Local bounds on the difference of two solutions

˜

w=w−x,z˜=z−x and then use Lemma 9.3.5 (b) to obtain Z

R^q

Z

R^q

|p_t⁰_−s,l(w−x⁰)pt−s,l(z−x)| |w−x|^2p/η1|z−x|^2p/η1

×1{|w−x|>(t⁰−s)^1/2−η0 ∨2|x−x⁰|}e^2r/η1(|w−x|+|z−x|)(|w−z|^−α+ 1)dw dz

≤c Z

R^q

Z

R^q

|p_t⁰_−s,l( ˜w+x−x⁰)|(t−s)^−1/2p2(t−s)(˜z)|w|˜ ^2p/η1|˜z|^2p/η1

×1{|w|˜ >(t⁰−s)^1/2−η0 ∨2|x−x⁰|}e^{2r/η1(|w|+|˜˜ z|)}(|w˜−z|˜^−α+ 1)dw d˜˜ z

≤c(R) Z

R^q

Z

R^q

exp(−₆₄¹ (t⁰−s)^−2η0)p_16(t⁰−s)( ˜w)p2(t−s)(˜z)

× |w|˜ ^2p/η1|˜z|^2p/η1(t−s)^−1/2e^{2r/η1(|w|+|˜˜ z|)}(|w˜−z|˜^−α+ 1)dw d˜˜ z.

≤c(R, η1, K)(t−s)^−1/2exp(−₆₄¹ (t⁰−s)^−2η0)

e^{32r2η−21 (t0−s)}(t⁰−s)^pη1−1(t−s)^pη−11 ((t⁰−s)^−α2 + 1).

≤c(R, K)(t−s)^−1/2exp(−₁₂₈¹ (t⁰−s)^−2η0),

where we used Lemma 9.3.6, first part, in the next to last line and (t⁰ −s) ≤ K.

The other summands are similar, we use Lemma 9.3.5 (with t = t⁰ for lines 1 and 3) and can use the exponential of t⁰ −s (t−s in lines 1 and 3) to control all of the negative exponents. Putting this back in (9.39) gives the result, since (1 +α/2)(1−η₁/2) + (1/2)(η₁/2)≤1 +α/2.

9.4. Local bounds on the difference of two solutions

Within this section we present the extension of Theorem 5.3.3, i.e. the results showing “H¨older-continuity of order 2”. This chapter is very similar in its ideas to Chapter 5 of [MP11].

Let us recall that forn∈N0,

an= exp(−n(n+ 1)/2) and for (t, x),(t⁰, x⁰)∈R+×R^q:

d((t, x),(t⁰, x⁰)) =p

|t−t⁰|+|x−x⁰| (| · |Euclidean norm).

Define for N, K, n∈N,β ∈[0,1/2] the random set

Z(N, n, K, β)(ω) ={(t, x)∈[0, T_K]×[−K, K]^q ⊂R+×R^q : there is a

(ˆt₀,xˆ₀)∈[0, T_K]×R such thatd((t, x),(ˆt₀,xˆ₀))≤2^−N,

|u(ˆt0,xˆ0)| ≤an∧(√

an2^−N), and |∇u_1,an(ˆt0,xˆ0)| ≤a^β_n}, (9.40)

118 Pathwise Uniqueness For β = 0 define Z(N, n, K,0) = Z(N, n, K) as above, but with the condition on

∇u_1,an omitted.

Note that (t, x) ∈ Z(N, n, K, β) always implies t ≤K. For γ <1 define recur-sivelyγ₀ = 1 and

γ_m+1 =γγ_m+ 1−α

2. (9.41)

This gives the explicit formula

γ_m= 1 +(γ−α/2)(1−γ^m)

1−γ . (9.42)

Since α <2(2γ−1) we have γ∞= ^1−α/2_1−γ >2 so there will be an ¯m ∈N such that γm+1¯ >2≥γm¯. Set ˜γm:=γm∧2,0≤m≤m¯ + 1.

Definition 9.4.1. A collection of [0,∞]-valued random variables {N(α) :α∈A}

will be called stochastically bounded uniformly inα, iff

Mlim→∞sup

α∈AP[N(α)≥M] = 0.

Form∈Z+, we will let (P_m) denote the following property:

Property (Pm). For any n ∈ N, ξ, ε0 ∈ (0,1), K ∈ N^≥K1 and β ∈ [0,1/2], there is anN₁(ω) =N₁(m, n, ξ, ε₀, K, β)∈Na.s., such that for all N ≥N₁, if (t, x)∈Z(N, n, K, β),t⁰ ≤T_K andd((t, x),(t⁰, x⁰))≤2^−N, then

|u(t⁰, x⁰)| ≤a^−ε_n ⁰2^{−N ξ}[(√

an∨2^−N)^γm−1+a^β_n1{m >0}]. (9.43) Moreover, N₁ is stochastically bounded uniformly in (n, β).

Proposition 9.4.2. Property (Pm) holds for any m≤m¯ + 1.

We first give a proof of the induction start, meaning that we prove (P0):

Proof of Proposition 9.4.2, first part. We apply Theorem 5.3.3. Set Z(N, K) :=

Z(N,0, K,0), meaning that a₀ = 1. Let ξ ∈(0,1) and ξ⁰ := (1 +ξ)/2. If (t, x) ∈ Z(N, n, K, β) then (t, x) ∈ Z(N, K). Theorem 5.3.3 gives that there exists an N0(ξ⁰, K+ 1) a.s. such that for N ≥N0(ξ⁰, K+ 1) we have |u(t, x)−u(ˆt0,xˆ0)| ≤ 2^{−N ξ0}. Setting N₁(0, ξ, K) := N₀(ξ⁰(ξ), K + 1)∨4(1−ξ)⁻¹, we obtain for N ≥ N₁(0, ξ, K),

|u(t, x)| ≤2^{−N ξ0}+|u(ˆt0,xˆ0)| ≤2^{−N ξ0}+ 2^−N ≤2^{1−N ξ0}.

And therefore, for (t⁰, x⁰) witht⁰ ≤T_K and d((t, x),(t⁰, x⁰))≤2^−N we obtain

|u(t⁰, x⁰)| ≤ |u(t⁰, x⁰)−u(t, x)|+|u(t, x)| ≤2^{−N ξ0}+ 2^{1−N ξ0} ≤2^{2−N ξ0} ≤2^{−N ξ} where the last inequality holds since N ≥4(1−ξ)⁻¹.

9.4 Local bounds on the difference of two solutions 119 The induction step from (P_m) to (P_m+1) is a bit more technical and needs some preparation. It will be completed at the end of this section on page 146.

To get there we first write down a lemma, which tells us what we can get out of Property (P_m):

Lemma 9.4.3. Let 0≤m≤m¯ + 1. Assume that (P_m) holds. Let η, ξ, ε₀, K, β as in (P_m). If d¯_N = 2^−N ∨d((s, y),(t, x)) and C_9.4.3(ω) = (4a^−ε_n ⁰ + 2^2N1(ω)2Ke^K)², then for any fixed N ∈N. On the event

{ω:N ≥N₁(m, n, ξ, ε₀, K, β),(t, x)∈Z(N, n, K, β)}, we have

|u(s, y)| ≤p

C_9.4.3e^|y−x|d¯^ξ_N

×[(√

a_n∨d¯_N)^γm−1+1{m >0}a^β_n] (9.44) for all s < T_K and y∈R^q.

Proof. There are two cases to consider.

Case 1: d=d((s, y),(t, x))≤2^−N1. Choose an N⁰ ∈ {N₁, . . . , N}such that

ifd >2^−N : 2^−N0−1< d≤2^−N0, ifd≤2^−N :N⁰ :=N.

Then, (t, x)∈Z(N⁰, n, K, β) andd≤2^−N0 ≤2d∨2^−N ≤2 ¯d_N and hence, Property (P_m) yields:

|u(s, y)| ≤a^−ε_n ⁰2^−N0ξ[(√

an∨2^−N0)^γm−1+a^β_n1{m >0}]

≤2a^−ε_n ⁰d¯^ξ_N[2(√

an∨d¯_N)^γm−1+a^β_n1{m >0}].

Case 2: d >2^−N1. If K≥K1 and s≤TK:

|u(s, y)| ≤2Ke^|y|≤2Ke^|y|(d2^N1)^ξ+γm−1

≤2Ke^Ke^|y−x|2^2N1d¯^ξ+γ_N ^m−1.

This lemma gives control onu(s,·) spatially close to points in Z(N, n, K, β). To do the induction step we want to use this control in|D(r, w)| ≤R₀e^R1|w||u(r, w)|^γ, which played a role for

Fδ,l(s, t, x) =

Z (s−δ)⁺ 0

pt−r,l(w−x)D(r, w)W(dr dw), (9.45)

120 Pathwise Uniqueness δ >0,0≤s≤t,x∈R^q and for 1≤l≤q. This was the derivative ofu_1,δ, as given in Lemma 9.2.1. This will lead to an even better bound on u1,δ and this iterative procedure will lead to the induction step. Later, we will also give estimates foru_2,δ. To estimate F_δ,l we use the following decomposition fors≤t≤t⁰,s⁰ ≤t⁰:

|F_δ,l(s, t, x)−F_δ,l(s⁰, t⁰, x⁰)|

≤ |F_δ,l(s, t, x)−F_δ,l(s, t, x⁰)|+

+|F_δ,l(s, t, x⁰)−F_δ,l(s, t⁰, x⁰)|

+|F_δ,l(s, t⁰, x⁰)−Fδ,l(s⁰, t⁰, x⁰)|

=|

Z (s−δ)⁺ 0

(pt−r,l(w−x⁰)−pt−r,l(w−x))D(r, w)W(dr dw)|

+|

Z (s−δ)⁺ 0

(p_t−r,l(w−x⁰)−p_t⁰_−r,l(w−x⁰))D(r, w)W(dr dw)|

+|

Z (s∨s⁰−δ)⁺ (s∧s⁰−δ)⁺

p_t⁰−r,l(w−x⁰)D(r, w)W(dr dw)|, 1≤l≤q.

(9.46) All of these three expressions are martingales in the upper integral bound, where the rest of the values x, x⁰, t, t⁰,(s∧s⁰−δ)⁺ stay fixed. We want to consider the quadratic variations of these martingales and use the Dubins-Schwarz theorem.

Remark 9.4.4. As for any spatial dimension 1 ≤l ≤q we would execute the same calculations we restrict ourselves now to l= 1 for the estimates on F_δ,l.

Dependence of constants on the universal constants α, q, γ, R₀ and R₁ will not be mentioned in the following lemmas.

In order to calculate the first two of these quadratic variations, we need to intro-duce the following partition of R^q (for fixed values ofx, x⁰, η0):

A^η₁⁰(r, t) =1{y ∈R^q:|y−x| ≤(t−r)^1/2−η0 ∨2|x−x⁰|}, A^η₂⁰(r, t) =1{y ∈R^q:|y−x|>(t−r)^1/2−η0 ∨2|x−x⁰|},

whenever 0 ≤ r < t. Most of the times we will just write A1 and A2 instead of A1(r, t), A2(r, t) if the values ofr, tare clear from the context. Following (9.46), we introduce the following square functions for i, j∈ {1,2}:

Q^i,j_X,δ,η

0(s, t, x, t, x⁰) =

Z (s−δ)⁺ 0

dr Z

A^η_i⁰(r,t)

dw Z

A^η_j⁰(r,t)

dz

|(p_t−r,1(w−x⁰)−pt−r,1(w−x))(pt−r,1(z−x⁰)−pt−r,1(z−x))|

R²₀e^R1(|w|+|z|)|u(r, w)|^γ|u(r, z)|^γ(|w−z|^−α+ 1),

9.4 Local bounds on the difference of two solutions 121

Q^i,j_T,δ,η

0(s, t, x⁰, t⁰, x⁰) =

Z (s−δ)⁺ 0

dr Z

A^η_i⁰(r,t⁰)

dw Z

A^η_j⁰(r,t⁰)

dz

|(p_t−r,1(w−x⁰)−p_t⁰−r,1(w−x⁰))(pt−r,1(z−x⁰)−p_t⁰−r,1(z−x⁰))|

R²₀e^R1(|w|+|z|)|u(r, w)|^γ|u(r, z)|^γ(|w−z|^−α+ 1) and

Q_S,δ(s, s⁰, t⁰, x⁰) =

Z (s∨s⁰−δ)⁺ (s∧s⁰−δ)⁺

dr Z

R^q

dw Z

R^q

dz|p_t⁰_−r,1(w−x⁰)p_t⁰_−r,1(z−x⁰)|

R²₀e^R1(|w|+|z|)|u(r, w)|^γ|u(r, z)|^γ(|w−z|^−α+ 1).

Now, we want to establish an upper bound for

Q^tot_δ,η0(s, t, x, s⁰, t⁰, x⁰) =Q_S,δ(s, s⁰, t⁰, x⁰) +

2

X

i,j=1

(Q^i,j_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰) +Q^i,j_X,δ,η

0(s, t, x, t, x⁰))

(9.47)

when s, t, x, s⁰, t⁰, x⁰ are subject to some restrictions. Then, (9.47) is clearly an upper bound itself for the quadratic variation of each of the three martingales in (9.46).

We first consider the cases (i, j) = (1,2),(2,1) or (2,2), so i+j ≥ 3. We start with a spatial estimate.

Lemma 9.4.5. For all K ∈N^≥K1, R >2 there exist c9.4.5(K, R), N9.4.5(K, ω) al-most 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: For ω ∈ {(t, x) ∈ Z(N, n, K, β), N ≥ N9.4.5}, we have

Q^i,j_X,δ,η

0(s, t, x, t, x⁰)≤c_9.4.52^4N9.4.5|x−x⁰|^2−η1 (9.48) for all 0≤s≤t,|x⁰| ≤K+ 1.

Proof. We will just give the proof for i= 2 without taking into accountj, i.e. the restriction onz. This suffices by symmetry.

Use the estimate on D, take ξ = 3/4 and set N9.4.5(K, ω) = N1(0,3/4, K) and w.l.o.g. δ < s. Then, in Lemma 9.4.3 for the case m = 0 we can take ε₀ = 0

122 Pathwise Uniqueness

where we used Lemma 9.3.1 in the next to last line.

Slightly more difficult are the time estimates:

Lemma 9.4.6. For all K ∈ N^≥K1, R > 2 there exist c9.4.6(K, R), N9.4.6(K, ω)

9.4 Local bounds on the difference of two solutions 123 Proof. As in the previous lemma we only consider the i= 2 case.

We use just the same proof as in Lemma 9.4.5, i.e. ε0= 0,ξ = 3/4 andN9.4.6(K, ω) = N1(0,3/4, K). The only difference is that we get time-differences instead of space differences (and when using Lemma 9.6.2 witht≤t⁰):

Q^i,j_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰)

≤C9.4.3c0(K, R) Z s−δ

0

dr(t−r)^−1−α/2exp(−η1(t⁰−r)^−2η0

256 )[1∧|t−t⁰|

t−r ]^1−η1/2. Now, we use the trivial inequality

exp(−η₁(t⁰−r)^−2η0

256 )≤exp(−η₁(t⁰−t)^−2η0

512 ) + exp(−η₁(t−r)^−2η0

512 )

and hence, have to consider two summands:

Q^i,j_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰)

=C9.4.3c0(K, R)(I1exp(−η1(t⁰−t)^−2η0

512 ) +I2).

We bound the first one by using Lemma 4.1 of [MP11]:

I₁= Z s−δ

0

dr(t−r)^−1−α/2[1∧ |t−t⁰| t−r ]^1−η1/2

≤ 2

2/α(|t−t⁰| ∧δ)^1−η1/2δ^{−α/2−1+η1/2}

≤c(|t−t⁰| ∧δ)^1/2−η1/2δ^−1−α/2 and by Lemma 9.3.1

exp(−η1(t⁰−t)^−2η0

512 )≤c(R)(512R)^R(|t−t⁰|^1/2∧1)^4γ. The second one

I2 ≤ Z s−δ

0

dr(t−r)^−1−α/2exp(−η1(t−r)^−2η0

512 )[1∧|t−t⁰| t−r ]^1−η1/2

≤ |t−t⁰|^1−η1/2sup

r≤t

[(t−r)^{−2−α/2+η1/2}exp(−η₁(t−r)^−2η0

512 )](s−δ)

≤ |t−t⁰|^1−η1/2(512R)^4RK.

SettingC9.4.6(K, R) =C9.4.3c0(K, R)(cc(R)(512R)^R+ (512R)^4RK) gives the result.

124 Pathwise Uniqueness Next we need to consider the distances for the cases i=j = 1. We start again with the distance in space:

Lemma 9.4.7. Let 0 ≤ m ≤ m¯ + 1 and assume that (P_m) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε0 ∈ (0,1), there exist c9.4.7(K, R) and N9.4.7(m, n, R, ε0, K, β)(ω) ∈ N almost surely such that for all η1 ∈ (1/R,1/2), η₀∈(0, η₁/32),δ ∈[a_n,1], N ∈N,(t, x)∈R+×R^q the following holds.

For ω∈ {(t, x)∈Z(N, n, K, β), N ≥N_9.4.6}: Q^1,1_X,δ,η

0(s, t, x, t, x⁰)≤c_9.4.7[a^−2ε_n ⁰+ 2^4N9.4.7]

×h

|x−x⁰|^2−η1(¯δ_N^{(γγm−1−α/2)∧0}+a^2βγ_n δ¯(γ−1−α/2)∧0

N )

+ (d∧√

δ)^2−η1δ^−1−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N] i

∀0≤s≤t,|x⁰| ≤K+ 1.

(9.50)

Hered¯N =|x−x⁰|∨2^−N andδ¯N =δ∨d¯²_N.Moreover,N9.4.7 is stochastically bounded uniformly in (n, β).

Proof. Let ξ = 1−(8R)⁻¹ ∈(15/16,1) and set N9.4.7 = N1(m, n, ξ, ε0, K, β). We can assume thats > δand therefore, we always haved((r, w),(t, x))∧d((r, z),(t, x))≥

√a_n in the integral. A use of Lemma 9.4.3 and the bound on |w−x|,|z−x| re-spectively, gives

Q^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,1(w−x⁰)−pt−r,1(w−x))(pt−r,1(z−x⁰)−pt−r,1(z−x)) e^4R1Ke^4γKR^2γ₀

[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(t−r)^1/2−η0 (9.51)

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

Q^1,1_X,δ,η

0(s, t, x, t, x⁰)≤C_9.4.3c₁(K) Z s−δ

0

dr(t−r)^−1−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 ].

9.4 Local bounds on the difference of two solutions 125 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], we can bound Q^1,1_X,δ,η

0(s, t, x, t, x⁰) by C9.4.3c1(K)

Z s−δ 0

dr(t−r)^−1−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 ]

≤4C9.4.3c1(K) Z s−δ

0

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

+ 4C_9.4.3c₁(K) Z s−δ

0

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

=C9.4.3c1(K)(I1+I2). (9.52)

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. (9.53) Use that to start with

I₁ ≤ Z t−δ¯N

0

dr(t−r)^{−1−α/2+γ0ξ+γ0(γm−1)}[1∧|x−x⁰|² t−r ] + (t−r)^{−1−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+γ(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) (9.54) which is true, since 1−x≤ −logx, x≥0.

In the −1-case the integral is bounded by logK + log(1/δ¯_N). Hence, using that

126 Pathwise Uniqueness 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^{(−1−α/2+γ}

0ξ+γ⁰(γm−1))∧0

N + ¯δ^{(−1−α/2+γ}

0ξ)∧0

N a^2βγ_n ).

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

Moreover, by Lemma 4.1(c) in [MP11] we bound I2 ≤ 2

2/α(δ∧ |x−x⁰|²)δ^−1−α/2d¯^2γ_N^0ξ[ ¯d^2γ_N^0(γm−1)+a^2βγ_n ]. (9.55) Therefore,

Q^1,1_X,δ,η

0(s, t, x, t, x⁰)≤c9.4.3c3(K, R)

|x−x⁰|^2−η1/2(¯δ_N^{(−1−α/2+γ0(γm+ξ−1))∧0}+ ¯δ_N^{(−1−α/2+γ0ξ∧0)}a^2βγ_n ) + (δ∧ |x−x⁰|²)δ^−1−α/2d¯^2γ_N^0ξ[ ¯d^2γ_N^0(γm−1)+a^2βγ_n

To finish the proof, replace ξ = 1−(8R)⁻¹ by 1 and γ⁰ =γ(1−2η0) by γ at the cost of multiplying by d^−η1/2 ≥δ¯_N^−η1/4, since:

ξγ⁰=≥γ(1−η₁/4), hence ξγ⁰−γ ≥ −γη₁/4≥ −η₁/4.

and

γ⁰(γm+ξ−1) =γ(1−2η0)(γm− 1

8R)≥γγm−η1

4 This holds, since η₁ ≥32η₀ gives

γ(1−2η₀)(1−(8R)⁻¹)≥γ(1− η₁

16)(γ_m− 1 8R)

≥γγ_m−γ 1

8R −γγ_mη₁ 16

≥γγ_m− 1

8R −2η1

16

≥γγ_m−η₁ 8 −η₁

8 (usingη₁ > R⁻¹)

≥γγm−η1

4.

A similar result can be obtained for the temporal distances.

9.4 Local bounds on the difference of two solutions 127 Lemma 9.4.8. Let 0 ≤ m ≤ m¯ + 1 and assume that (P_m) holds. For all K ∈ N^≥K1, R > 2, n ∈ N, β ∈ [0,1/2], ε0 ∈ (0,1), there exist c9.4.8(K, R) and N9.4.8(m, n, R, ε0, K, β)(ω)∈Nalmost surely such that

∀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 ≥N_9.4.8} Q^1,1_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰)≤c_9.4.8[a^−2ε_n ⁰+ 2^4N9.4.8]

[|t−t⁰|^1−η1/2(¯δ^(γγ_N ^{m−1−α/2)∧0}+a^2βγ_n δ¯_N^{γ−1−α/2}) + (|t−t⁰| ∧δ)^1−η1/2δ^−1−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N]]

∀0≤s≤t,|x⁰| ≤K+ 1.

Moreover, N_9.4.8 is stochastically bounded uniformly in (n, β).

Proof. Choose ξ = 1−(8R)⁻¹.

We do just the same proof as before and use

√t−r≤K^η0(t⁰−r)^1/2−η0

2^{−N γ}∨(t⁰−r)^γ0/2 ≤2^{−N γ0}+ (t⁰−t)^γ0/2+ (t−r)^γ0/2

≤2[ ¯d^γ_N⁰∨(t−r)^γ0/2] to get:

Q^1,1_{T ,δ,η}

0(s, t, x⁰, t⁰, x⁰)≤c_9.4.3c₁(K) Z s−δ

0

dr(t−r)^−1−α/2[1∧|t−t⁰| 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)[I1+I2d¯^2γ_N^0ξ( ¯d^2γ_N^0(γm−1)+a^2βγ_n )].

This means we are just in the same situation as in (9.52) and we can continue the proof as we did there.

The next lemma describes the quadratic variation of the last martingale in (9.46):

Lemma 9.4.9. 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 c9.4.9(K, R, γ) and N_9.4.9(m, n, R, ε₀, K, β)(ω)∈Nalmost surely such that

128 Pathwise Uniqueness

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

For ω∈{(t, x)∈Z(N, n, K, β), N ≥N9.4.9} QS,δ(s, s⁰, t⁰, x⁰)≤c9.4.9[a^−2ε_n ⁰ + 2^4N9.4.9]

|s−s⁰|^1−η1/2(¯δ_N^{(γγm−1−α/2)∧0}+a^2βγ_n δ¯(γ−1−α/2)∧0

N )

+ (|s⁰−s| ∧δ)^1−η1/21{δ <d¯²_N}δ^−1−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N]]

∀0≤s≤t, s⁰ ≤t⁰,|x⁰| ≤K+ 1.

Here d¯_N = (|t−t⁰|^1/2 +|x−x⁰|) ∨2^−N and δ¯_N = δ ∨d¯²_N. Moreover, N_9.4.9 is stochastically bounded uniformly in (n, β).

Proof. Choose ξ= (3/2−(2γ)⁻¹)∨(1−(4γR)⁻¹).

Choose N9.4.9 = N1(m, n, ξ, ε0, K, β). We have for r ≤ s∨s⁰ −δ : √

an < √ 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¯_N+ (t⁰−r)^1/2+|w−x⁰|.

This bound, Lemmas 9.3.2, 9.4.3 and 9.3.6 give:

Q_S,δ,η0(s, s⁰, t⁰, x⁰)≤c0(K, R)[a^−2ε_n ⁰+ 2^4N9.4.9]×

Z (s∨s⁰−δ)⁺ (s∧s⁰−δ)⁺

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

And we split this up in two integrals

≤4c0(K, R)[a^−2ε_n ⁰ + 2^4N9.4.9] {

Z (s∨s⁰−δ)⁺

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

=c(J1+J2).

For J2, the same estimate as in the proof of Lemma 5.6 in [MP11] holds:

J₂ ≤c(α)1{δ <d¯²_N}δ^−1−α/2(|s⁰−s| ∧δ)^1−η1/2[ ¯d^2γγ_N ^m+a^2βγ_n d¯^2γ_N].

For the other integral we note that since

−1−α/2 +ξγ >−1−(2γ−1) + (3 2 − 1

2γ)γ=−1 +γ

2 >−1, (9.56)

9.4 Local bounds on the difference of two solutions 129 the integral is well-defined. Set p=γ(γ_m+ξ−1)−1−α/2 orγξ−1−α/2. By (9.56), p∈(−(1 +γ)/2,1). If p⁰ = 0∧pand ε∈[0,−p⁰], then:

I(p) :=

Z (s∨s⁰−δ)⁺ (s∧s⁰−δ)⁺

1{r≤t⁰−d¯²_N}(t⁰−r)^pdr

≤

Z (s∨s⁰−δ)⁺

(s∧s⁰−δ)⁺ 1{r≤t⁰−d¯²_N}K(t⁰−r)^p0dr

≤Kmin(|s⁰−s|δ¯_N^p0,

Z |s⁰−s|

0

u^p0du) (compare (9.53))

≤ 2

1−γK|s⁰−s|^p0+1min((|s⁰−s|

δ¯N

)^−p0,1) (using (9.56))

≤ 2

1−γK|s⁰−s|^p0+1(|s⁰−s|

δ¯N

)^−p0−ε

≤C(γ, K)|s⁰−s|^1−ε¯δ^ε+p_N ⁰.

Then, there can be two cases for v =p+γ(1−ξ), such that v =γγ_m−1−^α₂ or v=γ−1−^α₂.

Case 1: v <0.

Then p⁰ = p < 0. Choose ε = γ(1−ξ) ≤ (4R)⁻¹ < η₁/4, then ε+p⁰ = v ≤ 0.

Therefore,

I(p)≤C(γ, K)|s⁰−s|^1−η1/4¯δ_N^v. Case 2: v≥0.

Thenp⁰ = (v−γ(1−ξ))∧0≥ −γ(1−ξ). Chooseε=−p⁰ ≤γ(1−ξ)< η₁/4. Thus, 1−ε≥1−^η₄¹ and

I(p)≤C(K)|s⁰−s|^1−ε¯δ_N⁰ < C₁(γ, K)|s⁰−s|^1−η1/4. In either case I(p)≤C₁(K)|s⁰−s|^1−η1/2δ¯^v∧0_N . Hence,

J1≤c4(K)|s⁰−s|^1−η1/2[¯δ_N^{(γγm−1−α/2)∧0}+a^2βγ_n δ¯^{γ−1−α/2}_N ]. (9.57) And this completes the proof.

Notation: d((s, t, x),˜ (s⁰, t⁰, x⁰)) := p

|s−s⁰|+p

|t−t⁰|+|x −x⁰|, s, t, s⁰, t⁰ ∈ R+, x, x⁰ ∈R^q.

As a corollary of all the previous calculations, we get a bound on Q^tot_δ,η

0 as defined in (9.47).

Corollary 9.4.10. 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_9.4.10(K, R),

130 Pathwise Uniqueness N_9.4.10(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 the following holds.

Forω ∈ {(t, x)∈Z(N, n, K, β), N ≥N9.4.10} Q^tot_δ,η0(s, t, x, s⁰, t⁰, x⁰)≤c_9.4.10(a^−2ε_n ⁰ + 2^4N9.4.10) ˜d^2−η1

δ^−1−α/2d¯^γγ_N^m+δ^−1−α/2a^2βγ_n d¯^2γ_N + ¯δ_N^{(γγm−1−α/2)∧0}+a^2βγ_n δ¯^{γ−1−α/2}_N

∀0≤s≤t≤t⁰≤T_K,|x⁰| ≤K+ 1.

(9.58) Here d˜ = ˜d((s, t, x),(s⁰, t⁰, x⁰)), d¯_N = d((t, x),(t⁰, x⁰))∨2^−N and δ¯_N = δ ∨d¯_N. Moreover, N_9.4.10 is stochastically bounded uniformly in(n, β).

Proof. The proof simply consists of putting together the last lemmas. LetN_9.4.10= N9.4.5∨N9.4.6∨N9.4.7∨N9.4.8∨N9.4.9, which is then clearly uniformly bounded in (n, β) and c₀=c_9.4.5∨c_9.4.6∨c_9.4.7∨c_9.4.8∨c_9.4.9.Then,

Q^tot_δ,η0(s, t, x, s⁰, t⁰, x⁰)≤5c₀(a^−2ε_n ⁰ + 2^4N9.4.10){d˜^2−η1[¯δ^(γγ_N ^{m−1−α/2)∧0}+a^2βγ_n δ¯(γ−1−α/2)∧0

N ]

+ ˜d^2−η_N ¹δ^−1−α/2[ ¯d^2γγ_N ^m+a^2βγ_n d^2γ_N + (|t−t⁰|^1/2∧1)^4γ]}

≤c_9.4.10(a^−2ε_n ⁰ + 2^4N9.4.10) ˜d^2−η1[δ^−1−α/2d¯^γγ_N^m+δ^−1−α/2a^2βγ_n d¯^2γ_N + ¯δ^(γγ_N ^{m−1−α/2)∧0}+a^2βγ_n δ¯_N^{γ−1−α/2}].

Notation: In the following, the α of [MP11] is replaced byλ∈[0,1]. Introduce

∆¯_u⁰

1(m, n, λ, ε₀,2^−N) =a^−ε_n ⁰[a−λ/2(1+α/2)

n 2^{−N γγm}+ (a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0} +a−λ/2(1+α/2)+βγ

n (a^λ/2_n ∨2^−N)^γ].

(9.59) Proposition 9.4.11. Let 0≤m ≤m¯ + 1 and assume that (Pm) holds. Then, for all n ∈ N, η₁ ∈ (0,1/2], ε₀ ∈ (0,1), K ∈ N^≥K1, λ ∈ [0,1], β ∈ [0,1/2] there is an N_9.4.11=N_9.4.11(m, n, η₁, ε₀, K, λ, β)(ω)∈N^≥2 almost surely such that for allN ≥ N9.4.11, (t, x) ∈ Z(N, n, K, β), s≤t ≤t⁰, s⁰ ≤t⁰ ≤TK and d((s, t, x)(s⁰, t⁰, x⁰))<

2^−N, it holds that

|F_aλ

n,l(s, t, x)−F_a^λ

n,l(s⁰, t⁰, x⁰)| ≤2⁻⁸⁶q⁻⁴d˜^1−η1∆¯_u⁰

1(m, n, λ, ε0,2^−N) , l= 1, . . . , q.

(9.60) Moreover, N_9.4.11 is stochastically bounded uniformly in(n, λ, β).

9.4 Local bounds on the difference of two solutions 131 Proof. We do the proof for l = 1 only, see Remark 9.4.4. Let R = 33η₁⁻¹, η₀ ∈ (R⁻¹, η1/32) and consider the cases≤tin the beginning only. Set

d=d((t, x),(t⁰, x⁰)), d˜=p

|s⁰−s|+d, d¯_N =d∨2^−N,

¯δ_n,N =a^λ_n∨d¯²_N.

By Corollary 9.4.10 for (t, x)∈Z(N, n, K, β), N ≥N9.4.10 it holds that:

Q^tot_aλ

n,η0(s, t, x, s⁰, t⁰, x⁰)^1/2 ≤c_9.61(K, η₁)(a^−ε_n ⁰+ 2^2N9.4.10) ˜d^1−η1/2 [(a^λ_n)−1/2(1+α/2)[ ¯d^γγ_N^m+a^βγ_n d¯^γ_N]

+ ¯δ^(γγm−1−α/2)/2∧0

n,N +a^βγ_n δ¯(γ−1−α/2)/2

n,N ]

∀s≤t≤t⁰, s⁰≤t⁰≤TK,|x⁰| ≤K+ 2.

(9.61)

Therefore, define

∆(m, n,d¯N) = 2⁻⁹⁶a^−ε_n ⁰{a−λ/2(1+α/2)

n [ ¯d^γγ_N^m+a^βγ_n d¯^γ_N] + (

qδ¯n,N)^{(γγm−1−α/2)∧0}+a^βγ_n (

qδ¯n,N)^{γ−1−α/2}}(q⁵4^c(q))⁻¹, which satisfies

∆(m, n,2^−N+1)≤(2^γγm∨2^γ∨2⁰∨2^{γ−1−α/2})∆(m, n,2^−N)≤4∆(m, n,2^−N).

ChooseN3= ³³_η

1[N9.4.10+N4(K, η1)] + +_η⁴

1(8 + 10 logq), whereN4is chosen in such a way that

q⁷4^c(q)c₁(K, η₁)[a^−ε_n ⁰ + 2^2N9.4.10]2^−η1N3/4≤c_9.61(K, η₁)[a^−ε_n ⁰ + 2^2N9.4.10]2^{−8N9.4.10−8N4}

≤a^−ε_n ⁰2⁻¹⁰⁰,

i.e. N₄ = N₄(a_n, ε₀, N_9.4.10, c_9.61) and hence, N₃ = N₃(n, ε₀, N_9.4.10, K, η₁), which is stochastically uniformly bounded in (n, λ, β).

Let N⁰ ∈ N be such that ˜d ≤ 2^−N0, which implies ˜d^1−η1/2 ≤ 2^−N0η1/4d˜^1−3η1/4. Then, it is true that on the event

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

Q^tot_aλ

n,η0(s, t, x, s⁰, t⁰, x⁰)^1/2 ≤c1(K, η1)(a^−ε_n ⁰ + 2^2N9.4.10)

2^−N0η1/4d˜^1−3η1/42¹⁰⁰a^ε_n⁰∆(m, n,d¯_N)(q⁷4^c(q))

≤d˜^1−3η1/4 1

16∆(m, n,d¯_N),

132 Pathwise Uniqueness whenever s ≤ t ≤ t⁰, s⁰ ≤ t⁰ ≤ T_k,|x⁰| ≤ K+ 2. Remembering the decomposition of Fδ,1 in (9.46) into the sum of 3 martingales and applying the Dubins-Schwarz-Theorem (Dubins-Schwarz-Theorem 3.2.8) we can write as long as s≤t≤t⁰, s⁰ ≤t⁰ and ˜d≤2^−N:

P[|F_aλ

n,1(s, t, x)−F_aλ

n,1(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≤T_K]

≤3P[ sup

u≤d˜^2−3η1/2(∆(m,n,d¯N)/16)²

|B(u)| ≥d˜^1−η1∆(m, n,d¯_N)/3]

≤3P[sup

u≤1

|B(u)| ≥d˜^−η1/4]

≤c Z ∞

d˜^−η1/4

exp(−y²/2)dy

≤c0exp(−d˜^−η1/2/2), (9.62)

where we used the Reflection Principle in the next to 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 = 0, 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) =E∩ {0} × {0≤t≤TK} ×R^q, s= 1, α1 = 1,∆1(ˆn,2^−N) = ∆(m, n,2^−N), k1 = 4, c(α1)≤4, η=η1, Y_nˆ(y) =F_aλ

n,1(s, t, x) withy = (s, t, x), N₀(η, K,n) =ˆ N₃(n, ε₀, N_9.4.10, K, η₁).

Note that the N₀ is uniformly bounded in ˆn. Then, we obtain forN ≥N_9.4.11:=

N1(η, K,n) and (t, x)ˆ ∈Z(N, n, K, β)(ω), ˜d=d((s, t, x),(s⁰, t⁰, x⁰))≤2^−N,s≤ K and s≤t, s⁰≤t⁰ ≤T_K:

|F_aλ

n,1(s, t, x)−F_aλ

n,1(s⁰, t⁰, x⁰)| ≤32(q+ 2)4^c(q)+1∆(m, n,2^−N) ˜d^1−η1. (9.63) Thus,

|F_aλ

n,1(s, t, x)−F_aλ

n,1(s⁰, t⁰, x⁰)| ≤2⁻⁸⁸q⁻⁴d˜^1−η1∆¯_u⁰

1(m, n, λ, ε₀,2^−N). (9.64) However, if t⁰ ≤ t, then (t⁰, x⁰) ∈ Z(N −1, n, K+ 1, β) interchange (s, t, x) with (s⁰, t⁰, x⁰) and we give the same estimate as (9.61) to bound

Q^tot_aλ

n(s⁰, t⁰, x⁰, s, t, x)≤4RHS of (9.61),

as always “RHS” stands for “the right hand side of”. Proceeding as in the case t≤t⁰, we end up with (9.64) replaced by:

|F_aλ

n,1(s⁰, t⁰, x⁰)−F_aλ

n,1(s, t, x)| ≤2⁻⁸⁶q⁻⁴d˜^1−η1∆¯_u⁰

1(m, n, λ, ε₀,2^−N).

9.4 Local bounds on the difference of two solutions 133 This does the proof for the first coordinate and clearly the constant c_9.4.11 and N9.4.11 can be chosen such that the result holds uniformly for all dimensions 1 ≤ l≤q.

So, applying Proposition 9.4.11 for any dimension l= 1, . . . , q, we get for the gra-dient ∇u_1,δ(t, x) = (F_δ,l(t, t, x))1≤l≤q (remember the Stochastic Fubini Theorem):

Corollary 9.4.12. Let 0≤m≤m¯ + 1 and assume that (P_m) holds. Let n, η₁, ε₀, K, λ and β be as in Proposition 9.4.11. For allN ≥N_9.4.11,(t, x)∈Z(N, n, K, β), x⁰∈R^q and t⁰ ≤TK:

d((t, x),(t⁰, x⁰))≤2^−N implies that

|∇u_1,aλ

n(t, x)− ∇u_1,aλ

n(t⁰, x⁰)| ≤2⁻⁸⁵q⁻²d((t, x),(t⁰, x⁰))^1−η1∆¯_u⁰

1(m, n, λ, ε₀,2^−N).

This result gives us something like a H¨older regularity of the gradient∇u_1,δ with δ =a^λ_n. This will be helpful later.

Recalling the definition of Jn,i, however, we just “know” the range of the gradients of u_1,δ for δ =a_n. But it will be helpful to find a result relating this range to the gradients of u_1,δ for δ = a^λ_n. The definition of F_δ,l allows us to relate these two gradients, since forδ ≥an and s=t−δ+an:

∂_xlu_1,δ(t, x) =∂_xlP_δ(u_(t−δ)+)(x) =∂_xlPt−s+a_n(u_(s−an)+)(x)

=−F_an,l(s, t, x) (9.65)

=−F_an,l(t−δ+a_n, t, x).

Note the last equality holds for anyt, δ, an≥0, where they are trivial if t−δ≤0.

Again we will give the proof for l= 1 only, see Remark 9.4.4.

Lemma 9.4.13. 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_9.4.13(K), N9.4.13(m, n, R, ε0, K, β)(ω)∈N almost surely such that ∀η₁ ∈ (1/R,1/2), N ∈N, (t, x)∈R+×R^q the following holds.

For ω∈{(t, x)∈Z(N, n, K, β), N ≥N_9.4.13} Q_S,an(s, t, t, x)≤c_9.4.13(K)[a^−2ε_n ⁰+ 2^4N9.4.13]

h|t−s|^1−η1/4(((t−s)∨a_n)^{(γγm−1−α/2)}+a^2βγ_n ((t−s)∨a_n)^{(γ−1−α/2)}) +1{a_n<2^−2N}((t−s)∧a_n)a^−1−α/2_n 2^{N η1/2}[2^{−2N γγm}+a^2βγ_n 2^{−2N γ}]i

∀0≤s≤t.

(9.66)

134 Pathwise Uniqueness Use this in Lemma 9.4.3 to obtain

Q_S,an(s, t, t, x) =

By Lemma 9.3.6, we can evaluate the spatial integrals Q_S,an(s, t, t, x)≤C_9.4.3c(K)

As in Lemma 9.4.9 there are two integrals to consider now. For J₂ we do J2= 2^{−2N γξ}[2^{−2N γ(γm−1)}+a^2βγ_n ]

9.4 Local bounds on the difference of two solutions 135 And for J₁ we first refer to the estimate (9.56) showing that all exponents in the integrands stay above −1. Before we start, note that for p∈(−(1 +γ)/2,1).

Moreover, N9.4.14 is stochastically bounded uniformly in (n, β).

Proof. Let R= 2/η1. We give a proof for l= 1 only. Then, Lemma 9.4.13 implies

136 Pathwise Uniqueness Now, we proceed similarly as in the proof of Proposition 9.4.11. Choose N₀ = N0(K) such that

c₁(K)[a^−ε_n ⁰ + 2^2N9.4.13]2^{−2N9.4.13−2N0} ≤2⁻¹⁰⁰a^−ε_n ⁰ and then observe that for N₂(m, n, η₁, ε₀, K, β) = _η⁸

1[N_9.4.13+N₀(K) + 3 log(q+ 2)(log 2)⁻¹]:

(q+ 2)³c₁(K, R₀)[a^−ε_n ⁰ + 2^2N9.4.13]2^−N2η1/4≤c₁(K, R₀)[a^−ε_n ⁰ + 2^2N9.4.13]2^{−2N9.4.13−2N0}

≤2⁻¹⁰⁰a^−ε_n ⁰. In the case √

t−s≤2^−N2, we clearly have (t−s)^η1/8 ≤2^−η1N2/4. For N ≥N2 it follows that:

Q_S,an(s, t, t, x)^1/2≤(q+ 2)⁻³2⁻¹⁰⁰a^−ε_n ⁰{√

t−s^1−η1/2[(√

t−s∨√

a_n)^{γγm−1−α/2} +a^βγ_n (√

t−s∨√

an)^{γ−1−α/2}] +a^−α/4_n 2^{N η1/2}[2^{−N γγm}+a^βγ_n 2^{−N γ}]}

=:√

t−s^1−η1/2∆1(m, n,√

t−s∨√

an) + 2^{N η1/2}∆2(m, n,2^−N).

(9.67) With the Dubins-Schwarz Theorem (Theorem 3.2.8) we obtain

P[|F_an,1(s, t, x)−F_an,1(t, t, x)| ≥√

t−s^1−η1∆₁(m, n,√

t−s∨√ a_n) + 2^{N η1}∆₂(m, n,2^−N),(t, x)∈Z(N, n, K, β), N ≥N₂,√

t−s≤2^−N2]

≤P[sup

u≤1

|B(u)| ≥(t−s)^−η1/4∧2^{N η1/2}]1{t−s≤1}

≤c₀exp(−1

2((t−s)^−η1/2∧2^{N η1})). (9.68) Let

`_N = 22(N+c(q)+1)N^−8/η1, where (9.69) c(q) = (log 2)⁻¹log(q+ 2)≤q+ 1. (9.70) If N ≥N₃:= 2^N2 that easily leads to

N ≥2^N2 ⇒N^−4/η1 ≤2^−4N2/η1 ≤2^−8N2 ≤2^−N2−1 and (9.71) p`N2^−N−c(q)= 2^N+c(q)+1N^−4/η12^−N−c(q)≤2N^−4/η1(≤2^−N2). (9.72) We are going to consider

M_N = max{

|F_an,1(i2^−2(N+c(q)), j2^−2(N+c(q)), k2^−(N+c(q)))−F_an,1(j2^−2(N+c(q)), j2^−2(N+c(q)), k2^−(N+c(q)))|

(√

j−i2^−(N+c(q)))^1−η1∆₁(m, n,√

j−i2^−(N+c(q))∨√

a_n) + 2^{N η1}∆₂(m, n,2^−N) , 0≤j−i≤`_N,(j2^−2(N+c(q)), k2^−(N+c(q)))∈Z(N, n, K, β), i, j∈Z+, k∈Z^q}

9.4 Local bounds on the difference of two solutions 137 for which it holds by decomposing into the different possible choices for i, j and k and (9.68):

P[M_N ≥1, N ≥N₃]

≤(K+ 1)²2^4(N+c(q))(2K+ 1)^q2^q(N+c(q)) c0exp(−1

2(`N2<sup>−2(N+c(q))</sup>)<sup>−η1/2</sup>∧2<sup>N η1</sup>)1{`_N ≥1}

=c₂(q, K)2^(4+q)Nexp(−1 2(p

`<sub>N</sub>2<sup>−(N+c(q))</sup>)<sup>−η1</sup> ∧(2<sup>(N+c(q))η1</sup>2<sup>−c(q)η1</sup>))1{`_N ≥1}

≤c₂(q, K)2^(4+q)Nexp(−1

22^−c(q)η1(p

`<sub>N</sub>2<sup>−(N+c(q))</sup>)<sup>−η1</sup>∧(2<sup>−(N+c(q))</sup>)<sup>−η1</sup>)1{`_N ≥1}

≤c2(q, K)2^(4+q)Nexp(−2^{−1−c(q)η1}(p

`_N2^−(N+c(q)))^−η1)

≤c2(q, K)2^(4+q)Nexp(−2^−2−q/2N⁴) (by (9.70),(9.72) and η1 ≤1/2 ).

Set

AN ={M_N ≥1, N ≥N3} and

N₄ = min{N :ω ∈

∞

\

N⁰=N

A^c_N0}.

Then, we have forN ≥N5 = (log 2)^1/32^1+q/6(q+ 4)^1/3 that P[N₄ ≥N] =P[

∞

[

N⁰=N

A_N⁰]≤c₂(q, K)

∞

X

N⁰=N

2^(4+q)N0exp(−2^−2−q/2N⁰⁴)

≤c3(q, K) exp(−2^−3−q/2N⁴).

Next, choose N₆ such that 2^1−N ≤N^−4/η1 for all N ≥N₆. Then set N9.4.14:= (N9.4.11∨N3∨N4∨N5∨N6) +c(q), which is uniformly bounded in (n, β) as all of the components are.

From now on, consider the event

N ≥N_9.4.14,(t, x)∈Z(N, n, K, β), s≤t,√

t−s≤N^−4/η1. (9.73) Then, there are two cases:

Case 1: t−s≥2^2(1−N)

AsN ≥N9.4.14we have N−c(q)≥N4, i.e. ω∈A^c_N−c(q). Additionally,N−c(q)≥ N₃ and therefore

M_N−c(q)<1. (9.74)

138 Pathwise Uniqueness To use this result we need to introduce discrete versions of s, tand x. We set s<sub>`</sub> = b2<sup>2`</sup>sc2<sup>−2`</sup>, t` = b2^{2`</sup>tc2<sup>−2`}, x` = Pq

k=1sgn(xk)b2^{`</sup>|x<sub>k</sub>|c2<sup>−`}ek, where (ek) are the standard unit vectors inR^q. Thend((tN, xN),(ˆt0,xˆ0))≤2^−N+d((tN, xN),(t, x))≤ 2^{−(N−c(q))}. Hence due to the choice of c(q) in (9.70):

(tN, xN)∈Z(N −c(q), n, K, β). (9.75) Now, split the expressions

|F_an,1(s, t, x)−F_an,1(t, t, x)|

≤[|F_an,1(s, t, x)−Fan,1(s_N, t_N, x_N)|+|F_an,1(t_N, t_N, x_N)−Fan,1(t, t, x)|]

+|F_an,1(sN, tN, xN)−Fan,1(tN, tN, xN)|

=T1+T2.

Since (t, x)∈Z(N, n, K, β), ˜d((t, t, x),(tN, tN, xN))∨d((s, t, x),˜ (sN, tN, xN))≤(2+

q)2^−N = 2^{−(N−c(q))} and as N −c(q) ≥N_9.4.11 we can apply Proposition 9.4.11 for both terms in T₁ and obtain

T₁ ≤2⁻⁸⁵2^{−(N−c(q))(1−η1)}∆¯_u⁰

1(m, n,1, ε₀,2^{−(N−c(q))})q⁻⁴

≤2⁻⁸⁵2^c(q)4^c(q)q⁻⁴2^{−N(1−η1)}∆¯_u⁰

1(m, n,1, ε₀,2^−N)

≤2⁻⁸⁵q⁻⁴q³3³2^{−N(1−η1)}∆¯_u⁰

1(m, n,1, ε0,2^−N)

≤2⁻⁸⁰q⁻¹2^{−N(1−η1)}∆¯_u⁰

1(m, n,1, ε0,2^−N), where we used ¯∆_u⁰

1(. . . ,2^−N+c(q)) ≤ 4^c(q)∆¯_u⁰

1(. . . ,2^−N), which can be seen from (9.59). ForT₂ we use N ≥N₆:

q

b2^2Ntc − b2^2Nsc2^−N ≤√

t_N −s_N ≤2·2^−N +√ t−s

≤2N^−4/η1

= 2^N+1N^−4/η12^−N

≤2^N+1(N−c(q))^−4/η12^−N

=q

`<sub>N−c(q)</sub>2<sup>−N</sup>(recall (9.69)), henceb2<sup>2N</sup>tc − b2<sup>2N</sup>sc ≤`N−c(q).

Therefore, by (9.74), (9.75) and the definition of M_N−c(q): T2 ≤M_N−c(q)[(√

tN −sN)^1−η1∆1(m, n,√

tN −sN ∨√ an) + 2^{(N−c(q))η1}∆2(m, n,2^{−(N−c(q))})]

≤√

t_N −s_N^1−η1∆₁(m, n,√

t_N−s_N∨√

a_n) + 2^{(N−c(q))η1}∆₂(m, n,2^{−(N−c(q))}).

9.4 Local bounds on the difference of two solutions 139 Since t−s≥2^2−2N, we can bound t−sby

(t−t_N) + (t_N −s_N) + (s_N −s)≤2^1−2N + (t_N −s_N)≤ 1

2(t−s) + (t_N −s_N), (9.76) and hence,

1

2(t−s)≤(t_N−s_N) (9.77)

and similarly,

(t_N −s_N)≤t−s+ 2^−2N+1≤2(t−s). (9.78) Therefore, we get by adding up T₁ and T₂ and writing out the definition of the several ∆’s, defined in (9.59) and (9.67):

|F_an,1(s, t, x)−Fan,1(t, t, x)| ≤2⁻⁸⁰q⁻¹2^{−N(1−η1)}∆¯_u⁰

1(m, n,1, ε0,2^−N) + 2⁻⁹⁹a^−ε_n ⁰√

t−s^1−η1[(√

t−s∨√

an)^{γγm−1−α/2}+a^βγ_n (√

t−s∨√

an)^{γ−1−α/2}](q+ 2)⁻³ + 2⁻¹⁰⁰a^−ε_n ⁰a^−α/4_n 2^{(N−c(q))η1}[2−(N−c(q))γγm+a^βγ_n 2^{−(N−c(q))γ}](q+ 2)⁻³

≤2⁻⁸⁰q⁻¹2^{−N(1−η1)}a^−ε_n ⁰[a−1/2(1+α/2)

n 2^{−N γγm}+ (a^1/2_n ∨2^−N)^{(γγm−1−α/2)∧0} +a−1/2(1+α/2)+βγ

n (a^1/2_n ∨2^−N)^γ] + 2⁻⁹⁹(q+ 2)⁻³a^−ε_n ⁰√

t−s^1−η1[(√

t−s∨√

an)^{γγm−1−α/2}+a^βγ_n (√

t−s∨√

an)^{γ−1−α/2}] + 2⁻¹⁰⁰(q+ 2)⁻¹a^−ε_n ⁰a^−α/4_n 2^{N η1}[2^{−N γγm}+a^βγ_n 2^{−N γ}].

We can put together the first and third term in the first bracket, and the last bracket using

2^{−N(1−η1)}a^{−(1+α/2)/2}_n + 2^{N η1}a^−α/4_n =a^−α/4_n 2^{N η1}[2^−N

√an

+ 1].

This gives,

|F_an,1(s, t, x)−F_an,1(t, t, x)| (9.79)

≤2⁻⁷⁸q⁻¹a^−ε_n ⁰{a⁻

α

n42^{N η1}[2^−N

√an

+ 1][2^{−N γγm}+a^βγ_n (√

a_n∨2^−N)^γ] + (√

a_n∨2^−N)^{(γγm−1−α/2)∧0}2^{−N(1−η1)}} + 2⁻⁹⁹q⁻¹a^−ε_n ⁰√

t−s^1−η1[(√

t−s∨√

a_n)^{(γγm−1−α/2)}+a^βγ_n (√

t−s∨√

a_n)^{γ−1−α/2}].

140 Pathwise Uniqueness Case 2. t−s≤2^2(1−N)

Our assumptions (9.73) imply that (t, x)∈Z(N −1, n, K, β), s≤t≤K, N−1≥ N9.4.11 and (in this case)

d((s, t, x),˜ (t, t, x))≤2^−(N−1). Now, just use Proposition 9.4.11 and obtain

|F_an,1(s, t, x)−F_an,1(t, t, x)| ≤2⁻⁸⁶q⁻⁴(√

t−s)^1−η1∆¯_u⁰

1(m, n,1, ε₀,2^−(N−1))

≤2⁻⁸³q⁻¹2^{−N(1−η1)}a^−ε_n ⁰ h

a^−α/4_n 2^{−N γγm}+ (a^1/2_n ∨2^−N)^{(γγm−1−α/2)∧0}+a^βγ_n a^{−(1+α/2)/2}_n (2^−N∨a^1/2_n )^γi

≤2⁻⁸³q⁻¹a^−ε_n ⁰{a^−α/4_n 2^{N η1}(2^−N

√a_n)(2^{−N γγm}+a^βγ_n (2^−N ∨a^1/2_n )^γ) + 2^{−N(1−η1)}(a^1/2_n ∨2^−N)^{(γγm−1−α/2)∧0}}.

And this term is bounded by the first terms on the right-hand side of (9.79).

Remark 9.4.15. We could have tried to obtain a more general version of Lemma 9.8.1 to be applied in the previous proof, but note that we used Proposition 9.4.11 just right after (9.75). So there needed to be some a-priori regularity assumption in Lemma 9.8.1 which made it quite more complicated.

There is a similar result for G_δ in a special case, which will be needed later:

Proposition 9.4.16. Let 0 ≤ m ≤ m¯ + 1 and assume that (Pm) holds. For any n ∈ N, η₁ ∈ (0,1/2), ε₀ ∈ (0,1), K ∈ N^≥K1, λ ∈ [0,1] and β ∈ [0,1/2], there is an N_9.4.16 = N_9.4.16(m, n, η₁, ε₀, K, λ, β) ∈ N a.s. such that for all N ≥ N_9.4.16, (t, x)∈Z(N, n, K, β), s≤t and √

t−s≤2^−N,

|G_aλ

n(s, t, x)−G_aλ

n(t, t, x)| ≤2⁻⁹⁵a^−ε_n ⁰(t−s)^(1−η1)/2a^−λα/4_n [(a^λ/2_n ∨2^−N)^γγm+a^βγ_n (√

an∨2^−N)^γ].

Moreover, N9.4.16 is stochastically bounded uniformly in(n, λ, β).

The proof of this Proposition is put in Chapter 9.6. Most of the calculations are similar and even easier than the ones before: we have to deal directly with the heat kernels instead of their derivatives. Recall that the goal of this chapter was to do the induction step of (Pm), i.e. to get good H¨older-estimates on u = u1,δ +u2,δ. We require some notation to give a result for u_1,δ:

Notation:

∆¯_u1(m, n, λ, ε₀,2^−N) =a^−ε0−λ(1+α/2)/2

n [aβ+λ(1+α/2)/2

n +1(m≥m)(a¯ ^λ/2_n ∨2^−N) + (a^λ/2_n ∨2^−N)^γγm+1+a^βγ_n (a^λ/2_n ∨2^−N)^γ+1]

9.4 Local bounds on the difference of two solutions 141 and

N_9.4.17⁰ (η) = min{N ∈N: 2^1−N ≤N^−8/η} for a constant η >0.

Proposition 9.4.17. Let0≤m≤m¯+ 1and assume that(P_m)holds. Then for all n∈N, η1 ∈(0,1/2∧(2−α)/2), ε₀, ε1 ∈(0,1), K ∈N^≥K1,λ∈[ε1,1−ε₁],β ∈[0,1/2]

there is an N_9.4.17 =N_9.4.17(m, n, η₁, ε₀, K, λ, β)(ω) ∈N^≥2 almost surely such that

∀N ≥N_9.4.17 and n, λ that satisfy

an≤2^{−2(N9.4.14+1)}∧2^{−2(N9.4.170 (η1ε1)+1)} and λ≥ε1, (9.80) and (t, x)∈Z(N, n, K, β), t⁰≤T_K, x⁰ ∈R^q s.t.d((t, x),(t⁰, x⁰))≤2^−N it holds that

|u_1,aλ

n(t, x)−u_1,aλ

n(t⁰, x⁰)| ≤2⁻⁹⁰d((t, x),(t⁰, x⁰))^1−η1∆¯_u1(m, n, λ, ε₀,2^−N).

Moreover, N_9.4.17 is stochastically bounded uniformly in (n, λ, β).

Proof. Let

N_9.4.17⁰⁰ = 2N_9.4.11(m, n, η₁/2, ε₀, K+ 1, λ, β)∨N_9.4.16(m, n, η₁/2, ε₀, K+ 1, λ, β) + 1, which is uniformly bounded in (n, λ, β). Assume

N ≥N_9.4.17⁰⁰ ,(t, x)∈Z(N, n, K, β), t⁰ ≤TK, d((t, x),(t⁰, x⁰))≤2^−N

and that (9.80) holds. Then (t⁰, x⁰) ∈ Z(N −1, n, K + 1, β) and without loss of generality t⁰ ≤ t (otherwise, we interchange the role of t⁰ and t in the following).

We have by definition G_aλ

n(t⁰, t, x) =P_t−t⁰_+aλ

n(u((t⁰−a^λ_n)⁺,·))(x) =Pt−t⁰(u_1,aλ

n(t⁰,·))(x). (9.81) And therefore, we can write

|u_1,aλ

n(t⁰, x⁰)−u_1,a^λ

n(t, x)| ≤ |u_1,aλ

n(t⁰, x⁰)−u_1,a^λ

n(t⁰, x)|

+|u_1,aλ

n(t⁰, x)−Pt−t⁰(u_1,aλ

n(t⁰,·))(x)|

+|G_aλ

n(t⁰, t, x)−G_aλ

n(t, t, x)|

≡T₁+T₂+T₃. (9.82)

We start with T₁. If y ∈ [[x, x⁰]] = ×^q_i=1[x_i∧x⁰_i, x_i∨x⁰_i], then by Corollary 9.4.12 withη1/2 instead of η1 for 1≤l≤q and (9.65):

|∇u_1,aλ

n(t⁰, y)| ≤ |∇u_1,aλ

n(t⁰, y)− ∇u_1,aλ

n(t, x)|+|∇u_1,aλ

n(t, x)− ∇u_1,aλ

n(ˆt₀,xˆ₀)|

+|∇u_1,aλ

n(ˆt₀,xˆ₀)− ∇u_1,an(ˆt₀,xˆ₀)|+a^β_n.

≤2⁻⁸⁴q⁻²2^{−N(1−η1/2)}∆¯_u⁰

1(m, n, λ, ε0,2^−N) +q max

l=1,...,q(|F_an,l(ˆt₀−a^λ_n+a_n,tˆ₀,xˆ₀)−F_an,l(ˆt₀,ˆt₀,xˆ₀)|) (9.83) +a^β_n.

142 Pathwise Uniqueness

The second term can now be bounded by the third one, since √

a_n≤a^λ/2_n : Now we put together the estimate for (9.83):

|∇u_1,aλ

9.4 Local bounds on the difference of two solutions 143 With this estimate and the Mean Value Theorem in R^q we obtain

T₁ ≤ |x−x⁰|(a^β_n+ 2⁻⁷³∆˜_u1(m, n, λ, ε₀, η₁, a^λ/2_n ∨2^−N)). (9.87) Additionally, Proposition 9.4.16 gives

T₃ ≤2⁻⁹²|t−t⁰|^12(1−η1)a^−ε0−

λα

n 4 [(a^λ/2_n ∨2^−N)^γγm+ (a^λ/2_n ∨2^−N)^γa^βγ_n ]. (9.88) For T2 = |u_1,aλ

n(t⁰, x)−Pt−t⁰(u_1,a^λ

n(t⁰,·))(x)| a bit more work is required. Let (B(s) : s ≥ 0) be a q-dimensional Brownian Motion started in x ∈ R^q under the law Px.

First, assume that

|B(t−t⁰)−x| ≤2^−32N9.4.11. Then,

√

t−t⁰ ≤D:=d((t⁰, B(t−t⁰)),(t, x))≤√

t−t⁰+ 2^−32N9.4.11

≤2^−N + 2^−32N9.4.11

≤2^−2N9.4.11+ 2^−32N9.4.11 ≤2^−N9.4.11.

(9.89)

Next, choose N⁰∈ {N_9.4.11, . . . , N}in such a way that (a) ifD≤2^−N thenN⁰ :=N

(b) ifD >2^−N then 2^−N0−1 < D≤2^−N0. In Case (b) we have

2^−N0−1≤2^−N +|B(t−t⁰)−x|, hence 2^−N0 ≤2^1−N + 2|B(t−t⁰)−x|

and trivially in Case (a):

2^−N0 = 2^−N ≤2^1−N + 2|B(t−t⁰)−x|.

If y∈R^q such that |y−x| ≤ |B(t−t⁰)−x|, then for (t, x)∈Z(N⁰, n, K, β) do the estimate of (9.86) to obtain:

|∇u_1,aλ

n(t⁰, y)| ≤2⁻⁷³a^−ε_n ⁰(a^λ/2_n ∨2^−N0)¹⁻

η1 2

[(a^λ/2_n ∨2^−N0)^{(γγm−1−α/2)∧0}+a−λ(1+α/2)/2

n (a^λ/2_n ∨2^−N0)^γγm +a−λ(1+α/2)/2

n a^βγ_n (a^λ/2_n ∨2^−N0)^γ] +a^β_n

≤2⁻⁷⁰a^−ε_n ⁰(a^λ/2_n + 2^−N +|B(t−t⁰)−x|)^1−η21 [(a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}

+a−λ(1+α/2)/2

n (a^λ/2_n + 2^−N +|B(t−t⁰)−x|)^γγm +a−λ(1+α/2)/2

n a^βγ_n (a^λ/2_n + 2^−N +|B(t−t⁰)−x|)^γ] +a^β_n.

144 Pathwise Uniqueness If we use the multidimensional Mean Value Theorem we can write where expectation is taken overB alone:

Ex[1{|B(t−t⁰)−x| ≤2^−32N9.4.11}|u_1,aλ

n(t⁰, x)−u_1,a^λ

n(t⁰, B(t−t⁰))|]

≤E0[|B(t−t⁰)|{a^−ε_n ⁰(a^λ/2_n + 2^−N +|B(t−t⁰)|)^1−η21 [(a^λ/2_n ∨2^−N|)^{(γγm−1−α/2)∧0}+a−λ(1+α/2)/2

n (a^λ/2_n + 2^−N +|B(t−t⁰)|)^γγm +a−λ(1+α/2)/2

n a^βγ_n (a^λ/2_n + 2^−N +|B(t−t⁰)|)^γ] +a^β_n}]

≤c₁√

t−t⁰{a^−ε_n ⁰(a^λ/2_n + 2^−N+√

t−t⁰)^1−η21 [(a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}+a−λ(1+α/2)/2

n (a^λ/2_n + 2^−N +√

t−t⁰)^γγm +a−λ(1+α/2)/2

n a^βγ_n (a^λ/2_n + 2^−N +√

t−t⁰)^γ] +a^β_n} using |t−t⁰| ≤2^−2N bound this by

≤c₂√

t−t⁰{a^−ε_n ⁰(a^λ/2_n + 2^−N)^1−η21

[(a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}+a−λ(1+α/2)/2

n (a^λ/2_n + 2^−N)^γγm +a−λ(1+α/2)/2

n a^βγ_n (a^λ/2_n + 2^−N)^γ] +a^β_n}

≤c₃√

t−t⁰{a^β_n+a^−ε_n ⁰(a^λ/2_n ∨2^−N)^1−η21 [a−λ(1+α/2)/2

n (a^λ/2_n ∨2^−N)^γγm+ (a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}+ aβγ−λ(1+α/2)/2

n (a^λ/2_n ∨2^−N)^γ]}

=:c3

√

t−t⁰{a^β_n+ ˜∆u1(m, n, λ, ε0, η1, a^λ/2_n ∨2^−N)}.

This was the estimate for the “good”B. Turning to the “bad”B we observe that forK ≥K1 and t⁰ ≤T_K it is always true that

|u_1,aλ

n(t⁰, y)| ≤Ey[|u((t⁰−a^λ_n)⁺, B(a^λ_n))|]≤2KEy[e^|B(aλn)|]≤2Ke^1+|y|. Since t−t⁰ ≤2^−4N9.4.11 it is true that

2^−32N9.4.11 ≥(t−t⁰)³⁸. Therefore,

Ex[1{|B(t−t⁰)−x|>2^−32N9.4.11}|u_1,aλ

n(t, x)−u_1,a^λ

n(t⁰, B(t−t⁰))|]

≤P0[|B(t−t⁰)−x|>2^−32N9.4.11]^1/28eK(Ex[e^2|B(t−t0)|+e^2|x|])^1/2

≤P0[B(1)>(t−t⁰)^−1/2(t−t⁰)^3/8]^1/2c(K)

≤c((t−t⁰)^16/8E0[|B(1)|¹⁶])^1/2

≤c(t−t⁰)≤c√

t−t⁰∆˜_u1.

9.4 Local bounds on the difference of two solutions 145 Hence we can conclude that:

T2≤c√

t−t⁰{a^β_n+ ˜∆u1(m, n, λ, ε0, η1, a^λ/2_n ∨2^−N)}. (9.90) Now, put (9.87), (9.88) and (9.90) together to obtain

|u_1,aλ

n(t⁰, x⁰)−u_1,aλ n(t, x)|

≤cd((t, x),(t⁰, x⁰))[a^β_n+ ˜∆u1(m, n, λ, ε0, η1, a^λ/2_n ∨2^−N)]

+ 2⁻⁹²|t−t⁰|^12(1−η1)a^−ε0−

λα

n 4 [(a^λ/2_n ∨2^−N)^γγm+ (a^λ/2_n ∨2^−N)^γa^βγ_n ].

=cd((t, x),(t⁰, x⁰)){a^β_n+a^−ε_n ⁰(a^λ/2_n ∨2^−N)^1−η1/2a⁻

λ 2(1+α/2) n

[(a^λ/2_n ∨2^−N)^γγm+a

λ 2(1+α/2)

n (a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}+a^βγ_n (a^λ/2_n ∨2^−N)^γ]}

+ 2⁻⁹²d((t, x),(t⁰, x⁰))^1−η1a^−ε_n ⁰a⁻

λ 2(1+α/2)

n a

λ

n2[(a^λ/2_n ∨2^−N)^γγm+a^βγ_n (a^λ/2_n ∨2^−N)^γ].

Next use d((t, x),(t⁰, x⁰)) ≤ 2^−N, such that d^η1/2(a^λ/2_n ∨2^−N)^−η1/2 ≤ (2^−N)⁰ = 1 and a^ε_n⁰ ≤1 to bound the above by:

≤(c2^−Nη21 + 2⁻⁹²)d((t, x),(t⁰, x⁰))^1−η1a^−ε0−λ(1+α/2)/2 n

{a^β_na^λ(1+α/2)/2_n + (a^λ/2_n ∨2^−N)[(a^λ/2_n ∨2^−N)^γγm+a^βγ_n (a^λ/2_n ∨2^−N)^γ +a^λ(1+α/2)/2_n (a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}]}.

Choose N1(K, η1) so large that 2^−η21N1c ≤2⁻⁹², and N9.4.17= N_9.4.17⁰⁰ ∨N1 which then clearly is stochastically bounded in n, λ, β. For N ≥N_9.4.17 and m < m¯ we have

a^λ(1+α/2)/2_n (a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0} ≤(a^λ/2_n ∨2^−N)^1+α/2(a^λ/2_n ∨2^−N)^{(γγm−1−α/2)∧0}

≤(a^λ/2_n ∨2^−N)^γγm.

If m≥m¯ the left hand side is bounded bya^λ/2(1+α/2)_n Therefore, we can write

|u_1,aλ

n(t⁰, x⁰)−u_1,aλ

n(t, x)| ≤

≤2⁻⁹⁰d^1−η1a^−ε_n ⁰a−λ(1+α/2)/2

n +1(m≥m)(a¯ ^λ/2_n ∨2^−N)

[a^β_na^λ(1+α/2)/2_n + ((a^λ/2_n ∨2^−N)^γγm+1+a^βγ_n (a^λ/2_n ∨2^−N)^γ+1]

= 2⁻⁹⁰d((t, x),(t⁰, x⁰))^1−η1∆¯u1(m, n, λ, ε0,2^−N).

And this is what we wanted to show.

146 Pathwise Uniqueness We also would like to obtain a similar result for u_2,δ. We postpone its proof which is simpler than the previous calculations to Section 9.7.

Notation:

∆¯1,u2(m, n, ε0,2^−N) =a^−ε_n ⁰2^{−N γ}[(a^1/2_n ∨2^−N)^γ(γm−1)+a^βγ_n ], (9.91)

∆¯_2,u2(m, n, λ, ε₀) =a^−ε_n ⁰[a

λ

2(˜γm+1−1)

n +a^βγ_n a

λ 2(γ−α/2)

n ]. (9.92)

Proposition 9.4.18. Let 0 ≤m ≤m¯ + 1 and assume that (Pm) holds. Then for all n ∈N, η₁ ∈ (0, α/4∧(2−α)/2), ε₀ ∈(0,1), K ∈ N^≥K1, λ∈ [0,1], β ∈ [0,1/2]

there is an N_9.4.18=N_9.4.18(m, n, η₁, ε₀, K, λ, β)(ω)∈N almost surely such that

∀N ≥N9.4.18,(t, x)∈Z(N, n, K, β), t⁰ ≤TK

d:=d((t, x),(t⁰, x⁰))≤2^−N implies that

|u_2,aλ

n(t, x)−u_2,aλ

n(t⁰, x⁰)| ≤2⁻⁹⁴(d(1−α/2)(1−η1)∆¯_1,u2 +d^1−η1∆¯_2,u2).

Moreover, N9.4.18 is stochastically bounded uniformly in(n, λ, β).

Now, we are ready to finish the chapter, that means to do the induction step in the proof of Proposition 9.4.2:

Proof of Proposition 9.4.2:

Let 0 ≤ m ≤ m¯ and assume that (P_m) holds. We want to show (P_m+1). Let therefore ε₀ ∈ (0,1), M = d_ε²

0e, ε₁ = _M¹ ≤ ε₀/2 and λ_i = iε₁ for i = 1, . . . , M. Then, clearly λi∈[ε1,1] for alli= 1, . . . , M.

Letn, ξ, K, βbe as in (P_m) and w.l.o.g. ξis sufficiently large, such that forη₁ := 1−ξ we have η₁ < ^α₄ ∧(1− ^α₂). Set ξ⁰= (1 +ξ)/2∈(ξ,1) and

N₂(m, n, ξ, ε₀, K, β)(ω) =

M

_

i=1

N_9.4.17(m, n, η₁, ε₀/2, K+ 1, λ_i, β).

N3(m, n, ξ, ε0, K, β)(ω) =

M

_

i=1

N9.4.18(m, n, η1, ε0/2, K+ 1, λi, β).

N4(m, n, ξ, ε0, K, β)(ω) =d 2

1−ξe(N_9.4.14∨N9.4.17⁰ + 1)

=: 1 1−ξN5.

N₁ =N₂∨N₃∨N₄∨N₀(ξ⁰, K) + 1∈N,

where N0(ξ⁰, K) is the constant we obtained from Theorem 5.3.3. By the results on each of the single constants we know that N₁ is then stochastically bounded uniformly in (n, β). Let now

N ≥N₁,(t, x)∈Z(N, n, K, β), t⁰ ≤T_K and d:=d((t, x),(t⁰, x⁰))≤2^−N.

9.4 Local bounds on the difference of two solutions 147 There are two cases for the values ofn to consider.

We start with the case of smalln:

a_n>2^{−N5(m,n,η1,ε0,K,β)}, (9.93)

This already completes the first case.

For large n, i.e.:

an≤2^−N5 (9.94)

let N⁰ = N −1 ≥ N2∨N3, which gives (t⁰, x⁰) ∈ Z(N⁰, n, K + 1, β) by triangle inequality. As (9.94) holds we can apply Proposition 9.4.17 with (ε₀/2, K + 1) instead of (ε₀, K).

Additionally one can also use Proposition 9.4.18. So we can estimate |u(ˆt0,xˆ0)− u(t⁰, x⁰)|. Before doing so we have to choose which partition of u to take in the

148 Pathwise Uniqueness Furthermore, we have byξ−α/2≤(2−α)^ξ₂

2^{−N0(2−α)ξ2} ≤2^−N0ξ2^N0α2 and 2^{−N0((2−α)ξ2+γ)} = 2^−N0ξ2^{−N0(γ−α/2)}

≤2^−N0ξ(a^1/2_n ∨2^−N0)^γ−α/2 (γ > α/2).

Using the aforementioned propositions, the special i, (9.95) and the just previous lines, we get:

Now, apply (9.96) to bound this by

≤2⁻⁸⁹a⁻

Now, we can proceed to the last step of this statement and obtain

|u(t⁰, x⁰)| ≤ |u(ˆt0,xˆ0)|+|u(ˆt0,xˆ0)−u(t⁰, x⁰)|

9.5 Proof of Proposition 9.2.2 149