Verification of the hypotheses of Proposition 9.1.2

We follow Chapter 3 in [MP11]. First, recall the canonical distance for the heat-equation:

Notation. Fort, t⁰ ≥0 and x, x⁰ ∈R^q let d((t, x),(t⁰, x⁰)) =p

|t⁰−t|+|x⁰−x|.

Assume the setting of the beginning of the last chapter. That means thatX¹and X² are two solutions of the SPDE (5.1) with the same noise ˙W and u:=X¹−X² is the difference of the two, i.e.

u(t, x) = Z t

0

Z

R^q

pt−s(y−x)D(s, y)W(ds dy) a.s. for all t≥0, x∈R^q, (9.16) wherept(x) is theq-dimensional heat kernel andD(s, y) =σ(X¹(s, y))−σ(X²(s, y)), which by (9.4) obeys

|D(s, y)| ≤R₀e^R1|y||u(s, y)|^γ. (9.17) Let (P_t)t≥0 be the heat-semigroup acting on C_tem. To verify (H₂) of Propositon 9.1.2, we need to consider (9.15) and hence need to know what happens close to (t, x) ∈ R+×R^q where |u(t, x)| ≈ a_n. It will be a key idea to split up u into a smooth partu₁ and the differenceu₂ =u−u₁.The smooth partu₁ will allow to be differentiated and follow more or less the heuristics given in Chapter 5. Therefore, forδ ≥0 set

u_1,δ(t, x) :=P_δ(u((t−δ)⁺,·))(x), u_2,δ :=u−u_1,δ. (9.18) We will show that for u_1,δ there is more or less a H¨older-continuous derivative in the space-coordinate and for u_2,δ we can find good estimates for small δ. By uniform continuity of the heat kernelP_δ:C_tem→C_tem it is true that bothu_1,δ and sou_2,δ have sample paths inC(R+, Ctem). By definition, we have

u_1,δ(t, x) = Z

R^q

Z (t−δ)⁺ 0

Z

R^q

p_(t−δ)+−s(y−z)D(s, y)W(ds dy)p_δ(z−x)dz

9.2 Verification of the hypotheses of Proposition 9.1.2 99 and with the help of the Stochastic-Fubini-Formula (Proposition 3.4.9, where (5.7) is used for the expectation condition) reformulate that forδ ≤t to

u_1,δ(t, x) = the spatial derivative of the heat kernel. The following result holds, which is anal-ogous to Lemma 3.1 in [MP11] and has essentially the same proof:

Lemma 9.2.1. The random fields G_δ and F_δ,l are jointly continuous in (s, t, x) ∈ R²₊×R^q and

Note that for the special choice of s=t in the previous lemma:

∂_xlu_1,δ(t, x) =

be the set of points with the smallest u-values in a certain neighborhood close tox and let

ˆ x_n(t, x)

be a measurable choice of a point inBn(t, x) (e.g. with the smallest first coordinate, if this does not suffice to uniquely select a point, take the smallest second coordinate and so on). Let us fix two positive but very small constants ε₀, ε₁ throughout the paper

100 Pathwise Uniqueness Let L=L(ε₀, ε₁) =bε⁻¹₀ (1/2−6ε₁)c ∈N and set fori= 0, . . . , L

βi =iε0 ∈[0,1

2 −6ε1], λi= 2(βi+ε1)∈[0,1] (9.21) and β_L+1 = ¹₂−ε₁.So alltogether fori= 0, . . . , L+ 1:

βi ∈[0,1

2 −ε1]. (9.22)

We define the following subsets ofR^q:

Jn,0(s) :={x∈R^q :|x| ≤K0,|hu_s,Φ^m_xⁿ⁺¹i| ≤an,|∇u_1,an(s,xˆn(s, x))| ≥ ^aεn₄⁰]},

J_n,L(s) :={x∈R^q :|x| ≤K₀,|hu_s,Φ^m_xⁿ⁺¹i| ≤a_n,|∇u_1,an(s,xˆ_n(s, x))| ≤ ^aβLn₄ ]}

and for i= 1, . . . , L−1:

J_n,i(s) :={x∈R^q:|x| ≤K₀,|hu_s,Φ^m_xⁿ⁺¹i| ≤a_n,|∇u_1,an(s,xˆ_n(s, x))| ∈[^aβi+1n₄ ,^aβin₄ ]}.

Recall (9.15) and observe that for t≥0, n∈N: Iⁿ(t)≤a^−1−2/n_n

L(ε0,ε1)

X

i=0

Z t 0

ds Z

R^3q

dxdwdz1Jn,i(s)(x)|u(s, w)|^γ|u(s, z)|^γ

×Φ^m_xⁿ⁺¹(w)Φ^m_xⁿ⁺¹(z)(|w−z|^−α+ 1)Ψs(x)

=:

L(ε0,ε1)

X

i=0

I_n,i(t). (9.23)

To verify the hypotheses of Proposition 9.1.2, it suffices to show the existence of stopping times UM,n,K satisfying (H1) as well as fori= 0, . . . , L,

(H2,i) for all M, K ∈Nwith K≥K1 lim

n→∞E(In,i(t0∧U_M,n,K)) = 0.

We will get to the definition of these stopping times in Chapter 9.5. We now define σx :=σx(n, s) :=∇u_1,an(s,xˆn(s, x))(|∇u_1,an(s,xˆn(s, x))|)⁻¹

as the direction of the gradient ∇u_1,an at the point ˆx_n(s, x) close to x.We also set

¯l_n(β_i) =a^β_n^i+5ε1,

where dependence onβ_i is not written out explicitly if there are no disambiguities.

9.2 Verification of the hypotheses of Proposition 9.1.2 101 To get (H_2,i) we want to derive some properties of points inJ_n,i. Therefore, set J˜n,0(s) :={x∈R^q:|x| ≤K0,|hu_s,Φ^m_xⁿ⁺¹i| ≤an,

σ_x· ∇u_1,a2ε0

n (s, x⁰)≥a^ε_n⁰/16 for allx⁰ ∈R^q s. t. |x⁰−x| ≤5¯l_n(β₀) and |u

2,a^λn⁰(s, x⁰)−u_2,a^λ0

n (s, x⁰⁰)| ≤2⁻⁷⁵a^β_n¹(|x⁰−x⁰⁰| ∨a

2

α(γ−β1−ε1)

n ∨an)

for allx⁰, x⁰⁰∈R^q s.t. |x⁰−x| ≤4√

an,|x⁰⁰−x⁰| ≤¯ln(β0) and |u(s, x⁰)| ≤3a^(1−ε_n ^0)/2 for all x⁰ ∈R^q s.t.|x⁰−x| ≤√

a_n}, J˜_n,L(s) :={x∈R^q:|x| ≤K₀,|hu_s,Φ^m_xⁿ⁺¹i| ≤a_n,

|∇u1,a^λLn (s, x⁰)| ≤a^β_n^L for all x⁰ ∈R^q s.t.|x⁰−x| ≤5¯l_n(β_L) and |u

2,a^λLn (s, x⁰)−u

2,a^λLn (s, x⁰⁰)| ≤2⁻⁷⁵a^β_n^L+1(|x⁰−x⁰⁰| ∨a

2

α(γ−β_L−ε₁)

n ∨a_n)

for allx⁰, x⁰⁰∈R^q s.t. |x⁰−x| ≤4√

an,|x⁰⁰−x⁰| ≤¯ln(βL)}

and for i= 1, . . . , L−1:

J˜_n,i(s) :={x∈R^q :|x| ≤K₀,|hu_s,Φ^m_xⁿ⁺¹i| ≤a_n,

|∇u1,a^λLn (s, x⁰)| ≤a^β_n^L and σ_x· ∇u

1,a^λin (s, x⁰)≥a^β_nⁱ⁺¹/16 for all x⁰ ∈R^q s.t. |x⁰−x| ≤5¯ln(βi)

and |u

2,a^λin (s, x⁰)−u_2,aλi

n (s, x⁰⁰)| ≤2⁻⁷⁵a^βnⁱ⁺¹(|x⁰−x⁰⁰| ∨a

2

α(γ−βi+1−ε1)

n ∨an)

for all x⁰, x⁰⁰∈R^q s.t.|x⁰−x| ≤4√

an,|x⁰⁰−x⁰| ≤¯ln(βi)}.

We also define two deterministic constants

nM(ε1) = inf{n≥1 :a^ε_n¹ ≤2^−M−8}, n₀(ε0, ε1) = sup{n∈N:√

an<2^−a

−ε0ε1/4

n }

and will from now on always assume that

n > n_M(ε₁)∨n₀(ε₀, ε₁). (9.24) The next proposition shows that we can ultimately estimate the size of the sets J˜_n,i(s) instead of that ofJ_n,i(s) :

Proposition 9.2.2. J˜n,i(s) is a compact set for all s ≥ 0, i ∈ {0, . . . , L}. There exist stopping timesU_M,n,K satisfying(H₁)such that for alln≥n_M,i∈ {0, . . . , L}

and s≤U_M,n,K:

J_n,i(s)⊂J˜_n,i(s).

102 Pathwise Uniqueness The proof of this proposition can be found in Chapter 9.5. We will use this propo-sition to show (H2,i) at the end of this chapter. We need the following notation for i∈ {0, . . . , L}:

l_n(β_i) := (129a^1−β_n ⁱ⁺¹)∨a

2

α(γ−βi+1−ε1)

n ,

where we omit the dependence on β_i if there are no disambiguities and obtain:

Lemma 9.2.3. If i∈ {0, . . . , L} and n > n_M(ε₁), then

by (9.20), (9.22) and because a^ε_n¹ < 2⁻⁸ by (9.24). This gives the first inequality.

For the second one, use β_i≤ ¹₂ −6ε₁ and (9.24) to see that

√a_n¯l_n(β_i)⁻¹ =a

1

2−β_i−5ε₁

n ≤a^ε_n¹ <1/2.

We give some elementary properties of the sets ˜Jn,i(s).

Lemma 9.2.4. Assume s≥0, i∈ {0, . . . , L}, x∈J˜n,i(s), x⁰ ∈ R^q and |x⁰−x| ≤

9.2 Verification of the hypotheses of Proposition 9.1.2 103

the distance toxof any point on the line betweenx⁰ andx⁰⁰ is bounded from above by 5¯l_n(β_i). By the Mean Value Theorem and the definition of ˜J_n,i(s),we get

Finally, prove (d) much in the same way as the previous claims: We have

|ˆxn(s, x)−w| < |ˆxn(s, x)−x|+|x−w| ≤ 2√

104 Pathwise Uniqueness

Now, define

Fn(s, x) :=hΦ^m_xⁿ⁺¹, usi= Z

B(0,√ an)

Φ^mn+1(z)u(s, x+z)dz and recall that we write xky ifx, y∈R^q are collinear.

Lemma 9.2.5. Assume i∈ {0, . . . , L−1}, s∈R+.

(a) If x ∈J˜n,i(s), x˜ ∈R^q with (˜x−x) kσx and ln(βi) <|(˜x−x)·σx| ≤¯ln(βi), then

F_n(s,x)˜ −F_n(s, x)

(≥2⁻⁵a^βnⁱ⁺¹|˜x−x| , if(˜x−x)·σ_x≥0,

≤ −2⁻⁵a^β_nⁱ⁺¹|˜x−x|, if (˜x−x)·σ_x<0.

(b) If x, y∈J˜n,i(s), |x−y| ≤¯ln(βi). Then for y˜∈R^q, such that(y−y)˜ kσx and l_n(β_i)<|y−y|˜ <¯l_n(β_i) it holds that

˜

y /∈J˜n,i(s).

(c) If x∈J˜_n,i(s),z∈R^q and |x−z| ≤¯l_n(β_i)/2, then Z

(−¯ln/2,¯ln/2)

db1{z+σ_xb∈J˜_n,i(s)∩B(x,¯l_n/2)} ≤2l_n(β_i).

Proof. For (a) assume (˜x−x)·σx∈[ln(βi),¯ln(βi)]. Then Fn(s,x)˜ −Fn(s, x) =

Z

B^q(0,√ an)

Φ^mn+1(z)(u(s,x˜+z)−u(s, x+z))dz.

Clearly,|z| ≤√

an and for x⁰⁰= ˜x+z, x⁰ =x+z, we have

|x⁰−x| ≤√

a_n,(x⁰⁰−x⁰) = (˜x−x)kσ_x,|x⁰−x⁰⁰| ∈[l_n(β_i),¯l_n(β_i)].

Therefore, we can apply Lemma 9.2.4 (b) to obtain:

F_n(s,x)˜ −F_n(s, x)≥ Z

B^q(0,√ an)

Φ^mn+1(z)|˜x−x|2⁻⁵a^βnⁱ⁺¹dz

≥2⁻⁵a^βnⁱ⁺¹|˜x−x|. (9.25) The same can be done in the case (˜x−x)·σ_x<0.

9.2 Verification of the hypotheses of Proposition 9.1.2 105 To do (b) use the same ideas as before, where Lemma 9.2.4 (b) is replaced by Lemma 9.2.4 (c), to obtain

|F_n(s,y)| ≥ |F˜ _n(s,y)˜ −Fn(s, y)| − |F_n(s, y)|

≥2⁻⁵a^βnⁱ⁺¹ln(βi)−an

≥ 97 32an. Hence, ˜y /∈J˜_n,i(s).

For (c) assumey=z+σxb∈J˜n,i(s) for a certainb∈[−¯ln/2,¯ln/2] (otherwise the integral is 0 anyway). Observe that

|y−x| ≤ |y−z|+|b| ≤¯l_n. So, we can apply (b) for x, y∈J˜_n,i(s) to obtain that

Z

(−¯ln/2,¯ln/2)

db1{z+σxb∈J˜n,i(s)∩B(x,¯ln/2)}

≤ Z

(−¯ln,¯ln)

db1{y+σxb∈J˜n,i(s)} ≤2ln(βi).

Let Σ_x be a q×(q−1) dimensional matrix consisting of an orthonormal basis of the orthogonal space σ_x^ortho={y∈R^q:σx·y= 0}and let|A|denote the Lebesgue measure of a measurable setA⊂R^q.

Lemma 9.2.6. For i ∈ {0, . . . , L −1} and s ≥ 0, n ∈ N there is a constant c_9.2.6 =c_9.2.6(q) such that

|J˜_n,i(s)| ≤c_9.2.6K₀^ql_n(β_i)¯l_n(β_i)⁻¹.

Proof. Set B_x = B^q(x,¯l_n(β_i)/4) and cover the compact set ˜J_n,i(s) with a finite number of these balls, say B_x⁰¹, . . . , B_x0Q0. If |x^0j −x^0k| ≤ ¯ln(βi)/4, then B_x^0j ⊂ B^q(x^0k,¯ln(βi)/2). So, if we increase the radius of the balls around x⁰¹, . . . , x^0Q0 to

¯l_n(β_i)/2, it suffices to use those balls whose centers have at least distance ¯l_n(β_i)/4, which we denote by x¹, . . . , x^Q.If we consider B^q(x^k,¯l_n(β_i)/8), k = 1, . . . , Q, then all of these balls are disjoint. Thus, we have

Q≤K₀^q(¯ln(βi)/8)^−q (9.26) and also

J˜n,i(s)⊂

Q

[

k=1

B^q(x^k,¯ln(βi)/2)∩J˜n,i(s). (9.27)

106 Pathwise Uniqueness Next we want to consider the Lebesgue measure of the sets on the right-hand-side using some kind of Cavalieri decomposition and Lemma 9.2.5 (c). Fix k ∈ {1, . . . , Q} and denote byc(q) the volume of theq-dimensional Euclidean ball. We have

|B^q(x^k,¯l_n(β_i)/2)∩J˜_n,i(s)|= Z

B^q(0,¯ln/2)

dz1{x^k+z∈J˜_n,i(s)}

≤ Z

B^q−1(0,¯ln/2)

dz Z

(−¯ln/2,¯ln/2)

db1{x^k+ Σ_xkz+σ_xkb∈J˜_n,i(s)}

≤ Z

B^q−1(0,¯ln/2)

dz2l_n(β_i)

= 2c(q−1)(¯l_n(β_i)/2)^q−1l_n(β_i),

since|x^k+ Σ_xz−x^k|=|Σ_xz|=|z| ≤¯l_n/2. And therefore, by (9.26) and (9.27) for c9.2.6 = 4·4^qc(q−1):

|J˜_n,i(s)| ≤c_9.2.6K₀^ql_n(β_i)(¯l_n(β_i))⁻¹.

We are now in the position to complete the

Verification of the Hypothese (H2) in Proposition 9.1.2 Let n∈N,t >0 and M ∈Nfixed.

First, consider i = 0. For x ∈ J_n,0(s) and |y−x| ≤ √

a_n we have |u(s, y)| ≤ 3a^(1−εn ^0)/2 due to Proposition 9.2.2. So, we obtain in (9.23) for n large enough so that ε₁ > _n² :

I₀ⁿ(t0∧UM,n,K)≤a^−1−2/n_n 3^2γa^γ(1−ε_n ⁰⁾

Z t0∧UM,n,K

0

ds Z

R^q

dxΨs(x)1Jn,0(s)(x) (9.28) Z

R^q

dw Z

R^q

dzΦ^m_xⁿ⁺¹(w)Φ^m_xⁿ⁺¹(z)(|w−z|^−α+ 1)

≤C1(kΨk∞,kΦk∞)a^−1−ε_n ¹a^γ(1−ε_n ⁰⁾

a^−α/2_n t₀K₀^ql_n(β₀)¯l_n(β₀)⁻¹ (by (9.24) and Lemma 9.2.6)

≤C₁⁰t₀K₀^qa−1−α/2+γ(1−ε0)

n (a^1−ε_n ^0−5ε1 ∨a

2

α(γ−ε₁−ε₀)−5ε₁

n ).

And this expression tends to zero asn→ ∞ since, using thatε0 < ε1 by (9.20),

−1−α/2 +γ(1−ε₀) + 1−ε₀−5ε₁ ≥γ−α/2−7ε₁ >9ε₁ >0

9.2 Verification of the hypotheses of Proposition 9.1.2 107 use Lemma 9.2.4 (d) to get that

|u(s, y)| ≤5a^βi+ To treat the integral inw and z, we use Lemma 9.11.1 to obtain:

I_iⁿ(t₀∧U_M,n,K)≤25ca^−1−2/n_n a^2β_n ^iγ+γm^α_n+1

Hence, it suffices to check for positivity ofρ_1,i andρ_2,i to obtain the desired result.

ρ1,i=−1−α

108 Pathwise Uniqueness by (9.20). Additionally, note that by (9.20),

2

α(γ−β_i+1−ε₁) = 2γ−1

α +1−2β_i+1−2ε₁ α

≥ 1 2 +1

2(1−2β_i−4ε₁)

= 1−βi−2ε1. So we can calculate

ρ_2,i=−1−α

2 + 2β_iγ+γ+ 2

α(γ−β_i+1−ε₁)−β_i−6ε₁

≥ −1−α

2 + 2βiγ+γ+ 1−βi−2ε1−βi−6ε1

= 2γ−1−α

2 −8ε1> ε1 >0 by (9.20).

To finish the proof, we note that in the case i=Lit suffices to use a trivial bound on the integral in (9.31) and obtain with βL≥ ¹₂ −6ε1−ε0≥ ¹₂ −7ε1 from (9.20):

I_Lⁿ(t₀∧U_M,n,K)≤C₃a^−1−ε_n ¹a^2β_n ^Lγ+γa⁻

α

n2t₀K₀^q

≤C₄a^{γ−1−α/2+2(}

1

2−7ε₁)γ−ε₁ n

=C4a2γ−1−α/2−15ε1

n ≤C4a^ε_n¹

by (9.20). And so, we are done with the proof of Propostion 9.1.2. 2

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