Proof of Proposition 9.4.16

9.6. Proof of Proposition 9.4.16

Using the stochastic Fubini Formula we obtain G_δ(s, t, x) = Hence, we define the following square functions fori, j ∈ {1,2}:

Qˇ^i,j_X,δ,η The sum of these estimates of square functions is denoted by

Qˇ^tot_δ =

160 Pathwise Uniqueness All of the proofs are similar to the ones in Chapter 9.4. First, we present some analogues of the bounds on the heat kernels.

Lemma 9.6.1. There is a positive constant C > 0, such that for 0 < t < t⁰, w, v ∈R^q the following holds:

(a) Setting vˆ₀:= 0 andvˆ_i:= ˆvi−1+v_ie_i,1≤i≤q, where e_i is the i-th unit vector in R^q,we have for the spatial differences

|p_t(w+v)−pt(w)| ≤Ct⁻¹²

q

X

i=1

Z |vi| 0

dri p2t(w+ ˆvi−1+riei). (9.118)

(b) We obtain for the time differences

|p_t(w)−p_t⁰(w)| ≤C|t−t⁰|¹²t⁻¹²(p_t(w) +p_2t⁰(w)). (9.119) And as in Lemma 9.3.4 we can derive:

Lemma 9.6.2. There is a positive constant C =C(α, q) < ∞ such that for any x, x⁰ ∈R^q, 0< t≤t⁰:

Z

R^q

Z

R^q

|(p_t(w−x)−p_t⁰(w−x⁰))(p_t(z−x)−p_t⁰(z−x⁰))|(|w−z|^−α+ 1)dw dz

≤Ct^−α/2

1∧ |x−x⁰|²+|t−t⁰| t

.

(9.120) Both of these lemmas can also be found in a slightly different way (the temporal exponents differ) in Lemma 5.3 of [MPS06], so we omit their proof here.

Lemma 9.6.3. LetR >2andη₀ ∈(1/R,1/2). Then there is a constantC =C(η₀) such that for y,y˜∈R^q and 0< t≤t⁰:

(a) 1{|˜y|> t^01/2−η0∨2|y−y|} |p˜ _t(y)| ≤Cexp(−₁₆¹t^−2η0)p_2t(y).

(b) 1{|˜y|> t^01/2−η0∨2|y−y|} |p˜ _t(y)| ≤2^qCexp(−₁₆¹t^−2η0)p_8t(˜y).

The proof of this lemma is almost the same as the proof of Lemma 9.3.5 and we omit it.

9.6 Proof of Proposition 9.4.16 161 Lemma 9.6.4. For all R > 2, there is a constant C =C(K, R) such that for all 0≤p, r≤R, η0, η1∈(1/R,1/2), 0≤s < t≤t⁰ < R and x, x⁰∈[−K, K]^q:

Z

R^q

Z

R^q

|w−x|^p|z−x|^p(pt−s(w−x)−p_t⁰_−s(w−x⁰))(pt−s(z−x)−p_t⁰_−s(z−x⁰))

×1{|w−x|>(t⁰−s)^1/2−η0∨2|x−x⁰|}

×er|w−x|+r|z−x|(|w−z|^−α+ 1)dw dz

≤C(t−s)^−α/2exp(−η₁(t⁰−s)^−2η0/64)[1∧(|x−x⁰|²+|t−t⁰|

t−s )]^1−η1/4. (9.121) This lemma can be proven in the same way as Lemma 9.3.7.

Lemma 9.6.5. For all K ∈ N^≥K1, R > 2 there exist c_9.6.5(K, R), N_9.6.5(K, ω) almost surely such that

∀η₀, η₁ ∈(1/R,1/2),δ ∈(0,1],β∈[0,1/2],N, n∈N,(t, x)∈R+×R^qthe following holds for i+j≥3:

For ω∈ {(t, x)∈Z(N, n, K, β), N ≥N_9.6.5} Qˇ^i,j_X,δ,η

0(s, t, x, t⁰, x⁰)≤c9.4.52^4N9.6.5[(d∧√

δ)^2−η1/2δ^−α/2(d∧1)^2γ+d^2−η1/2]

∀0≤s≤t≤t⁰, x⁰ ∈R^q,

(9.122) where d=d((t, x),(t⁰, x⁰)).

This lemma has almost the same proof as Lemma 9.4.5.

Lemma 9.6.6. For all K ∈ N^≥K1, R > 2 there exist c9.6.6(K, R), N9.6.6(K, ω) almost surely such that ∀η₀, η1 ∈ (1/R,1/2), δ ∈ (0,1], β ∈ [0,1/2], N, n ∈ N, (t, x)∈R+×R^q the following holds for i+j≥3:

Forω ∈ {(t, x)∈Z(N, n, K, β), N ≥N_9.6.6} Qˇ^i,j_T,δ,η

0(s, t, x⁰, t⁰, x⁰)≤c_9.6.62^4N9.6.6[|t−t⁰|^1−η1/2+|t−t⁰|^1−η1/2δ^−α/2(|t−t⁰| ∧1)^γ]

∀0≤s≤t, x⁰∈R^q.

(9.123) This lemma has almost the same proof as Lemma 9.4.6. The next lemma gets endowed with a proof, as an example for the one of the proofs, but also for some technical differences to Lemma 9.4.7.

Lemma 9.6.7. Let 0 ≤ m ≤ m¯ + 1 and assume that (Pm) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε₀ ∈ (0,1), there exist c_9.6.7(K, R) and

162 Pathwise Uniqueness N_9.6.7(m, n, R, ε₀, K, β)(ω)∈Nalmost surely such that

∀η₁∈(1/R,1/2), η0∈(0, η1/32),δ ∈[an,1],N ∈N, (t, x)∈R+×R^q the following holds.

For ω∈ {(t, x)∈Z(N, n, K, β), N ≥N_9.6.7} Qˇ^1,1_X,δ,η

0(s, t, x, t⁰, x⁰)≤c_9.6.7[a^−2ε_n ⁰ + 2^4N9.6.7][d^2−η1/2

+d^2−η1/2δ^−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N]]

∀0≤s≤t≤t⁰, x⁰ ∈R^q.

(9.124) Hered¯_N =|x−x⁰|+√

t⁰−t∨2^−N andδ¯_N =δ∨d¯²_N.Moreover,N_9.6.7is stochastically bounded uniformly in (n, β).

Proof. Let ξ = 1−(8R)⁻¹ ∈(15/16,1) and set N_9.6.7 = N₁(m, n, ξ, ε₀, K, β). We can assume that s > δ and therefore, we have always d((r, w),(t, x))≥√

an in the integral. A use of Lemma 9.4.3 and the bound on|w−x|,|z−x|respectively, gives

Qˇ^1,1_X,δ,η

0(s, t, x, t, x⁰)≤C_9.4.3 Z s−δ

0

dr Z

R^q

dw Z

R^q

dz

(pt−r(w−x⁰)−pt−r(w−x))(pt−r(z−x⁰)−pt−r(z−x)) R^2γ₀ e^4R1Ke^4γK

[2^−N ∨((t−r)^1/2+ (t−r)^1/2−η0∨2|x−x⁰|)]^2γξ

{[2^−N ∨((t−r)^1/2+ (t−r)^1/2−η0∨2|x−x⁰|)]^γm−1+a^β_n}^2γ (|w−z|^−α+ 1).

Let γ⁰ =γ(1−2η0) and observe the trivial inequalities

√

t−r≤K^η0/2(t−r)^1/2−η0, (9.125)

|x−x⁰| ≤c(q)K|x−x⁰|^1−2η0. Then, Lemma 9.3.4 allows the following bound

Qˇ^1,1_X,δ,η

0(s, t, x, t, x⁰)≤C9.4.3c1(K) Z s−δ

0

dr(t−r)^−α/2[1∧ |x−x⁰|² t−r ] [2^{−2N γ}∨(t−r)^γ0∨ |x−x⁰|^2γ0]^ξ

[2^{−N γ0(γm−1)}∨(t−r)^γ0(γm−1)∨ |x−x⁰|^2γ(γm−1)+a^2βγ_n ].

Using

2^{−2N γ}∨(t−r)^γ0∨ |x−x⁰|^2γ0 ≤2^{−2N γ0}∨ |x−x⁰|^2γ0+ (t−r)^γ0

≤2[ ¯d^2γ_N⁰∨(t−r)^γ0],

9.6 Proof of Proposition 9.4.16 163 we can bound the above by

Qˇ^1,1_X,δ,η

0(s, t, x, t, x⁰)≤C_9.4.3c₁(K) Z s−δ

0

dr(t−r)^−α/2[1∧|x−x⁰|²

t−r ] 2^ξ( ¯d^2γ_N^0ξ∨(t−r)^γ0ξ) 2^γm−1[( ¯d²_N ∨(t−r))^γ0(γm−1)+a^2βγ_n ]

≤C9.4.3c2(K) Z s−δ

0

dr1{t−r ≥d¯²_N}(t−r)^{−α/2+γ0ξ}[1∧ |x−x⁰|² t−r ] [(t−r)^γ0(γm−1)+a^2βγ_n ]

+C_9.4.3c₂(K) Z s−δ

0

dr1{t−r <d¯²_N}(t−r)^−α/2[1∧|x−x⁰|² t−r ] ¯d^2γ_N^0ξ [ ¯d^2γ_N^0(γm−1)+a^2βγ_n ]

=C9.4.3c2(K)(I1+I2). (9.126)

We start with an estimate on I1. Ifr ≤s−δ and t−r≥d¯²_N, then

r≤t−d¯²_N ∧s−δ≤t−d¯²_N ∧t−δ =t−δ¯_N. (9.127) Use that to start with

I₁ ≤ Z t−δ¯N

0

dr(t−r)^{−α/2+γ0ξ+γ0(γm−1)}[1∧|x−x⁰|² t−r ] + (t−r)^{−α/2+γ0ξ}[1∧|x−x⁰|²

t−r ]a^2βγ_n . We want to drop the minimum with 1 to consider

|x−x⁰|² Z t

δ¯_N

du u^{−2−α/2+γ0ξ+γ0(γm−1)}+u^{−2−α/2+γ0ξ}a^2βγ_n

and then face three cases for the exponents: < −1,= −1, > −1. In the first and third case use the following inequality forp∈(−1,1), p6= 0,0< a < b:

Z b a

u^p−1du= 1

|p||a^p−b^p| ≤log(b/a)(a^p+b^p). (9.128) This is true, since 1−x≤ −logx, forx≥0 with x= (b/a)^p orx= (a/b)^p.

In the −1-case the integral is bounded by logK+ log(1/δ¯N). Hence, using that t ≤ K (therefore t0∨(−1−α/2+γ⁰ξ+γ⁰(γm−1)) ≤ K¹), in any of the cases there is a constantc(K) such that :

I₁≤K¹|x−x⁰|²log(K/δ¯_N)(¯δ^(−α/2+γ_N ^{0ξ+γ0(γm−1))∧0}+ ¯δ^(−α/2+γ_N ^0ξ)∧0a^2βγ_n ),

164 Pathwise Uniqueness The log-term is bounded by c(K, R)|x−x⁰|^−η1/2 using Lemma 9.3.1.

Moreover, by Lemma 4.1(b) in [MP11] we bound

I₂ ≤c(α)|x−x⁰|²(|x−x⁰|²∨δ)^−α/2d¯^2γ_N^0ξ[ ¯d^2γ_N^0(γm−1)+a^2βγ_n ]. (9.129) Therefore, we can bound ˇQ^1,1_X,δ,η

0(s, t, x, t, x⁰) by

c_9.4.3c₃(K, R)[|x−x⁰|^2−η1/2(¯δ_N^{(−α/2+γ0(γm+ξ−1))∧0}+ ¯δ_N^{(−α/2+γ0ξ∧0)}a^2βγ_n ) + (δ∧ |x−x⁰|²)δ^−α/2d¯^2γ_N^0ξ[ ¯d^2γ_N^0(γm−1)+a^2βγ_n ]

Now we can replace ξ = 1−(8R)⁻¹ by 1 andγ⁰ =γ(1−2η₀) byγ at the cost of multiplying byd^−η1/2≥¯δ_N^−η1/4. This is true, since

ξγ⁰ =γ(1−2η₀)(1−(8R)⁻¹)≥γ(1−η₁/4), hence ξγ⁰−γ ≥ −γη₁/4≥ −η₁/4 and

γ⁰(γm+ξ−1) =γ(1−2η0)(γm− 1 8R)x

≥γ(1− η1

16)(γm− 1

8R) (by η1 ≥32η0)

≥γγm−γ 1

8R −γγm

η₁ 16

≥γγ_m− 1

8R −2η₁ 16

≥γγm−η1

8 −η1

8 (byη1 > R⁻¹)

≥γγ_m−η1

4. Use

γγ_m−α/2≥γ −α/2>1−γ >0 to put all things together to

Qˇ^1,1_X,δ,η

0(s, t, x, t, x⁰)≤c9.4.3c3(K, R)[|x−x⁰|^2−η1/2 + (√

δ∧ |x−x⁰|)^2−η1/2δ^−1−α/2[ ¯d^2γγ_N ^m+ ¯d^2γ_Na^2βγ_n ]

Lemma 9.6.8. Let 0 ≤ m ≤ m¯ + 1 and assume that (Pm) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε0 ∈ (0,1), there exist c9.6.8(K, R) and N_9.6.8(m, n, R, ε₀, K, β)(ω)∈Nalmost surely such that

9.6 Proof of Proposition 9.4.16 165

∀η₁ ∈(1/R,1/2),η₀∈(0, η₁/32),δ ∈[a_n,1],N ∈N,(t, x)∈R+×R^q the following holds.

Forω ∈ {(t, x)∈Z(N, n,K, β), N ≥N9.6.8} Qˇ^1,1_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰)≤c_9.6.8[a^−2ε_n ⁰ + 2^4N9.6.8] [|t−t⁰|^1−η1/2

+|t−t⁰|^1−η1/2δ^−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N]]

∀0≤s≤t, x⁰ ∈R^q.

(9.130)

Hered¯_N =|x−x⁰|+√

t⁰−t∨2^−N andδ¯_N =δ∨d¯²_N.Moreover,N9.6.8 is stochastically bounded uniformly in (n, β).

For this lemma, the proof would just be the same as Lemma 9.4.8 using some ideas of the proof just before. And finally, we state a lemma about the distance in thes-variable:

Lemma 9.6.9. Let 0 ≤ m ≤ m¯ + 1 and assume that (Pm) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε₀ ∈ (0,1), there exist c_9.6.9(K, R, γ) and N_9.6.9(m, n, R, ε₀, K, β)(ω) ∈ N almost surely such that ∀η₁ ∈ (1/R,1/2), δ ∈ [an,1], N ∈N, (t, x)∈R+×R^q the following holds.

Forω∈{(t, x)∈Z(N, n, K, β), N ≥N_9.6.9} Qˇ_S,δ,η0(s, s⁰, t⁰, x⁰)≤c9.6.9[a^−2ε_n ⁰ + 2^4N9.6.9]|s⁰−s|δ^−α/2

{|s⁰−s|^−η1/2[(|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γγm +a^2βγ_n (|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γ]

+ 2^{N η1}[ ¯d^2γ(γ_N ^m−1)+a^2βγ_n d¯^2γ_N]}

∀0≤s≤t, x⁰ ∈R^q.

(9.131)

Here d¯_N = (|t−t⁰|^1/2 +|x−x⁰|)∨2^−N and δ¯_N = δ ∨d¯²_N. Moreover, N_9.6.9 is stochastically bounded uniformly in (n, β).

Proof. Choose ξ = 1−(2γR)⁻¹.

ChooseN_9.6.9 =N₁(m, n, ξ, ε₀, K, β). It is forr ≤s∨s⁰−δ:√

a_n<√ δ <√

s−r <

√t−r, thus

2^−N ∨((t⁰−r)^1/2+|w−x|)≤2^−N ∨ |x−x⁰|+ (t⁰−r)^1/2+|w−x⁰|

≤d¯²_N+ (t⁰−r)^1/2+|w−x⁰|.

166 Pathwise Uniqueness This bound, Lemmas 9.6.2, 9.4.3 and 9.3.6 give:

QˇS,δ,η0(s, s⁰, t⁰, x⁰)≤c0(K, R)[a^−2ε_n ⁰ + 2^4N9.6.9] Z (s∨s⁰−δ)⁺

(s∧s⁰−δ)⁺

dr(t⁰−r)^−α/2[ ¯d^2ξγ_N + (t⁰−r)^ξγ][ ¯d^2γ(γ_N ^m−1)+ (t⁰−r)^γ(γm−1)+a^2βγ_n ]

≤4c0(K, R)[a^−2ε_n ⁰ + 2^4N9.6.9]{

Z (s∨s⁰−δ)⁺

(s∧s⁰−δ)⁺ 1{r ≤t⁰−d¯²_N}(t⁰−r)^−α/2+ξγ[(t⁰−r)^γ(γm−1)+a^2βγ_n ]dr +

Z (s∨s⁰−δ)⁺

(s∧s⁰−δ)⁺ 1{r > t⁰−d¯²_N}(t⁰−r)^−α/2drd¯^2ξγ_N [ ¯d^2γ(γ_N ^m−1)+a^2βγ_n ]}

=c(J1+J2).

Both integrals are bounded by integral length times the maximal integrand:

J1≤ |s⁰−s|[(t⁰−(s−δ)⁺)^{γ(γm+ξ−1)−α/2}+a^2βγ_n (t⁰−(s−δ)⁺)^γξ−α/2, (9.132) J₂≤ |s⁰−s|(t⁰−(¯s−δ))^−α/2[ ¯d^2γ(γ_N ^m−1)+a^2βγ_n ]. (9.133) Observe the following estimates

(t⁰−(¯s−δ)⁺)≥(t⁰−(t⁰−δ)⁺)≥δ, (t⁰−(¯s−δ)⁺)^−α/2 ≤δ^−α/2,

t⁰−(s−δ)⁺ ≤ |t⁰−t|+ ((t−s)∨(t⁰−s⁰)) +δ,

(t⁰−(s−δ)⁺)^γγm−α/2 ≤8δ^−α/2(|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γγm d¯^2γ(ξ−1)_N = (2^−N ∨d)^−1/R≤2^{N η1},

|s⁰−s|=|¯s−s| ≤ |t⁰−s| ≤ |t⁰−(s−δ)⁺|,

(t⁰−(¯s−δ)⁺)^γ(ξ−1) ≤(t⁰−(¯s−δ)⁺)^−(2R)−1 ≤ |s⁰−s|^−(2R)−1 ≤(|s⁰−s| ∧1)^−η1/2. With these estimates one can easily obtain:

QˇS,δ,η0(s, s⁰, t⁰, x⁰)≤c1(K, R)[a^−2ε_n ⁰ + 2^4N9.6.9]|s⁰−s|δ^−α/2 {|s⁰−s|^−η1/2(|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γγm

+a^2βγ_n (|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γ+ 2^{N η1}δ^−α/2[ ¯d^2γ(γ_N ^m−1)+a^2βγ_n ]}.

Notation: d((s, t, x),ˇ (s⁰, t⁰, x⁰)) :=|s−s⁰|^1/2+|t−t⁰|^1/6+|x−x⁰|^1/3.

The definition of this metric seems rather strange. The reason for doing so is that

9.6 Proof of Proposition 9.4.16 167 we only need a result for |t−t⁰|+|x−x⁰|very small (equal 0, in fact). The reader may already have noticed that however, without the derivatives on the heat kernels, the local bounds get better in the exponent by +1. Nevertheless, in Lemma 9.4.7 there is a∧0 which allows us not to be better thand² ∼ |x−x⁰|². So the only way to get better there is to punish large distances ind=|x−x⁰|. By setting ˇd≈d^1/3, we have in Lemma 9.6.7 (the analogue of Lemma 9.4.7)

d²=d^1/3d^2/3 ≤d^1/3d^γγm/3 = ˇd( ˇd^γγm).

Remark 9.6.10. There is a subtle point why we did not prescribe (j+f)2^{−6`</sup> − (i+e)2<sup>−2`} ≤ 2^{−2`</sup> instead of ≤ 2<sup>−2N</sup>. The reason is that 2<sup>−`} ≈ dˆis the asymp-totics only for the first factor. The expression in brackets is governed by the 2^−N expression, which is determined by the Z(N, n, K, β) expression, i.e. the distance d((t, x),(ˆt0,xˆ0)).

Next, define

∆¯u1(m, n, λ, ε0,2^−N) =a^−ε_n ⁰a^−λα/4_n [(a^λ/2_n ∨2^−N)^γγm+a^βγ_n (a^λ/2_n ∨2^−N)^γ] with ¯∆u1(. . . ,2^−N+1) ≤ 2 ¯∆u1(. . . ,2^−N). Then, we put together the various esti-mates on the quadratic variations for G_δ:

Corollary 9.6.11. Let 0 ≤ m ≤ m¯ + 1 and assume that (Pm) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε0 ∈ (0,1), there exist c9.6.11(K, R), N_9.6.11(m, n, R, ε₀, K, β)(ω)∈N almost surely such that

∀η₁ ∈ (1/R,1/2), η0 ∈ (1/R, η1/32) δ ∈ [an,1], N ∈ N, (t, x) ∈ R+ ×R^q and dˇ= ˇd((s, t, x),(s⁰, t⁰, x⁰))≤2^−N, |t−s| ∨ |t⁰−s⁰| ≤2^−2N the following holds:

For ω∈{(t, x)∈Z(N, n, K, β), N ≥N_9.4.10} Qˇ^tot_δ (s, t, x, s⁰, t⁰, x⁰)≤c9.6.11(a^−2ε_n ⁰ + 2^4N9.6.11) ˇd^2−32η1δ^−α/2

[(δ∨2^−N)^γγm+a^βγ_n (δ∨2^−N)^γ]²

∀0≤s≤t≤t⁰, x⁰ ∈R^q.

(9.134)

Moreover, N_9.6.11 is stochastically bounded uniformly in (n, β).

Proof. The proof simply consists of putting together the last lemmas. LetN9.6.11= N_9.6.5∨N_9.6.6∨N_9.6.7∨N_9.6.8∨N_9.6.9, which is then clearly uniformly bounded in (n, β) and

c₀ =c_9.6.11(a^−2ε_n ⁰+ 2^4N9.6.11) = (c_9.6.5∨c_9.6.6∨c_9.6.7∨c_9.6.8∨c_9.6.9)(a^−2ε_n ⁰+ 2^4N9.6.11).

168 Pathwise Uniqueness Then, we get for ˇd= ˇd((s, t, x),(s⁰, t⁰, x⁰))≤2^−N:

d=d((t, x),(t⁰, x⁰))≤2^−3N d¯N = 2^−N

|t−t⁰| ∨(t−s)∨(t⁰−s⁰)≤2^−6N ∨2^−2N = 2^−2N dˇ⁴≤2^−4N ≤2^{−2N γγm}

( ˇd^−32η1+ 2^{N η1})≤( ˇd^−32η1 + ˇd^−η1)≤2 ˇd^−32η1. and thus,

Qˇ^tot_δ (s, t, x, s⁰, t⁰, x⁰)≤3c₀{( ˇd³∧√

δ)^2−η1/2δ^−α/2( ˇd³∧1)^2γ+ ˇd^6−32η1 + ˇd^6−32η1 + ˇd^6−32η1δ^−α/2[2^{−2N γγm}+a^2βγ_n 2^{−2N γ}]

+|s⁰−s|^1−η1/2δ^−α/2(|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γγm +a^2βγ_n (|t⁰−t| ∨(t−s)∨(t⁰−s⁰)∨δ)^γ

+|s⁰−s|2^{N η1}δ^−α/2[ ¯d^2γ(γ_N ^m−1)+a^2βγ_n ]}.

≤3c0{dˇ²δ^−α/2(2^{−2N γγm}+a^2βγ_n )( ˇd^−32η1 + 2^{N η1})

+ ˇd^2−32η1δ^−α/2[2 ˇd⁴+ (2^−2N ∨δ)^γγm+a^2βγ_n (2^−2N ∨δ)^γ]

≤6c0dˇ^2−32η1δ^−α/2(2^{−2N γγm}+a^2βγ_n )

+ ˇd^2−32η1δ^−α/2[2^−4N+1+ (2^−2N∨δ)^γγm+a^2βγ_n (2^−2N ∨δ)^γ]

≤9c0dˇ^2−32η1δ^−α/2[(2^−N ∨√

δ)^2γγm+a^2βγ_n (2^−N∨√ δ)^2γ].

Finally, we can do the proof of Proposition 9.4.16 just in the same way as Propo-sition 9.4.11.

Proof of Proposition 9.4.16: LetR= 33η⁻¹₁ , η0∈(R⁻¹, η1/32) and consider the case s≤tin the beginning only. Set

dˇ=p

|s⁰−s|+|x−x⁰|^1/3+|t−t⁰|^1/6.

By Corollary 9.6.11 for (t, x)∈Z(N, n, K, β), N ≥N_9.6.11 it holds that:

Qˇ^tot_aλ

n(s, t, x, s⁰, t⁰, x⁰)^1/2 ≤c9.6.11(a^−ε_n ⁰ + 2^2N9.6.11) ˇd^η1/8dˇ^1−7η1/8[ ¯∆u1(m, n, λ, ε0,2^−N)]

∀s≤t≤t⁰, s⁰≤t⁰≤TK,|x⁰| ≤K+ 2,

(9.135)

9.6 Proof of Proposition 9.4.16 169 if ˇd((s, t, x),(s⁰, t⁰, x⁰))≤2^−N and |t⁰−s⁰| ∨ |t−s| ≤2^−2N.

Choose N3= ³³_η

1[N9.6.11+N4(K, η1)] +c5(q), whereN4 is chosen in such a way that (q+ 2)2^q+3c₁(K, η₁)[a^−ε_n ⁰+ 2^2N9.6.11]2^−η1N3/8 ≤c₁(K, η₁)[a^−ε_n ⁰+ 2^2N9.6.11]2^{−4N9.6.11−4N4}

≤a^−ε_n ⁰2⁻¹⁰²,

i.e. N4=N4(an, ε0, N9.6.11, c1(K, η1)) and hence N3 =N3(n, ε0, N9.6.11, K, η1). Set

∆(m, n,d¯_N) := (q+ 2)2^q+3)⁻¹a^−ε_n ⁰2⁻¹⁰⁰∆¯_u1(m, n, λ, ε₀,2^−N) and let N⁰∈Nsuch that ˇd≤2^−N0. Then it holds on

{ω: (t, x)∈Z(N, n, K+ 1, β), N ≥N3, N⁰ ≥N3} that

Qˇ^tot_aλ

n(s, t, x, s⁰, t⁰, x⁰)^1/2 ≤dˇ^1−(7η1/8)2⁻²∆(m, n,d¯_N).

Remembering the decomposition ofG_δin (9.114) into the sum of three martingales and applying the Dubins-Schwarz-Theorem (Theorem 3.2.8), we can write as long ass≤t≤t⁰, s⁰≤t⁰ and ˇd≤2^−N:

P[|G_aλ

n(s, t, x)−G_aλ

n(s⁰, t⁰, x⁰)| ≥d((s, t, x)(s⁰, t⁰, x⁰))^1−η1∆(m, n,d¯_N) (t, x)∈Z(N, n, K+ 1, β), N⁰∧N ≥N3, t⁰≤TK]

≤3P[ sup

u≤d˜^2−7η1/4(∆(m,n,d¯N)/4)²

|B(u)| ≥d˜^1−η1∆(m, n,d¯N)/3]

≤3P[sup

u≤1

|B(u)| ≥d˜^−η1/8]

≤c Z ∞

d˜^−η1/8

exp(−y²/2)dy

≤c₀exp(−d˜^−η1/4/2), (9.136)

where we used the Reflection Principle in the second last inequality.

Next, apply Lemma 9.8.1, where we should make clear what the parameters are.

We take

q₁=q₂ = 1, q₃=q, r= 3, E=R²+×R^q,

¯

q=q+ 2, v₁=v₂ = 2, v₃= 1, v₀ = 1, ˆ

n= (m, n, λ, β), S=N²×[0,1/2]×(0,1),

Σ(N, K,n) =ˆ Z(N, n, K, β),Σ⁰(N) ={0} × {0≤t≤T_K} ×R^q, s= 1, α₁ = 1,∆₁(ˆn,2^−N) = ∆(m, n,2^−N), k₁ = 2, c(α₁)≤4, η=η₁, Y_nˆ(y) =G_aλ

n(s, t, x) with y= (s, t, x), N₀(η, K,ˆn) =N₃.

170 Pathwise Uniqueness Note that theN₀ is uniformly bounded in ˆn= (n, λ, β). By Lemma 9.8.1 we know there exists a N9.4.16 stochastically bounded uniformly in (n, β, λ), for which we obtain for s≤t≤t⁰, s⁰ ≤t⁰,(t, x)∈Z(N, n, K, β), N ≥N9.4.16, |t−s| ∨ |t⁰−s⁰| ≤

In this chapter we consider spatial and temporal distances of u_2,δ(t, x). We will always assume that 0≤t≤t⁰.

Therefore, it seems reasonable to introduce the following square functions Qˆ_{T ,1,δ}(t, t⁰, x) =

Im Dokument Pathwise Uniqueness of the Stochastic Heat Equation with Hölder continuous o diffusion coefficient and colored noise (Seite 159-170)