Nontriviality

This section is devoted to the proof of nontriviality, that is, non-Gaussianity.

Theorem 5.4 If λ > 0 then any limit measure ν constructed via the procedure in Section 4 is non-Gaussian.

Proof In order to show that the limiting measure ν is non-Gaussian, it is sufficient to prove that the connected four-point function is nonzero, see [BFS83]. In other words, we shall prove that the distribution

U₄^ν(x1, . . . , x4) :=Eν[ϕ(x1)· · ·ϕ(x4)]

−Eν[ϕ(x1)ϕ(x2)]Eν[ϕ(x3)ϕ(x4)]−Eν[ϕ(x1)ϕ(x3)]Eν[ϕ(x2)ϕ(x4)]

−Eν[ϕ(x1)ϕ(x4)]Eν[ϕ(x2)ϕ(x3)], x1, . . . , x4 ∈R^d, is nonzero.

To this end, we recall that in Theorem4.9we obtained a limit measureµwhich is the joint law of (ϕ, X, X ) and thatν is the marginal corresponding to the first component. Let K_i=F⁻¹ϕ_i be a Littlewood–Paley projector and consider the connected four-point function U₄^ν convolved with(K_i, K_i, K_i, K_i) and evaluated at (x₁, . . . , x₄) = (0, . . . ,0), that is,

U₄^ν∗(Ki, Ki, Ki, Ki)(0,0,0,0) =Eν[(∆iϕ)⁴(0)]−3Eν[(∆iϕ)²(0)]²

=Eµ[(∆iϕ)⁴(0)]−3Eµ[(∆iϕ)²(0)]² =:L(ϕ, ϕ, ϕ, ϕ),

where L is a quadrilinear form. Since under the limit µ we have the decomposition ϕ = X− λX +ζ, we may write

L(ϕ, ϕ, ϕ, ϕ) =L(X, X, X, X)−4λL(X, X, X, X ) +R (5.3) where R contains terms which are at least bilinear in X or linear in ζ. Due to Gaussianity of X, the first term on the right hand side of (5.3) vanishes. Our goal is to show that the second term behaves like2ⁱ whereas the terms inR are more regular, namely, bounded by 2^i(1/2+κ). In other words, R cannot compensate 4λL(X, X, X, X ) and as a consequenceL(ϕ, ϕ, ϕ, ϕ) 6= 0if λ >0.

Let us begin with L(X, X, X, X ). To this end, we denote k_[123] =k1+k2 +k3 and recall that

(∆iX)(0) = Z

R^d

ϕi(k) Z 0

−∞

e^{−[m2+|k|2](−s)}ξ(ds,ˆ dk), (∆_iX )(0) =

Z 0

−∞

ds Z

R^d

Z

R^d

Z

R^d

ϕ_i(k_[123])e^{−[m2+|k[123]|2](−s)}

× u v

Y

l=1,2,3

Z _s

−∞

e^{−[m2+|kl|2](s−sl)}ξ(dsˆ _l,dk_l) }

~,

where J·Kdenotes Wick’s product. Hence denoting H := [4m²+|k_[123]|²+|k₁|²+|k₂|²+|k₃|²] we obtain

L(X, X, X, X ) =E h

(∆iX)(0)(∆iX)(0)(∆iX)(0)(∆iX )(0) i

= 3!

Z 0

−∞

ds Z

R^d

Z

R^d

Z

R^d

ϕi(k_[123])e^−H(−s) Y

l=1,2,3

Z s

−∞

e^{−2[m2+|kl|2](s−sl)}ϕi(kl)dsldkl

= 3!

8 Z 0

−∞

ds Z

R^d

Z

R^d

Z

R^d

ϕ_i(k_[123])e^−H(−s) Y

l=1,2,3

ϕ_i(k_l) dk_l m²+|k_l|²

= 3!

8 Z

R^d

Z

R^d

Z

R^d

ϕi(k_[123]) H

Y

l=1,2,3

ϕi(kl) dk_l m²+|k_l|²

≈2^i(−8+9)≈2ⁱ.

Let us now estimate various terms inR. The terms containing only combinations of X, X can be estimated directly whereas for terms where ζ appears it is necessary to use stationarity due to the limited integrability in space. For instance,

E

h

(∆iX)(0)(∆iX)(0)(∆iX )(0)(∆iX )(0) i

.2−2i(−1/2−κ)2^{−2i(1/2−κ)}E

h

kXk²_C−1/2−κ(ρ^σ)kX k²_C1/2−κ(ρ^σ)

i .2^i4κ

and similarly for the other terms without ζ which are collectively of order2^i4κ(λ²+λ⁴). For the remaining terms, we fix a weightρas above and use stationarity. In addition, we shall be careful about having the necessary integrability. For instance, for the most irregular term we have

E[(∆iX)³(0)(∆iζ)(0)] = Z

R^d

ρ⁴(x)E[(∆iX)³(x)(∆iζ)(x)]dx=Ehρ⁴,(∆iX)³(∆iζ)i and we bound this quantity as

|E[(∆_iX)³(0)(∆_iζ)(0)]|6E[k∆_iX_εk³_L∞(ρ^σ)k∆_iζk_L1(ρ^4−3σ)].E[k∆_iX_εk³_L∞(ρ^σ)k∆_iζk_L2(ρ²)] .2−3i(−1/2−κ)2^i(−1+2κ)E

h

kXk³_C−1/2−κ(ρ^σ)kζk_B1−2κ 2,2 (ρ²)

i

.2−3i(−1/2−κ)

2^i(−1+2κ)(E[kXk⁶_C−1/2−κ(ρ^σ)])^1/2(E[kζk²

B^1−2κ_2,2 (ρ²)])^1/2 .2^i(1/2+5κ)(λ+λ^7/2).

where we used Theorem 4.9. Next,

|E[(∆iX)²(0)(∆iζ)²(0)]|6E[k∆_iXk²_L∞(ρ^σ)k∆_iζk_L2(ρ^1+ι)k∆_iζk_L2(ρ²)] 62−2i(−1/2−κ)2^{−i(1−2κ)}E[kXk²_C−1/2−κ(ρ^σ)kζk_B0

4,∞(ρ)kζk_H^1−2κ_(ρ2)].2^i4κ(λ^5/4+λ⁵), and

|E[(∆iX)(0)(∆iζ)³(0)]|6E[k∆_iXk_L^∞_(ρσ)k∆_iζk³_L3(ρ(4−σ)/3)] 6E[k∆_iXk_L∞(ρ^σ)k∆_iζk³_L4(ρ)]

.2^{−i(−1/2−κ)}E

hkXk_C−1/2−κ(ρ^σ)kζk³_B0 4,∞(ρ)

i

.2^i(1/2+κ)(λ^3/4+λ^9/2),

|E[(∆iζ)⁴(0)]|=|Ehρ⁴,(∆iζ)⁴i|6Ek(∆_iζ)k⁴_L4(ρ)6E[kζk⁴_B0

4,∞(ρ)].(λ+λ⁶).

Proceeding similarly for the other terms we finally obtain the bound

|R|.2^i(1/2+5κ)(λ^3/4+λ⁷).

Therefore for a fixed λ >0 there exists a sufficiently largeisuch that E[(∆_iϕ)⁴(0)]−3(E[(∆_iϕ)²(0)²])² .−2ⁱλ <0,

and the proof is complete. 2

6 Integration by parts formula and Dyson–Schwinger equations

The goal of this section is twofold. First, we introduce a new paracontrolled ansatz, which allows to prove higher regularity and in particular to give meaning to the critical resonant product in the continuum. Second, the higher regularity is used in order to improve the tightness and to construct a renormalized cubic term Jϕ³K. Finally, we derive an integration by parts formula together with the Dyson–Schwinger equations and we identify the continuum dynamics.

6.1 Improved tightness

In this section we establish higher order regularity and a better tightness which is needed in order to define the resonant product JX²K◦φin the continuum limit. Recall that the equation (4.6) satisfied by φ_M,ε has the form

L εφ_M,ε=−3λJX_M,ε² Kφ_M,ε+U_M,ε, (6.1) where

UM,ε := −3λJX_M,ε² K4(YM,ε+φM,ε)−3λ²bM,ε(XM,ε+YM,ε+φM,ε)

−3λ(U₆^εJX_M,ε² K)Y_M,ε−3λX_M,ε(Y_M,ε+φ_M,ε)²−λY_M,ε³

−3λY_M,ε² φM,ε−3λYM,εφ²_M,ε−λφ³_M,ε. If we let

χ_M,ε:=φ_M,ε+ 3λX_M,ε φ_M,ε, (6.2) we obtain by the commutator lemma, LemmaA.14,

3λJX_M,ε² K◦φ_M,ε+ 3λ²b_M,εφ_M,ε=−3λJX_M,ε² K◦(3λX_M,ε φ_M,ε) + 3λ²b_M,εφ_M,ε + 3λJX_M,ε² K◦χ_M,ε

=−λ²Xe_M,εφM,ε+ 3λ²(bM,ε−˜bM,ε(t))φM,ε

+λ²C_ε(φ_M,ε,−3X_M,ε,3JX_M,ε² K) + 3λJX_M,ε² K◦χ_M,ε. Recalling thatZ_M,ε=−3λ⁻¹JX_M,ε² K◦Y_M,ε−3b_M,ε(X_M,ε+Y_M,ε) can be rewritten as (4.9) and controlled due to Lemma4.3, where we also estimated XM,εYM,ε and XM,εY_M,ε² , we deduce

U_M,ε = −λ²Xe_M,εφ_M,ε+ 3λ²(b_M,ε−˜b_M,ε(t))φ_M,ε+λ²C_ε(φ_M,ε,−3X_M,ε,3JX_M,ε² K) +3λJX_M,ε² K◦χM,ε

+λ²Z_M,ε−3λJX_M,ε² K≺(Y_M,ε+φ_M,ε)−3λ(U₆^εJX_M,ε² K)Y_M,ε−3λX_M,εY_M,ε²

−6λX_M,εY_M,εφ_M,ε−3λX_M,εφ²_M,ε−λY_M,ε³ −3λY_M,ε² φ_M,ε−3λY_M,εφ²_M,ε−λφ³_M,ε.

Consequently, the equation satisfied by χM,ε reads

L εχ_M,ε = L εφ_M,ε+ 3λJX_M,ε² Kφ_M,ε+ 3λX_M,εL εφ_M,ε−6λ∇_εX_M,ε ∇_εφ_M,ε

= U_M,ε+ 3λX_M,εL εφ−6λ∇_εX_M,ε ∇_εφ_M,ε

= UM,ε+ 3λX_M,ε(−3λJX_M,ε² KφM,ε+UM,ε)−6λ∇_εX_M,ε ∇_εφM,ε, (6.3) where the bilinear form ∇_εf ≺ ∇_εgis defined by

∇_εf ≺ ∇_εg:= 1

2(∆ε(f ≺g)−∆εf ≺g−f ≺∆εg) and can be controlled as in the proof of LemmaA.14.

Next, we state a regularity result for χM,ε, proof of which is postponed to Appendix A.6.

While it is in principle possible to keep track of the exact dependence of the bounds on λ we do not pursue it any further since there seems to be no interesting application of such bounds.

Nevertheless, it can be checked that the bounds in this section remain uniform over λbelonging to any bounded subset of [0,∞).

Proposition 6.1 Letρ be a weight such thatρ^ι ∈L^4,0 for someι∈(0,1). LetφM,ε be a solution to (6.1) and letχ_M,ε be given by (6.2). Then

kρ⁴χM,εk_L1

TB_1,1^1+3κ,ε 6C_{T ,m}²_,λQρ(XM,ε)(1 +kρ²φM,ε(0)k_L2,ε).

We apply this result in order to deduce tightness of the sequence(ϕM,ε)M,εas time-dependent stochastic processes. In other words, in contrast to Theorem4.8, where we only proved tightness for a fixed time t>0, it is necessary to establish uniform time regularity of (ϕ_M,ε)_M,ε. To this end, we recall the decompositions

ϕ_M,ε=X_M,ε+Y_M,ε+φ_M,ε=X_M,ε−λX_M,ε+ζ_M,ε with

ζM,ε=YM,ε+λX_M,ε+φM,ε=−L ⁻¹_ε [3λ(U_>^εJX_M,ε² KYM,ε] +φM,ε. (6.4) Theorem 6.2 Let β∈(0,1/4). Then it holds true that for all p∈[1,∞) and τ ∈(0, T)

sup

ε∈A,M >0Ekϕ_M,εk^2p

W_T^β,1B_1,1^{−1−3κ,ε}(ρ^4+σ)+ sup

ε∈A,M >0Ekϕ_M,εk^2p

L^∞_τ,TH^{−1/2−2κ,ε}(ρ²)6C_λ <∞, where L^∞_τ,TH^{−1/2−2κ,ε}(ρ²) =L^∞(τ, T;H^{−1/2−2κ,ε}(ρ²)).

Proof Let us begin with the first bound. According to Proposition 6.1 and Theorem 4.8 we obtain that

Ekχ_M,εk^2p

L¹_TB^1+3κ,ε_1,1 (ρ⁴)6C_{T ,λ}EQ_ρ(XM,ε)(1 +Ekρ²φ_M,ε(0)k^2p_L2,ε) 6C_{T ,λ}EQρ(Xε)(1 +Ekρ²(ϕM,ε(0)−XM,ε(0))k^2p_L2,ε+Ekρ²YM,ε(0)k^2p_L2,ε)

is bounded uniformly in M, ε. In addition, the computations in the proof of Proposition 6.1 imply that also EkL εχM,εk^2p

L¹_TB_1,1^−1+3κ,ε(ρ⁴) is bounded uniformly in M, ε. As a consequence, we deduce that

Ek∂_tχ_M,εk^2p

L¹_TB^−1+3κ,ε_1,1 (ρ⁴)6Ek(∆_ε−m²)χ_M,εk^2p

L¹_TB_1,1^−1+3κ,ε(ρ⁴)+EkL εχ_M,εk^2p

L¹_TB^−1+3κ,ε_1,1 (ρ⁴)

is also bounded uniformly in M, ε.

Next, we apply a similar approach to derive uniform time regularity ofφ_M,ε. To this end, we study the right hand side of (6.1). Observe that due to the energy estimate from Theorem4.5and the bound from Proposition6.1together with Theorem4.8the following are bounded uniformly inM, ε

EkJX_M,ε² KφM,εk^2p_L2

TH^−1−κ,ε(ρ^2+σ), EkJX_M,ε² K◦χM,εk^2p

L¹_TB^2κ,ε_1,1 (ρ^4+σ),

whereas all the other terms on the right hand side of (6.1) are uniformly bounded in better function spaces. Hence we deduce that

Ek∂_tφ_M,εk^2p

L¹_TB_1,1^{−1−3κ,ε}(ρ^4+σ)6Ek(∆_ε−m²)φ_M,εk^2p

L¹_TB_1,1^{−1−3κ,ε}(ρ^4+σ)+EkL εφ_M,εk^2p

L¹_TB^{−1−3κ,ε}_1,1 (ρ^4+σ)

is bounded uniformly inM, ε.

Now we have all in hand to derive a uniform time regularity ofζ_M,ε. Using Schauder estimates together with (6.4) it holds that

Ekζ_M,εk^2p

W_T^{(1−2κ)/2,1}B_1,1^{−1−3κ,ε}(ρ^4+σ)6E

L ⁻¹ε [3λ(U>^εJX_M,ε² KYM,ε]

2p

C_T^(1−κ)/2L^∞,ε(ρ^σ)

+Ekφ_M,εk^2p

W_T^1,1B_1,1^{−1−3κ,ε}(ρ^4+σ)

is bounded uniformly inM, ε.

Finally, since for all β∈(0,1)we have that both EkX_M,εk^2p

C_T^βC−1/2−κ−2β,ε(ρ^σ), EkX_M,εk^2p

C_T^βC^{1/2−κ−2β,ε}(ρ^σ)

are bounded uniformly inM, ε, we conclude that so isEkϕ_M,εk^2p

W_T^β,1B^{−1−3κ,ε}_1,1 (ρ^4+σ)forβ ∈(0,1/4), which completes the proof of the first bound.

In order to establish the second bound we recall the decompositionϕM,ε=XM,ε+YM,ε+φM,ε

and make use of the energy estimate from Corollary 4.7. Taking supremum over t∈[τ, T]and expectation implies

sup

ε∈A,M >0Ekφ_M,εk^2p_L∞

τ,TL^2,ε(ρ²) <∞.

The claim now follows using the bound forX_M,ε together with the bound forY_M,εin Lemma4.1.

2 Even though the uniform bound in the previous result is far from being optimal, it is sufficient for our purposes below.

Corollary 6.3 Let ρ be a weight such thatρ^ι∈L⁴ for someι∈(0,1). Letβ ∈(0,1/4)andα∈ (0, β). Then the family of joint laws of(E^εϕ_M,ε,E^εXM,ε)is tight onW_loc^α,1B^−1−4κ_1,1 (ρ^4+σ)×C_loc^κ/2X, where

X := Y

i=1,...,7

C^α(i)−κ(ρ^σ)

with α(1) =α(7) =−1/2, α(2) =−1, α(3) = 1/2, α(4) =α(5) =α(6) = 0.

Proof According to Theorem 6.31 in [Tri06] we have the compact embedding B_1,1^−1−3κ(ρ^4+σ)⊂B^−1−4κ_1,1 (ρ^4+2σ)

and consequently since α < β the embedding

W_loc^β,1B_1,1^−1−3κ(ρ^4+σ)⊂W_loc^α,1B_1,1^−1−4κ(ρ^4+2σ)

is compact, see e.g. Theorem 5.1 [Amm00]. Hence the desired tightness of E^εϕ_M,ε follows from Theorem 6.2 and LemmaA.15. The tightness of E^εXM,ε follows from the usual arguments and

does not pose any problems. 2

As a consequence, we may extract a converging subsequence of the joint laws of the processes (E^εϕ_M,ε,E^εXM,ε)_M,ε inW_loc^α,1B_1,1^−1−4κ(ρ^4+σ)×C_loc^κ/2X. Let µˆ denote any limit point. We denote by (ϕ,X) the canonical processes on W_loc^α,1B_1,1^−1−4κ(ρ^4+σ)×C_loc^κ/2X and let µ be the law of the pair (ϕ, X) under µˆ (or the projection of µˆ to the first two components). Observe that there exists a measurable mapΨ : (ϕ, X)7→(ϕ,X) such thatµˆ=µ◦Ψ⁻¹. Therefore we can represent expectations under µˆ as expectations under µ with the understanding that the elements of X are constructed canonically from X via Ψ. Furthermore, Y, φ, ζ, χ are defined analogously as on the approximate level as measurable functions of the pair (ϕ, X). In particular, the limit localizer U> is determined by the constant L₀ obtained in Lemma 4.1. Consequently, all the above uniform estimates are preserved for the limiting measure and the convergence of the corresponding lattice approximations to Y, φ, ζ, χ follows. In addition, the limiting processϕ is stationary in the following distributional sense: for allf ∈C_c^∞(R+) and allτ >0, the laws of

ϕ(f) and ϕ(f(· −τ)) on S⁰(R³)

coincide. Based on the time regularity of ϕ it can be shown that this implies that the laws of ϕ(t) and ϕ(t+τ) coincide for all τ >0 and a.e. t ∈[0,∞). The projection of µ on ϕ(t) taken from this set of full measure is the measure ν as obtained in Theorem4.9.

Im Dokument PDE construction of the Euclidean Φ 4 3 quantum field theory (Seite 38-43)