^εr). (3.D.2)

We state and prove some properties of this approximation, most of which are true for general subpotential operators. The usage of additional assumptions will be highlighted.

Proposition 3.D.1. We have

φ^ε(r)∈φ(J^εr). (3.D.3)

Furthermore,ψ^ε is continuous, convex and Gateaux differentiable, andφ^ε = (ψ^ε)⁰. In particular, φ^ε is also maximal monotone.

Proof. The first claim is clear by construction. The remaining statements are proved in [7, Theorem 2.9].

Lemma 3.D.2. The Yosida approximationφ^ε is Lipschitz continuous with Lipschitz constant ¹_ε. Proof. Fixx, y∈R. By definition ofJ^ε, we have

J^εx−J^εy+ε(φ^ε(x)−φ^ε(y)) =x−y.

By multiplying withφ^ε(x)−φ^ε(y) and keeping (3.D.3) in mind, we obtain ε(φ^ε(x)−φ^ε(y))²≤ |φ^ε(x)−φ^ε(y)| |x−y|, which immediately yields the claim.

Lemma 3.D.3. Defining|φ(r)|:= inf{|η|:η∈φ(r)}, we have |φ^ε(r)| ≤ |φ(r)|for allr∈R. Proof. By monotonicity ofφ, we get forη∈φ(r)

0≤(r−J^ε(r))(η−φ^ε(r)).

Noting thatr−J^ε(r) =εφ^ε(r), we can simplify

0≤ε|φ^ε(r)| |η| −ε(φ^ε(r))² to obtain the estimate.

The next lemma is proved in [7, Theorem 2.9]:

Lemma 3.D.4. For each r∈R, we have ψ^ε(r) = 1

2ε|r−J^ε(r)|²+ψ(J^εr), in other words, the infimum in (3.D.1) is assumed at J^εr.

As an immediate consequence, we get Corollary 3.D.5. For eachr∈R, we have

ψ(J^εr)≤ψ^ε(r)≤ψ(r).

Proof. The first inequality is clear by Lemma 3.D.4, the second one by settingr=sin (3.D.1).

Lemma 3.D.6. For each r∈R, we have

|ψ(r)−ψ^ε(r)| ≤ε|φ(r)|² for allr∈R. (3.D.4)

Proof. Fix an arbitraryr∈R. For anyη∈φ(r) we have, using Corollary 3.D.5 in the first step and the subdifferential inequality in the second step,

0≤ψ(r)−ψ(J^εr)≤ −η(J^εr−r)≤ |η|ε|φ^ε(r)|.

Sinceη∈φ(r) was arbitrary, we can pass to its infimum. Using Lemma 3.D.3, we obtain (3.D.4).

Corollary 3.D.7. With Lemma 3.D.6, under the additional assumption|φ(r)|²≤C(1 +|r|²), we obtain

|ψ(r)−ψ^ε(r)| ≤Cε(1 +r²) for allr∈R. Lemma 3.D.8. We have for all a, b∈R, ε1, ε2>0

(φ^ε1(a)−φ^ε2(b)) (a−b)≥ −C(ε1+ε2)

|φ^ε1(a)|²+|φ^ε2(b)|² . Proof. We compute

(φ^ε1(a)−φ^ε2(b)) (a−b) = (φ^ε1(a)−φ^ε2(b))(J^ε1a−J^ε2b)

+ (φ^ε1(a)−φ^ε2(b))(a−J^ε1a−(b−J^ε2b))

≥(φ^ε1(a)−φ^ε2(b))(ε₁φ^ε1(a)−ε₂φ^ε2(b))

≥ −1

2(ε1+ε2)

|φ^ε1(a)|²+|φ^ε2(b)|² ,

where the second step uses (3.D.3) for the first summand to be positive and (3.D.2) for the second summand. In the last step, we neglect the squared terms and use Young’s inequality for the mixed terms.

Corollary 3.D.9. Under the additional assumption|φ(r)|² ≤C(1 +|r|²), Lemma 3.D.3 immediately yields

(φ^ε1(a)−φ^ε2(b)) (a−b)≥ −C(ε₁+ε₂)

1 +|a|²+|b|² .

3.E Estimate on specific quadratic variations

Lemma 3.E.1. Let U, H be Hilbert spaces, Q : U → U linear, bounded, non-negative definite and symmetric, W a (possibly cylindrical) Q-Wiener process on U defined on a probability space (Ω,F,P) and normal filtration (F^t)_t≥0. Further let B : Ω×[0, T] →L2

Q¹²(U), H

such that B is predictable and

P Z T

0 kB(s)k_L

2

Q¹2(U),Hds <∞

!

= 1,

andf an(F^t)-adapted continuousH-valued process. Then, the quadratic variation of a stochastic integral onH of the form

Mt= Z t

0 hfr, BrdWriH

can be estimated from above by

hMi^t≤ Z t

0 kf_rk²HkB_rk²_L

2

Q¹2(U),Hdr.

Proof. IfQis of finite trace and thusW is a classical Wiener process, the statement follows from [112, Lemma 2.4.2] and [112, Lemma 2.4.3]. In case of a cylindrical Wiener process, we can compute, using the notation of Assumption 3.2.1 (A1),

hMi^t= Z ·

0

D

fr, Br◦J⁻¹d ˜Wr

E

H

t

≤ Z t

0 kfrk²H

Br◦J⁻¹

2 L₂

Q

1 12(U₁),H

dr

= Z t

0 kfrk²HkBrk²_L2Q¹_2(U),Hdr,

where in the second step, we use the Lemma for the classicalQ1-Wiener process ˜W onU1. The last step can be seen by the fact that for an orthonormal basis (ek)k∈N of Q¹²(U), we have that (J ek)k∈N is an orthonormal basis ofL₂ Q₁¹²(U₁), H

; see [112, section 2.5.2] for details.

Chapter 4

Ergodicity for singular-degenerate stochastic porous media equations

4.1 Introduction

We consider the singular-degenerate generalized stochastic porous medium equation dXt∈ ∆(φ(Xt))dt+BdWt,

X₀= x₀, (4.1.1)

on a bounded intervalO ⊆Rwith zero Dirichlet boundary conditions. The multi-valued function φis the maximal monotone extension of

R3x7→x1_{|x|>1}, (4.1.2) W is a cylindrical Wiener process on some separable Hilbert space U, and the diffusion coefficient B is anL²(O)-valued Hilbert-Schmidt operator satisfying a non-degeneracy condition (see (4.2.5) below).

Equation (4.1.1) is understood as an evolution equation on H⁻¹, the dual of H₀¹(O), where it can be solved uniquely in the sense of SVI solutions, as shown in Chapter 3. The main result of the present work is the existence and uniqueness of an invariant probability measure for solutions to (4.1.1).

The above form of stochastic porous media equations is motivated by the analysis of non-equilibrium systems, appearing in the context of self-organized criticality (for a survey, see e. g. [122]). Self-organized criticality is a statistical property of systems displaying intermittent events, such as earthquakes, which are activated when the underlying system locally exceeds a threshold. These dynamics are reflected by the discontinuity and degeneracy of the nonlinearityφabove. In order to get a better understanding of the long-time behaviour of these systems, we prove the existence of a unique non-equilibrium statistical invariant state for (4.1.1). Since this is the candidate to which the transition probabilities are expected to converge for long times, it is the key object for the statistical behaviour of the respective process.

A previous approach to the long-time behaviour of Markov processes stemming from monotone SPDEs with singular drift, by which the present article is inspired, is [77], which in turn uses the more abstract framework of [78]. In these works, the existence and uniqueness of invariant probability measures to stochastic local and non-local p-Laplace equations is proved, where the multivalued regime p = 1 is included. In one dimension, the paradigmatic case is the equation

dXt= ∆(sgn(Xt)) + dWt, (4.1.3)

where sgn denotes the maximal monotone extension of the classical sign function. The proof relies on sufficient criteria from [91], where the so-called lower bound technique has been extended to Polish spaces which are not necessarily locally compact. This technique relies on the existence of a state being an accessible point for the time averages of the transition probabilities uniformly in time, and the so-called “e-property”, which is a uniform continuity assumption on the Markov semigroup. To verify these criteria, the focus of [77] rests on energy estimates to first bound the mass of these averages toL^mballs for some suitably chosen m∈(2,3]. As a next step, the convergence to a chosen accessible state with

probability bounded below is shown, which is done by comparing the solution of (4.1.3) to a control process, which obeys the mere deterministic dynamics of (4.1.3), i. e.

d

dtXt= ∆(sgn(Xt)), X0=y,

(4.1.4)

fory ∈L^m,kykm ≤R for some R >0. In this, simpler setting than (4.1.1), there is a unique limiting state to (4.1.4) which is a natural candidate for the aforementioned accessible point.

In the present article, we aim to prove the existence and uniqueness of an invariant probability measure by similar ideas. While energy estimates for (4.1.1) are easier to obtain due to the linear growth of φ (cf. (4.1.2)) at ±∞, the degenerate form of the nonlinearity destroys the convergence of the noise-free system to a unique fixed point. This is why we have to add a forcing term to the control process and rely on a more refined deterministic analysis of the resulting inhomogeneous monotone evolution equation.

To guarantee the convergence of this modified control process, the forcing term has to be sufficiently non-degenerate, and as the connection of the solution to (4.1.1) to the control process only works if the noise is “close” to the deterministic forcing with non-zero probability, this relies on some non-degeneracy requirements on the noise. As in[77], it is important that the convergence of the deterministic process takes place uniformly for initial values in sets of bounded energy. We tackle this problem with the help

To guarantee the convergence of this modified control process, the forcing term has to be sufficiently non-degenerate, and as the connection of the solution to (4.1.1) to the control process only works if the noise is “close” to the deterministic forcing with non-zero probability, this relies on some non-degeneracy requirements on the noise. As in[77], it is important that the convergence of the deterministic process takes place uniformly for initial values in sets of bounded energy. We tackle this problem with the help

Im Dokument Stochastic partial differential equations arising in self-organized criticality (Seite 87-105)