Picard iteration

Having constructed the Besov area of the driving noise, we can now solve the fractional Burgers type equation. For this purpose, we first have to define what we mean by a solution.

Fort≥0 and α∈Rwe define the space C_T^α:=C([0, T], C^α(T^d,Rⁿ)) with norm

∥u∥_C^α

T := sup

s∈[0,T]

∥u_s∥_α, where we writeus :=u(s,·).

Definition 5.4.5. Letd∈N,σ−d/2>1/3,ε >0, andα∈(1/3, σ−d/2). Letu₀∈C^α. A function u∈C_T^α is called mild solution to

∂_tu(t, x) =−Au(t, x) +G(u(t, x))D_xu(t, x) + ˙W(t, x), (5.19) with initial conditionu₀, ifv:=u−ψ is inC_T^1+ε, and

v(t,·) =P_t(u₀−ψ₀)(x) +

 t 0

Pt−s[G(v(s,·) +ψ(s,·))D_x(v(s,·) +ψ(s,·))]ds +λ

 t 0

Pt−sψ(s,·)ds,

where the productG(v(s,·)+ψ(s,·))D_x(v(s,·)+ψ(s,·)) is as in Section 5.3: Sincev(s,·)∈ C^1+ε, we havev(s,·) +ψ(s,·)∈ D_ψ^α,α, and D_x(v(s,·) +ψ(s,·))∈ D^α−1,α_D

xψ .

Remark 5.4.6. A priori this definition depends on the constant λ > 0 that we intro-duced to obtain a stationary solution to the linear part of the equation. But ifψ solves

∂tψ(t, x) =−(A+λ)ψ(t, x) + ˙W(t, x), andψ^solves∂tψ(t, x) = −(A+λ) ψ(t, x) + ˙^ W(t, x), then

∂t(ψ−ψ)(t, x) = −A(ψ−ψ)(t, x) + λ^ψ(t, x)^ −λψ(t, x),

and from here it is easy to see that ψ−ψ is smooth. Therefore, anyw∈ D^α,α_ψ is also in D^α,α

ψ , which implies that the definition of the product Pt−sG(v(s,·) +ψ(s,·))D_x(v(s,·) + ψ(s,·)) does not depend on the special choice ofλ.

We also could have taken

ψ(t, x) =

 t 0

(Pt−sW˙ (s,·))(x)ds

as reference distribution, and not the stationary solutionψ. While it is harder to derive the regularity ofψdirectly, it is easy to show that the differenceψ−ψis smooth.

We could also define weak solutions. It should be no problem to show that mild and weak solutions coincide.

For further details on these questions see [HW13], Definition 3.1 and the following discussion.

Before proving the existence of a unique solution to (5.19), we establish some a priori estimates. To lighten the notation, we introduce a “rough path norm”.

Definition 5.4.7. Letα >1/3 andw∈C^α(T^d,Rⁿ) with associated Besov areawDxw∈ C^2α−1(T^d,Rⁿ⊗ L(T^d,Rⁿ)). In that case we writew:= (w, wD_xw)∈C^α, and we define the “norm”

∥w∥_Cα :=∥D_xw∥_α−1+∥w∥_α∥D_xw∥_α−1+∥π_◦(w,D_xw)∥_2α−1. Fort >0 we also introduce the spaceCt^α:=C([0, t],C^α) with “norm”

∥W∥_C^α

t := sup

s∈[0,t]

∥W(s,·)∥_C^α.

We first recall the smoothing properties of the semigroup generated by the fractional Laplacian.

Lemma 5.4.8. Let σ >1/2 if d is odd, and σ >1 if d is even. Let A=−(−∆)^σ with periodic boundary conditions on T^dand let(P_t)t≥0 be the semigroup generated by A. Let α∈R, and let u∈C^α, t >0, δ≥0. Then

∥P_tu∥_α+δ .t^−2σδ ∥u∥_α. Forα≤1, u∈C^α−1, v∈C^α, and ε≥0 we obtain

∥P_tu∥_1+ε.t^{−2+ε−α2σ} ∥u∥_α−1 and ∥P_tv∥_1+ε.t^{−1+ε−α2σ} ∥v∥_α.

Proof. The semigroup is given byP_t=ϕ(t^1/(2σ)D),t≥0, whereϕ(x) =e^−|x|2σ. Hence, it suffices to show thatϕsatisfies the assumptions of Corollary 5.3.21. Outside of every ball that contains the singularity 0 (whereϕis not infinitely differentiable),ϕbehaves like a Schwartz function. Let us show thatFϕ∈L¹. Let m be the smallest even integer that is strictly larger thand. Since∥Fϕ∥_L^∞ .∥ϕ∥_L1, it suffices to show that |·|^mFϕ∈L^∞. But

|x|^m|Fϕ(x)|=|F(∆^m/2ϕ)(x)|.∥∆^m/2ϕ∥_L1 . ^

η∈N^d:|η|≤m

∥∂^ηϕ∥_L1.

Ifdis odd, then 2σ−m >1−(d+ 1) =−d, and therefore the right hand side is finite.

The argument for evendis similar, and therefore the proof is complete.

Remark 5.4.9. We only used the lower bounds on σ to prove thatFϕ∈L¹. This holds for everyσ >0, in fact it is well known that|Fϕ(x)|.|x|^−d−2σ. But eventually we are only interested inσ−d/2>1/3, so the lower bounds in Lemma 5.4.8 do not impose any additional restrictions. On the other side they simplify the proof.

Based on these estimates, we can establish an a priori estimate for the action of the fractional heat kernel on the nonlinear part.

Lemma 5.4.10. Let α∈(1/3,1)and w= (w, wDxw)∈C^α. Letε >0and let v∈C^1+ε and G∈C_b². Then for all t∈[0, T] we have

∥P_t[G(v+w)D_x(v+w)]∥_1+ε.G,w t^{−2+ε−α2σ} (1 +∥v∥_2α) +t^−1+ε2σ ∥v∥_1+ε.

Proof. Since v+wis controlled by w, and D_x(v+w) is controlled by D_xw, it would be possible to directly apply Theorem 5.3.15 to defineG(v+w)D_x(v+w). But this would only give us a quadratic estimate inv, which would lead to problems when trying to construct global solutions. Therefore, we consider the terms G(v +w)D_xv and G(v +w)D_xw separately.

The semigroup estimate Lemma 5.4.8 implies for the smooth term

∥P_t[G(v+w)D_xv]∥_1+ε.t^−1+ε2σ ∥G(v+w)D_xv∥₀ .t^−1+ε2σ ∥G(v+w)D_xv∥_L^∞ .t^−1+ε2σ ∥G∥_L^∞∥D_xv∥_ε.t^−1+ε2σ ∥G∥_L^∞∥v∥_1+ε. (5.20) Theorem 5.3.11 applied withβ=α and γ=α−1 lets us estimate the rough term by

∥G(v+w)Dxw∥_α−1≤ ∥G(v+w)Dxw∥_{Dxw,α−1,α}.∥G(v+w)∥_w,α,α(1 +∥w∥_C^α).

(5.21) Since G∈C_b², Lemma 5.3.18 yields

∥G(v+w)∥_w,α,α.G(1 +∥w∥_α)(1 +∥v∥_2α)(1 +∥w∥_w,α,α)². (5.22) We combine (5.21) and (5.22) with the semigroup estimate Lemma 5.4.8 (where we take δ= 2 +ε−α), and obtain

∥P_t(G(v+w)D_xw)∥_1+ε.G,w t^{−2+ε−α2σ} (1 +∥v∥_2α). (5.23) The proof is completed by combining (5.20) and (5.23) and noting that α < 1 implies 2 +ε−α >1 +ε, so that we can replacet^−1+ε2σ byt^{−2+ε−α2σ} .

Next we establish a contraction property for the semigroup acting on the nonlinear part.

Lemma 5.4.11. Let α ∈ (1/3,1) and w ∈ C^α. Let ε > 0 and let v1, v2 ∈ C^1+ε and G∈C_b³. Then for all t∈[0, T]we have

∥P_t[G(v1+w)Dx(v1+w)]−Pt[G(v2+w)Dx(v2+w)]∥_1+ε

.G,w t^{−2+ε−α2σ} (1 +∥v₁∥_2α+∥v₂∥_2α)∥v₁−v₂∥_2α+t^−1+ε2σ (1 +∥v₁∥_1+ε)∥v₁−v₂∥_1+ε. Proof. We decompose

G(v₁+w)Dx(v₁+w)−G(v₂+w)Dx(v₂+w) =g₁+g₂+g₃, where

g₁ := (G(v₁+w)−G(v₂+w))D_xw, g₂:= (G(v₁+w)−G(v₂+w))D_xv₁, g₃ :=G(v₂+w)D_x(v₁−v₂).

Using Theorem 5.3.11 withβ =α and γ =α−1, the termg₁ can be estimated by

∥g₁∥_α−1 =∥(G(v₁+w)−G(v2+w))Dxw∥_α−1 .w ∥G(v₁+w)−G(v2+w)∥_w,α,α. We apply a Taylor expansion in the first step, Lemma 5.3.19 in the second step, and Lemma 5.3.18 in the third step, to obtain

∥G(v₁+w)−G(v2+w)∥_w,α,α

=











η∈Nⁿ:|η|=1

 1 0

(∂^ηG)(v₂+w+r(v₁−v₂))(v₁−v₂)^ηdr







w,α,α

.w



|η|=1

 1 0

∥(∂^ηG)(v₂+w+r(v₁−v₂))∥_w,α,α∥v₁−v₂∥_2αdr .G,w(1 +∥v₁∥_2α+∥v₂∥_2α)∥v₁−v₂∥_2α.

Hence

∥g₁∥α−1 .G,w (1 +∥v₁∥_2α+∥v₂∥_2α)∥v₁−v₂∥_2α. We apply a Taylor expansion tog₂ and obtain

∥g₂∥_L^∞+∥g₃∥_L^∞ .∥(G(v₁+w)−G(v₂+w))D_xv₁∥_L^∞+∥G(v₂+w)D_x(v₁−v₂)∥_L^∞ .G,w∥v₁−v2∥_L^∞∥v₁∥_1+ε+∥v₁−v2∥_1+ε.

The statement now follows from the semigroup estimate Lemma 5.4.8, which we apply withδ = 2 +ε−α to estimate Ptg1, and with δ = 1 +εto estimate Ptg2 and Ptg3. For the last two terms we also need that∥·∥₀ .∥·∥_L^∞.

We are now ready to prove the main result of this section.

Theorem 5.4.12. Let T > 0, d ∈ N, let σ ≥ 1 be such that σ−d/2 > 1/3, and let

α ∈ (1/3,(σ−d/2)∧1). Let u0 ∈ C^α and G ∈ C_b³. Then there exists a unique mild solution u∈C_T^α to equation (5.19).

Proof. Inspired by [Hai11], we subtract the contribution of the initial condition and solve forv_t=u_t−ψ_t−P_t(u₀−ψ₀).

We set up a Picard iteration in C^1+ε for some smallε >0, to be specified below. We definev⁰:= 0, and

vⁿ⁺¹_t :=

 t 0

Pt−s[G(vⁿ_s +ψs+Ps(u0−ψ0))D_x(v_sⁿ+ψs+Ps(u0−ψ0))]ds +λ

 t 0

Pt−sψ_sds.

In Lemma 5.4.2 and Lemma 5.4.3 we showed that (ψ, ψDxψ)∈CT^α. For Lemma 5.4.3 we needed that 1 +d/2−2σ <0. But sinceσ−d/2>1/3, this is satisfied.

Let β ∈(1/3, α) andt ∈[0, T]. We apply the a priori estimate Lemma 5.4.10 withβ in the place of α, to obtain

∥vⁿ⁺¹_t ∥_1+ε≤

 t 0

∥P_t−s[G(v_sⁿ+ψs+Ps(u0−ψ0))D_x(vⁿ_s +ψs+Ps(u0−ψ0))]∥_1+εds +λ

 t 0

∥P_t−sψ_s∥_1+εds .G,ψ

 t 0

(t−s)^{−2+ε−β2σ} (1 +∥v_sⁿ+Ps(u0−ψ0)∥_2β)ds +

 t 0

(t−s)^−1+ε2σ ∥v_sⁿ+P_s(u₀−ψ₀)∥_1+ε+λ∥P_t−sψ_s∥_1+ε^ds.

Now the semigroup estimate Lemma 5.4.8, applied with δ = 2β−α and δ = 1 +ε−α respectively, yields

∥v_tⁿ⁺¹∥_1+ε.G,ψ

 t 0

(t−s)^{−2+ε−β2σ} (1 +∥v_sⁿ∥_1+ε+s^{−2β−α2σ} ∥u₀−ψ0∥_α)ds +

 t 0

(t−s)^−1+ε2σ (∥v_sⁿ∥_1+ε+s^{−1+ε−α2σ} ∥u₀−ψ₀∥_α) +λ(t−s)^{−1+ε−α2σ} ∥ψ_s∥_α^ds.

Fora, b≥0 the integral^₀^t(t−s)^−as^−bdsconverges to zero ast tends to 0 if and only if a+b <1. So if we chooseε >0 small enough, then there existsT1∈(0, T], independent ofu₀, such that

∥vⁿ∥_C1+ε T1

≤ ∥u₀∥_α+ 1

2 (5.24)

for all n∈N.

Let us writeU_s:=P_s(u₀−ψ₀). The contraction estimate Lemma 5.4.11, applied with

β in the place of α, implies for allt∈[0, T] that

∥v_tⁿ⁺¹−v_tⁿ∥_1+ε≤

 t 0





Pt−s[G(vⁿ_s +ψs+Us)D_x(v_sⁿ+ψs+Us)]

−Pt−s[G(vⁿ⁻¹_s +ψ_s+U_s)D_x(v_sⁿ⁻¹+ψ_s+U_s)]^_

1+εds .G,ψ

 t 0

(t−s)^{−2+ε−β2σ} (1 +∥v_sⁿ+Us∥_2β+∥v_sⁿ⁻¹+Us∥_2β)∥v_sⁿ−vⁿ⁻¹_s ∥_2βds +

 t 0

(t−s)^−1+ε2σ (1 +∥v_sⁿ+U_s∥_1+ε)∥v_sⁿ−v_sⁿ⁻¹∥_1+ε

Now the same arguments as before, in combination with (5.24), prove the contraction property on [0, T₂] for a suitableT₂ >0. Hence, there exists a unique solution on [0, T₂].

It remains to show the existence of a global solution. Let Tmax ∈ (0, T] be the maximum time for which there exists a solution v on [0, Tmax). First assume that lim_t→Tmax∥v∥_C1+ε

t < ∞. Then we can iterate the construction of a local solution on small intervals of length T₃, where T₃ >0 is fixed, because the initial condition in each iteration will be bounded by sup_t<Tmax∥v∥_C1+ε

t <∞. Hence, Tmax =T. On the other side lim_t→Tmax∥v∥_C1+ε

t =∞ is impossible, because (5.24) yields

∥v∥_C1+ε Tmax

≤F^{◦⌈Tmax/T1⌉}(∥u₀∥_α)<∞,

whereF(x) := (x+ 1)/2, andF^◦m is the m–fold iterative application of F.

Remark 5.4.13. The continuity of all operators involved in the Picard iteration enables us to show that solutions to (5.19) depend continuously on the reference pathψ: If (ψ^j)_j∈N converges toψ, such that (π◦(ψ^j,D_xψ^j)) converges to π◦(ψ,D_xψ), then the solutionsv^j to

v^j(t,·) =P_t(u₀−ψ₀^j)(x) +

 t 0

Pt−s[G(v^j(s,·) +ψ^j(s,·))D_x(v^j(s,·) +ψ^j(s,·))]ds +λ

 t 0

Pt−sψ^j(s,·)ds

converge to v. This can be seen by similar arguments as the ones used in the proof of Proposition 8 in [Gub04].

As explained in Remark 5.4.4, we expect that if we approximate ˙W by suitable se-quences ( ˙W^j) of smooth functions, then (π◦(ψ^j,D_xψ^j)) converges to π◦(ψ,D_xψ). This indicates that our solution uis the limit of the classical solutions u^j to

∂_tu^j(t, x) =−Au^j(t, x) +G(u^j(t, x))D_xu^j(t, x) + ˙W^j(t, x).

Im Dokument Studies of robustness in stochastic analysis and mathematical finance (Seite 170-176)