Acknowledgements

To go back to the proof of Proposition 8. We define an Fn-measurable event An as the set of <sup>ω</sup> <sup>∈</sup> <sup>Ω</sup><sup>þ</sup> such that <sup>N</sup><sup>~</sup> <sup>n</sup>ð Þ <sup>ω</sup> contains some label <sup>m</sup> of length <sup>n</sup>. From the

Hence, for every x∈ F<sup>0</sup> and 0 ⩽t⩽T and ∀n∈ N0, we may apply Lemma 9 for

★ ð Þ <sup>ω</sup> h i

★ ð Þ� <sup>ω</sup> <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

★ ð Þ� <sup>ω</sup> <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

h i

h i

★ ð Þ� <sup>ω</sup> <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

� �

where the symbol Et,x½ � Xð Þ ω ; A denotes the integral of Xð Þ ω over a measurable

h i

<sup>∣</sup> <sup>þ</sup> <sup>∣</sup>Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

1An

Since ∩nAn ¼ ∅ by the binary critical tree structure [12], and since we have an

★ ð Þ <sup>ω</sup> <sup>1</sup>An h i

∣

� Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

★ ð Þ <sup>ω</sup> ; An

★ ð Þ <sup>ω</sup> ; <sup>A</sup><sup>c</sup>

★ ð Þ <sup>ω</sup>

ð A

★ ð Þ <sup>ω</sup> <sup>1</sup>An h i

<sup>∣</sup> <sup>þ</sup> <sup>∣</sup>Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

★ ð Þ <sup>ω</sup> , <sup>a</sup>:s:

★ ð Þ <sup>ω</sup> h i

n

∣

∣

∣

� 1An

Xð Þ ω Pt,xð Þ dω :

∣

★ ð Þ <sup>ω</sup> <sup>1</sup>An h i

★ ð Þ <sup>ω</sup> <sup>1</sup>An ð Þ¼ <sup>ω</sup> <sup>0</sup>, <sup>a</sup>:s: (40)

∣

∣ ¼ 0: (41)

∣ ! 0 as ð Þ n ! ∞ (42)

∣

(38)

(39)

★ ð Þ <sup>ω</sup> h i

,

★ ð Þ <sup>ω</sup> on <sup>Ω</sup><sup>þ</sup> An: (37)

definition, it holds immediately that

Recent Advances in Integral Equations

Mh i <sup>u</sup>0;<sup>f</sup>

★ ð Þ¼ <sup>ω</sup> <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

★ ð Þ <sup>ω</sup> h i

the expression below with the identity (Eq. (31)) to obtain

<sup>¼</sup> <sup>∣</sup>Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

<sup>⩽</sup>∣Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

<sup>þ</sup>∣Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

<sup>¼</sup> <sup>∣</sup>Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

★ ð Þ <sup>ω</sup> <sup>1</sup>An h i

> ★ ð Þ <sup>ω</sup> <sup>1</sup>An h i

and lim<sup>n</sup>!<sup>∞</sup> <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

∣Mh i <sup>u</sup>0;<sup>f</sup>

Furthermore, we continue computing

<sup>¼</sup> <sup>∣</sup>Et,x Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

<sup>¼</sup> <sup>2</sup>∣Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

ð Þ <sup>38</sup> <sup>⩽</sup>∣Et,x <sup>M</sup>n,h i <sup>u</sup>0;f;<sup>u</sup>

natural estimate

40

event A with respect to the probability measure Pt,xð Þ dω , namely,

Et,x½ �¼ Xð Þ ω ; A Et,x½ �¼ Xð Þ� ω 1<sup>A</sup>

★ ð Þj <sup>ω</sup> <sup>F</sup><sup>n</sup> h i

∣:

★ ð Þ <sup>ω</sup> <sup>1</sup>An ð Þ <sup>ω</sup> <sup>∣</sup> <sup>&</sup>lt; <sup>M</sup>h i <sup>U</sup>;<sup>F</sup>

it follows by the bounded convergence theorem of Lebesgue that

lim<sup>n</sup>!<sup>∞</sup> <sup>∣</sup>Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

★ ð Þ <sup>ω</sup> h i

holds for every ð Þ <sup>t</sup>; <sup>x</sup> <sup>∈</sup> ½ �� <sup>0</sup>; <sup>T</sup> <sup>F</sup>0. Thus, we attain that u tð Þ¼ ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

a.e.‐x∈F0. This finishes the proof of Proposition 8. □

Consequently, from Eq. (39) and Eq. (41), we readily obtain

Concurrently, this completes the proof of the uniqueness.

<sup>∣</sup>u tð Þ� ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

h i

<sup>∣</sup>u tð Þ� ; <sup>x</sup> Et,x <sup>M</sup>h i <sup>u</sup>0;<sup>f</sup>

This work is supported in part by the Japan MEXT Grant-in-Aids SR(C) 17 K05358 and also by ISM Coop. Res. Program: 2011-CRP-5010.
