1 Introduction

Let \(W=(W_{t}^{1},\ldots ,W_{t}^{d})_{t\ge 0}\) be a Brownian motion in the Euclidean space \(\mathbb R ^{d}\) of dimension \(d\ge 2\). The Stratonovich signature of \(W\) over the duration from time \(0\) to time \(T\), according to Chen [5] and Lyons [6], is the formal series with indeterminates \(X_{1},\ldots ,X_{d}\) whose coefficients are iterated Stratonovich’s path integrals of Brownian motion:

$$\begin{aligned} S(W)_{[0,T]}=\sum _{n=0}^{\infty }\sum _{j\in C_{n}}[j_{1}\ldots j_{n}]_{0,T}X_{j_{1}}\ldots X_{j_{n}} \end{aligned}$$
(1.1)

where \(C_{n}\) denotes the collection of all words with length \(n\) and letters in \(\{1,\ldots ,d\}, \sum _{j\in C_{n}}\) runs all words \( j=(j_{1},\ldots ,j_{n})\in C_{n}\), and the square bracket \([j_{1}\ldots j_{n}]_{s,t}\) of a word \(j\) denotes the multiple Stratonovich integral of Brownian motion over \([s,t]\), i.e.

$$\begin{aligned}{}[j_{1}\ldots j_{n}]_{s,t}=\int \limits _{s<t_{1}<\ldots <t_{n}<t}\circ dW_{t_{1}}^{j_{1}}\circ \ldots \circ dW_{t_{n}}^{j_{n}}. \end{aligned}$$
(1.2)

These stochastic integrals may be defined by means of Itô’s integration. In fact, multiple integrals may be defined inductively by

$$\begin{aligned}{}[j_{1}\ldots j_{n}]_{s,t}=\int _{s}^{t}[j_{1}\ldots j_{n-1}]_{[s,r]}\circ dW_{r}^{j_{n}} \end{aligned}$$

where \(\circ d\) indicates the integration in Stratonovich’s sense, which in turn can be expressed in terms of Itô’s and ordinary integrals.

If one is not concerned about the underlying algebraic structures defined by iterated integrals, it is not necessary to approach the Stratonovich signature through the formal series (1.1). We consider the collection of all possible iterated Stratonovich integrals \([j_{1}\ldots j_{n}]_{0,T}\), emphasizing the fact that they are all taken over a fixed time interval \([0,T]\), as the Stratonovich signature(s) of Brownian motion (over \([0,T]\)). Since we will work on signatures over a fixed interval, the lower script \(0\) and \(T\) will be omitted if no confusion may arise, for the sake of simplicity of notations. Without losing generality we may from now on assume that \(T=1\).

As the notion of signatures is so significant in this paper, we would like to give a formal definition.

Definition 1.1

Let \(W=(W_{t})_{t\ge 0}\) be a Brownian motion in \(\mathbb R ^{d}\) starting at \(0\). Then the Stratonovich signature (or signatures) of \(W\) over \([0,1]\) is the family of all iterated integrals (in the Stratonovich sense)

$$\begin{aligned}{}[j_{1}\ldots j_{n}]=\int \limits _{0<t_{1}<\ldots <t_{n}<1}\circ dW_{t_{1}}^{j_{1}}\circ \ldots \circ dW_{t_{n}}^{j_{n}} \end{aligned}$$

where \(n\) runs through \(1,2,\ldots ,\) and \(j_{1},\ldots ,j_{n}\in \{1,\ldots ,d\}\).

The interest for the signatures of paths has a long history. First of all, sequences of multiple iterated integrals arise naturally in Picard’s iteration of solving ordinary differential equations. Multiple iterated integrals of Brownian motion appeared already in early 1930s in Wiener’s celebrated work on harmonic analysis over the Wiener space. Itô [9] studied them in terms of his integration theory. Meanwhile, from 1950s to late 1970s, in a series of articles [25] etc. Chen demonstrated the usefulness of iterated integrals along piecewise smooth paths in manifolds. Chen showed the interesting algebraic structures defined by sequences of iterated integrals, developed a representation theory, and established a homotopy theory in terms of iterated integrals. The importance of multiple Stratonovich integrals, however, surprisingly was not recognized until the important contributions by Wong and Zakai [13], Ikeda and Watanabe [8], in which the convergence theorem for solutions to stochastic differential equations in Stratonovich’s sense was proved. The definite role played by iterated Stratonovich’s integrals was finally revealed in Lyons [11] (also see [10]) in which a universal limit theorem for solutions of Stratonovich’s stochastic differential equations was proved. Lyons has realized that the key elements for defining an integration theory along a continuous path which is not necessarily piecewise smooth is a sequence of iterated integrals that must be specified. This idea led to the discovery of the \(p\)-variation metric among continuous paths with finite variations, which allows to develop the theory of rough paths.

It has been conjectured that the signature of a path over a fixed time duration \([0,1]\), which can be read out at the terminal time \(1\), should be a good summary of information about the flow of timely ordered events, recorded in its path during time \(0\) to time \(1\). Chen [3] first proved that indeed it is possible to recover the whole path (up to tree-like components of the path which are not counted in its signature) by reading its signature as long as the path is smooth enough. Hambly and Lyons [6] quantified Chen’s result and extended it to rectifiable curves in multi-dimensional spaces. It is significantly more difficult to reconstruct a rectifiable path from its signature than for \( C^{1}\)-curves, and it requires delicate estimates on the growth of the signature for a continuous path with bounded variations. Hambly and Lyons [6] have developed their a priori estimates by exploiting deep results in hyperbolic geometry. Nevertheless, the preceding results are not applicable to interesting random curves, since, for example, almost all sample paths of a non-trivial diffusion process are not rectifiable.

In this article, we demonstrate that for \(d\ge 2\) almost all \(d\)-dimensional Brownian paths can be recovered from their Stratonovich’s signature. In other words, theoretically, all information recorded in Brownian motion from \(0\) to \(1\) can be read out from the Stratonovich signature over \([0,1]\).

To state our main result more precisely, we need to introduce more notations. Let \(\mathcal F _{t}^{0}=\sigma \{W_{s}:s\le t\}\) be the filtration generated by \(W\), and \(\mathcal F _{1}\) be the completion of \( \mathcal F _{1}^{0}\) (under the Wiener measure \(P\)), and \(\mathcal G _{1}\) be the complete \(\sigma \)-algebra generated by the Stratonovich signatures, i.e. the completion of the \(\sigma \)-algebra \(\sigma \{[j_{1}\ldots j_{n}]_{0,1}:j\in D_{n}\); \(n\in \mathbb N \}\).

Our main result may be stated as follows

Theorem 1.2

\(\mathcal F _{1}=\mathcal G _{1}\). Therefore the Stratonovich signature determines Brownian sample paths almost surely.

To prove this theorem, we need to develop a method of reconstructing almost all Brownian sample paths given their signatures. We will come to this point shortly.

In order to appreciate why Stratonovich signatures are able to represent the sample paths of Brownian motion, let us look at how to obtain iterated integrals of smooth differential forms along Brownian motion paths in terms of the Stratonovich signatures. The remarkable fact, which certainly goes back to Chen [3] for the deterministic case, is that any polynomials of Brownian motion (evaluated at a fixed time \(1\)) is a linear combination of the signatures over \([0,1]\). In fact

$$\begin{aligned} W_{t}^{j_{1}}\ldots W_{t}^{j_{n}}=\sum _{\pi \in S_{n}}[j_{\pi _{1}}\ldots j_{\pi _{n}}]_{0,t} \end{aligned}$$
(1.3)

where \(S_{n}\) denotes the group of permutations of \(\{1,\ldots ,n\}\). This formula can be proved by integrating by parts:

$$\begin{aligned} W_{t}^{j_{1}}W_{t}^{j_{2}}=\left[ j_{1}j_{2}\right] _{0,t}+\left[ j_{2}j_{1} \right] _{0,t} \end{aligned}$$

and for \(n\ge 2\)

$$\begin{aligned} W_{t}^{j_{1}}\ldots W_{t}^{j_{n}}W_{t}^{j_{n+1}}&= \sum _{\pi \in S_{n}}\int _{0}^{t}\left[ j_{\pi _{1}}\ldots j_{\pi _{n}}\right] _{0,s}\circ dW_{s}^{j_{n+1}} \\&+\sum _{\pi \in S_{n}}\int _{0}^{t}W_{s}^{j_{n+1}}\circ d\left[ j_{\pi _{1}}\ldots j_{\pi _{n}}\right] _{0,s} \\&= \sum _{\pi \in S_{n}}\left[ j_{\pi _{1}}\ldots j_{\pi _{n}}j_{n+1}\right] _{0,t} \\&+\sum _{\pi \in S_{n}}\int _{0}^{t}W_{s}^{j_{n+1}}\left[ j_{\pi _{1}}\ldots j_{\pi _{n-1}}\right] _{s}\circ dW_{s}^{j_{\pi _{n}}} \end{aligned}$$

so (1.3) follows. If \(\alpha ^{1}, \ldots , \alpha ^{k}\) are smooth differential forms on \(\mathbb R ^{d}\) with compact supports, then iterated Stratonovich integrals \([\alpha ^{1}\ldots \alpha ^{k}]_{s,t}\) are defined inductively by

$$\begin{aligned}{}[\alpha ^{1}\ldots \alpha ^{k}]_{s,t}=\int _{s}^{t}[\alpha ^{1}\ldots \alpha ^{k-1}]_{s,u}\alpha ^{k}(\circ dW_{u}). \end{aligned}$$

Since polynomials are dense in \(C^{k}\) functions for any \(k\) under uniform convergence over compact subsets, therefore all iterated Stratonovich integrals of \(1\)-forms against \(W\) are measurable functionals of the signatures. This is the context of the following lemma.

Lemma 1.3

If \(\alpha ^{1}, \ldots , \alpha ^{k}\) are smooth differential forms on \(\mathbb R ^{d}\) with compact supports, then \([\alpha ^{1}\ldots \alpha ^{k}]_{0,1}\) is \(\mathcal G _{1}\)-measurable.

Proof

If \(\alpha ^{l}\) have polynomial coefficients, then we have seen that \( [\alpha ^{1}\ldots \alpha ^{k}]_{0,1}\) is a linear combination of the Stratonovich signatures, so it is \(\mathcal G _{1}\)-measurable. In general case, we may approximate \(\alpha ^{1}, \ldots , \alpha ^{k}\) by polynomials \(\alpha _{n}^{1}, \ldots , \alpha _{n}^{k}\) in \(C^{k+1}( \mathbb R ^{d})\), so that

$$\begin{aligned}{}[\alpha _{n}^{1}\ldots \alpha _{n}^{k}]_{s,t}\rightarrow [\alpha ^{1}\ldots \alpha ^{k}]_{s,t} \end{aligned}$$

in probability. This yields that \([\alpha ^{1}\ldots \alpha ^{k}]_{0,T}\) is \( \mathcal G _{1}\)-measurable. \(\square \)

These iterated Stratonovich integrals \([\alpha ^{1}\ldots \alpha ^{k}]_{0,1}\) may be considered as “extended” signatures of \(W\) over \( [0,1]\), which were used by Chen (indeed it is Chen’s definition of signature of a path, though implicitly), and were used in the construction of a rectifiable path from its signature in Hambly and Lyons [6].

Since there are no essential differences in our proof of Theorem between dimension two and the higher dimensional case, we therefore concentrate on the case \(d=2\). The main idea and the key steps in the proof of Theorem 1.2 are described as follows.

To construct approximations of Brownian motion \(W\) in terms of a countable family of extended signatures, for each \(\varepsilon >0\) we construct an \( \varepsilon \)-grid so that \(\mathbb R ^{2}\) is divided into squares with center at \(\varvec{z}\varepsilon =(z_{1}\varepsilon ,z_{2}\varepsilon )\) and width \(\varepsilon \), and let

$$\begin{aligned} S_{\varvec{z}}=\left\{ (x_{1},x_{2}):|x_{1}-z_{1}\varepsilon |+|x_{2}-z_{2}\varepsilon |\le \frac{1}{2}\varepsilon (1-\varepsilon )\right\} \end{aligned}$$

which is strictly located inside the squares with the same center. We naturally construct an approximation by polygons which join the centers of the squares \(S_{\varvec{z}}\) which have been visited by Brownian motion \( W\). It is not very difficult to show these polygons converge to Brownian motion paths almost surely, and we want to show that these polygonal approximations are indeed determined by the Stratonovich signatures of \(W\). To this end, we construct a smooth differential \(1\)-form \(\phi ^{\varvec{ z}}\) which has a compact support inside the squares \(S_{\varvec{z}}\) so that for different indices \(\varvec{z}\in \mathbb Z ^{2}\), these differential 1-forms \(\phi ^{\varvec{z}}\) have disjoint supports. The key observation is that the Stratonovich integral \(\int \phi ^{\varvec{z} }(\circ dW)\) does not vanish almost surely over the duration that the Brownian motion has visited \(S_{\varvec{z}}\). This crucial fact allows us to identify those squares the Brownian motion has visited entirely in terms of the signatures of the Brownian motion.

2 Several technical facts

In this section we establish several technical facts which will be used in the proof of Theorem 1.2.

A planar square is a nice domain but its boundary has four corners and thus is not \(C^{1}\). For the technical reasons we consider a domain obtained from a square by replacing the portion of the boundary near each corner by a quarter of a small circle. More precisely, for a small \(\frac{1}{4} >\varepsilon >0\), and, as we will use this parameter \(\varepsilon \) for other constructions, for \(\beta \gg 1\) to be chosen late on, let

$$\begin{aligned} D=\left\{ (x_{1},x_{2}):0\le x_{1},x_{2}\le \frac{1}{2}\right\} \Bigg / \left\{ \left|x_{1}-\frac{1}{2}+\varepsilon ^{\beta }\right|^{2}+\left|x_{2}-\frac{1}{2}+\varepsilon ^{\beta }\right|^{2}\ge \varepsilon ^{2\beta }\right\} . \end{aligned}$$

The typical planar domain we will handle is

$$\begin{aligned} G=\{(x_{1},x_{2}):(|x_{1}|,|x_{2}|)\in D\}. \end{aligned}$$
(2.1)

For \(a>0, G_{a}\) denotes the similar planar domain \(aG\), i.e. \( G_{a}=\{x=(x_{1},x_{2}):(ax_{1},ax_{2})\in G\}\).

Let \(W_{t}=(W_{t}^{1},W_{t}^{2})\) be a two dimensional Brownian motion on a probability space \((\Omega ,\mathcal F ,P)\), and let \(a_{3}>a_{2}>a_{1}\). Let

$$\begin{aligned} S_{0}&= \inf \{t>0:W_{t}\in \partial G_{a_{3}}\},\\ S_{1}&= \inf \{t>S_{0}:W_{t}\in \partial G_{a_{1}}\} \end{aligned}$$

and

$$\begin{aligned} S_{2}=\inf \{t>S_{1}:W_{t}\in \partial G_{a_{2}}\} \end{aligned}$$

which are stopping times, finite almost surely. We are interested in the distribution of the random variable \(X=\int _{S_{0}}^{S_{2}}\phi (\circ dW_{s})\), where \(\phi \) is a differential \(1\)-form which coincides with \( x^{2}dx^{1}\) on \(G_{a_{2}}\), conditional to \(\{S_{1}<S_{2}\}\).

To this end, consider the diffusion process \(X=(X^{1},X^{2},X^{3})\) in \( \mathbb R ^{3}\) associated with the following stochastic differential equations

$$\begin{aligned} dX_{t}^{1}=dW_{t}^{1}; dX_{t}^{2}=dW_{t}^{2}; dX_{t}^{3}=X_{t}^{2}\circ dW_{t}^{1}. \end{aligned}$$
(2.2)

It is an easy exercise to calculate the infinitesimal generator of \(X\), which is \(L=\frac{1}{2}\left( A_{1}^{2}+A_{2}^{2}\right) \), where \(A_{1}= \frac{\partial }{\partial x_{1}}+x_{2}\frac{\partial }{\partial x_{3}}\) and \( A_{2}=\frac{\partial }{\partial x_{2}}\). In particular, the Lie bracket \( [A_{1},A_{2}]=-\frac{\partial }{\partial x^{3}}\), so that \(L\) is hypoelliptic (Theorem 1.1, p. 149, Hörmander [7]).

Lemma 2.1

Let \(W\) be Brownian motion in \(\mathbb R ^{2}\) on \((\Omega , \mathcal F ,P)\) started from a point at \(\partial G_{a_{1}}, S=\inf \{t>0:W_{t}\in \partial G_{a_{2}}\}\), and \(\xi =\) \(\int _{0}^{S}W_{s}^{2} \circ dW_{s}^{1}\). Then, for any \(y\in \partial G_{a_{2}}\), the conditional distribution \(P\{\xi \in dz|W_{S}=y\}\) has a continuous density function in \( z\).

Proof

Let \(D=G_{a_{2}}\times \mathbb R ^{1}\), and \(S=\inf \{t\ge 0:X_{t}\notin D\} \) the first exit time of the diffusion process \(X\). Then, \(D\) has a \( C^{1}\)-boundary (this is the reason for which we use rounded squares) and the condition required in [1] is satisfied, as the normal to the boundary belongs to the plane spanned by \(A_1\) and \(A_2\). Thus, according to a theorem of Ben Arous et al. (Theorem 1.22, p. 181, in [1]), the Poisson measure of \(L\) on the open domain \(D\) has a (smooth) density, which implies that the distribution of \(X_{S}\) has a continuous density function on \(\partial D\) with respect to the Lebesgue measure on \(\partial D\). Therefore the conditional distribution \(P\{\xi \in dz|W_{S}=y\}\) has a continuous density on \(\mathbb R ^{1}\) for \(y\in \partial G_{a_{2}}\). \(\square \)

Let \(f(x_{1},x_{2})\) be a smooth function on \(\mathbb R ^{2}\) with a support in \(G_{a_{3}}\) such that \(f(x_{1},x_{2})=x_{2}\) on \(G_{a_{2}}\). Consider the smooth differential 1-form \(\phi =f(x_{1},x_{2})dx_{1}\) on \(\mathbb R ^{2}\).

Lemma 2.2

Under above assumptions and notations. Let \( Z=\int _{S_{1}}^{S_{2}}\phi (\circ W_{t})\). Then the conditional distribution of \(Z\) given \(W_{S_{1}}=(x_{1},x_{2})\) and \(W_{S_{2}}=(y_{1},y_{2})\) has a continuous density function, i.e.

$$\begin{aligned} P\{Z\in dz|W_{S_{1}}=(x_{1},x_{2}),W_{S_{2}}=(y_{1},y_{2}) \}=p((x_{1},x_{2}),(y_{1},y_{2}),z)dz \end{aligned}$$
(2.3)

for some nonnegative function \(p\).

Proof

This follows from the Strong Markov property of \(X\) and the previous Lemma 2.1. \(\square \)

Lemma 2.3

Under conditions and notations described above. Let \(U\) be an open subset such that \(\overline{G_{a_{3}}}\cap U=\emptyset \) and \(\tau =\inf \{t>S_{0}:W_{t}\in \partial U\}\) be a hitting time. Let \(T=S_{2}+\tau \circ S_{2}\). Then the random variable \(\eta =\int _{S_{0}}^{T}\phi (\circ dW_{s})\ne 0\) almost surely on \(\{S_{1}<T\}\).

Proof

Write

$$\begin{aligned} \eta =\int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})+\int _{S_{0}}^{S_{1}}\phi (\circ dW_{s})+\int _{S_{2}}^{T}\phi (\circ dW_{s}). \end{aligned}$$

For any stopping time \(S\) we have two \(\sigma \)-fields, namely \(\mathcal F _{S}\) which is the \(\sigma \)-algebra of events happening before \(S\), and \( \mathcal F _{>S}\) the \(\sigma \)-algebra of events depending on the path after stopping time \(S\). By definition, \(1_{\{S_{1}<T\}}\int _{S_{0}}^{S_{1}} \phi (\circ dW_{s})\) is \(\mathcal F _{S_{1}}\)-measurable and \( 1_{\{S_{1}<T\}}\int _{S_{2}}^{T}\phi (\circ dW_{s})\) is \(\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\) measurable. Let \(Y=\int _{S_{0}}^{S_{1}} \phi (\circ dW_{s})+\int _{S_{2}}^{T}\phi (\circ dW_{s})\) for simplicity. By the strong Markov property

$$\begin{aligned} E\left\{ 1_{\{S_{1}<T\}}\int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})|\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\}&= 1_{\{S_{1}<T\}}E\left\{ \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})|\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\} \\&= 1_{\{S_{1}<T\}}E\left\{ \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})|W_{S_{1}},W_{S_{2}}\right\} \end{aligned}$$

so that

$$\begin{aligned} E\left\{ F(\eta )1_{\{S_{1}<T\}}|\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\} =1_{\{S_{1}<T\}}E\left\{ F\left( \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})+Y\right) |\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\} . \end{aligned}$$

Suppose \(F(z+y)=\sum _{j}H_{j}(z)K_{j}(y)\), then

$$\begin{aligned} E\left\{ F(\eta )|\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\}&= \sum _{j}K_{j}(Y)E\left\{ H_{j}\left( \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})\right) |\mathcal F _{S_{1}}\vee \mathcal F _{>S_{2}}\right\} \\&= \sum _{j}K_{j}(Y)E\left\{ H_{j}\left( \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})\right) |W_{S_{1}},W_{S_{2}}\right\} \end{aligned}$$

and therefore

$$\begin{aligned} E\left\{ F(\eta )1_{\{S_{1}<T\}}\right\} =\sum _{j}E\left\{ K_{j}(Y)E\left[ H_{j}\left( \int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})\right) |W_{S_{1}},W_{S_{2}}\right] 1_{\{S_{1}<T\}}\right\} . \end{aligned}$$

Since \(\int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})\) has a conditional probability density \(p(x,y,z)\)

$$\begin{aligned} E\left[ 1_{\{S_{1}<T\}}\int _{S_{1}}^{S_{2}}\phi (\circ dW_{s})\in dz|W_{S_{1}}=x,W_{S_{2}}=y\right] =p(x,y,z)dz \end{aligned}$$

and thus

$$\begin{aligned} E\left\{ F(\eta )1_{\{S_{1}<T\}}\right\}&= E\left\{ 1_{\{S_{1}<T\}}\int _{ \mathbb R }\sum _{j}K_{j}(Y)H_{j}\left( z\right) p(W_{S_{1}},W_{S_{2}},z)dz\right\} \\&= E\left\{ 1_{\{S_{1}<T\}}\int _\mathbb{R }F(Y+z)p(W_{S_{1}},W_{S_{2}},z)dz \right\} \!. \end{aligned}$$

In particular \(P\{\eta =0,S_{1}<T\}=0\). \(\square \)

3 Constructing approximations to Brownian paths

In this section, we construct polygonal approximations of the planar Brownian motion sample paths by tracing the sample paths of Brownian motion through prescribed \(\varepsilon \)-grids laid out in the plane. Our construction equally applies to higher dimensional Brownian motion with only minor modifications which we will leave to the reader. For \(d\ge 3\), squares can be replaced by hypercubes and the probabilities that the path threads its way between them can be evaluated as in Lemma 3.1 below. Note however one can also use the planar result, as one referee mentioned to us, the sample paths can be determined by their projections to the lower dimensional hyperspaces.

To make our arguments clear, let us work with the classical Wiener space \(( \varvec{W},\mathcal B ,P)\), where \(\varvec{W}\) is the space of all continuous paths in \(\mathbb R ^{2}\) started at \(0, \mathcal B \) is the Borel \(\sigma \)-algebra on \(\varvec{W}\) and \(P\) is the unique probability so that the coordinate process \(W=(W^{1},W^{2})\) is a planar Brownian motion on \((\varvec{W},\mathcal B ,P)\) started at \(0\).

Let \(\varepsilon \in (0,\frac{1}{4})\). Recall that \(G\) is the planar domain defined by (2.1) which is the planar square with corners rounded. For \(\varvec{z}=(z_{1},z_{2})\in \mathbb Z ^{2}\) we assign three boxes \( H_{\varvec{z}}^{\varepsilon }\subset K_{\varvec{z}}^{\varepsilon }\subset Z_{\varvec{z}}^{\varepsilon }\) which are all similar domains to \(G\), with a common center \(\varepsilon \varvec{z}\) lies on the \( \varepsilon \)-lattice \(\varepsilon \mathbb Z \):

$$\begin{aligned} H_{\varvec{z}}^{\varepsilon }&= \varepsilon \varvec{z}+\varepsilon (1-\varepsilon )G,\\ K_{\varvec{z}}^{\varepsilon }&= \varepsilon \varvec{z}+\varepsilon \left( 1-\varepsilon +\frac{\varepsilon \varphi (\varepsilon )}{2}\right) G ,\\ Z_{\varvec{z}}^{\varepsilon }&= \varepsilon \varvec{z}+\varepsilon \left( 1-\varepsilon +\varepsilon \varphi (\varepsilon )\right) G, \end{aligned}$$

and

$$\begin{aligned} V_{\varvec{z}}^{\varepsilon }=\varepsilon \varvec{z}+\varepsilon G \end{aligned}$$

where \(\varphi (\varepsilon )\ll \varepsilon ^{\alpha }\) (with \(\alpha \ge 11\)) but to be chosen late on.

Let us notice that the gap between \(Z_{\varvec{z}}^{\varepsilon }\) and the box \(V_{\varvec{z}}^{\varepsilon }\) has a magnitude \(\varepsilon ^{2}(1-\varphi (\varepsilon ))\), while the magnitude of the gap between \(H_{ \varvec{z}}^{\varepsilon }\) and \(K_{\varvec{z}}^{\varepsilon }\) is \( \frac{1}{2}\varepsilon ^{2}\varphi (\varepsilon )\). Since \(\varphi (\varepsilon )\ll \varepsilon ^{\alpha }\) so that

$$\begin{aligned} \varepsilon ^{2}(1-\varphi (\varepsilon ))\gg \frac{1}{2}\varepsilon ^{2}\varphi (\varepsilon ) \end{aligned}$$

a crucial fact we will use below.

If \(A\subset \mathbb R ^{2}\), then \(T_{A}\) denotes the hitting time of \(A\) by the Brownian motion \(W\).

Lemma 3.1

Given \(\beta \gg 10\), there is \(\varphi (\varepsilon )\ll \varepsilon ^{\alpha }\) (with \(\alpha \ge 11\)) such that for every \( \varvec{z}=(z_{1},z_{2})\in \mathbb Z ^{2}\) and \(x\in \partial Z_{ \varvec{z}}^{\varepsilon }\)

$$\begin{aligned} P\{T_{\partial V_{\varvec{z}}^{\varepsilon }}<T_{H_{\varvec{z} }^{\varepsilon }}|W_{0}=x\}\le \varepsilon ^{10}. \end{aligned}$$
(3.1)

Proof

We need to show that the probability on the left-hand side is dominated by the ratio of the distances between \(x\) to \(\partial H_{\varvec{z} }^{\varepsilon }\) and to \(\partial V_{\varvec{z}}^{\varepsilon }\) which is \(\frac{\varphi (\varepsilon )}{1-\varphi (\varepsilon )}\), which in turn yields the bound in (3.1) as \(\varepsilon <\frac{1}{4}\) by increasing \(\alpha \) to kill any possible constant appearing in the domination. This is standard for one dimensional Brownian motion. Similar estimates may be obtained by means of potential theory. Clearly the left-hand side of (3.1) does not depend on \(\varvec{z}\in \mathbb Z ^{2}\) so let us assume \(\varvec{z}=0\). Let \(u\) be the unique harmonic function on \(V_{\varvec{z}}^{\varepsilon }\setminus H_{ \varvec{z}}^{\varepsilon }\) such that \(u=1\) on \(\partial V_{\varvec{z }}^{\varepsilon }\) and \(u=0\) on \(\partial H_{\varvec{z}}^{\varepsilon }\) . Then, \(u(W_{t\wedge T_{\partial V_{\varvec{z}}^{\varepsilon }}\wedge T_{H_{\varvec{z}}^{\varepsilon }}})\) is a bounded martingale, so that

$$\begin{aligned} u(x)=P\{T_{\partial V_{\varvec{z}}^{\varepsilon }}<T_{H_{\varvec{z} }^{\varepsilon }}|W_{0}=x\}. \end{aligned}$$

By the uniform continuity of the potential \(u\) with respect to the distance of \(x\) to the interior boundary \(\partial H_{\varvec{z}}^{\varepsilon }\) (for example see sections 4.2 in Port and Stone [12]), we may chose \(\varphi (\varepsilon )\) small enough so that \(x\) is closer to \( \partial H_{\varvec{z}}^{\varepsilon }\) than to \(\partial V_{\varvec{ z}}^{\varepsilon }\), to ensure that \(u(x)\le \varepsilon ^{10}\) as long as \( x\in \partial Z_{\varvec{z}}^{\varepsilon }\).

In dimension two, we can identify the magnitude explicitly. Indeed, in dimension two, consider the harmonic function on the disk centered at \(0\) with radius \(\varepsilon \)

$$\begin{aligned} w(x_{1},x_{2})=\frac{1}{\log \left( 1+2\varepsilon -\varepsilon ^{2}\right) } \log \left( \frac{x_{1}^{2}+x_{2}^{2}}{\varepsilon ^{2}}+2\varepsilon -\varepsilon ^{2}\right) \end{aligned}$$

which vanishes on \(\rho \equiv \sqrt{x_{1}^{2}+x_{2}^{2}}=\varepsilon (1-\varepsilon )\) and is \(1\) on \(\rho =\varepsilon \). At \(\rho =\varepsilon (1-\varepsilon )+\varepsilon \varphi (\varepsilon )\)

$$\begin{aligned} w(x_{1},x_{2})&= \frac{1}{\log \left( 1+2\varepsilon -\varepsilon ^{2}\right) }\log \left( 1+2\varphi (\varepsilon )+\varphi (\varepsilon )^{2}-2\varepsilon \varphi (\varepsilon )\right) \\&\le C\frac{\varphi (\varepsilon )}{\varepsilon }. \end{aligned}$$

Note however similar estimates hold for our rounded squares in dimension \(2\). This can be done by a proper conformal transformation. \(\square \)

In what follows we choose such \(\varphi \) and \(\beta \) so that (3.1 ) holds for small \(\varepsilon \in (0,1/4)\).

For each path \(w\in \varvec{W}\), define a sequence \(\{\tau _{k}(w):k=0,1,2,\ldots \}\) of stopping times which trace the crossings of the path \(w\) through the \(\varepsilon \)-grid lattice \(\varepsilon \mathbb Z ^{2}\). Let \(\tau _{0}(w)=0\) and \(\varvec{n}_{0}(w)=(0,0)\), and define \( \tau _{k}(w)\) and \(\varvec{n}_{k}(w)\) inductively by

$$\begin{aligned} \tau _{k}(w)=\inf \left\{ t>\tau _{k-1}(w):w_{t}\in \bigcup \limits _{ \varvec{z}\ne \varvec{n}_{k-1}(w)}H_{\varvec{z}}^{\varepsilon }\right\} \end{aligned}$$

and \(\varvec{n}_{k}(w)\in \mathbb Z ^{2}\) such that \(w(\tau _{k}(w))\in H_{\varvec{n}_{k}(w)}^{\varepsilon }\) if \(\tau _{k}(w)<\infty \), and \( \varvec{n}_{k+1}(w)=\varvec{n}_{k}(w)\) if \(\tau _{k}(w)=\infty \). Then \(\{\tau _{k}:k=0,1,\ldots \}\) is a strictly increasing sequence of stopping times, and \(\tau _{k}\uparrow \infty \) almost surely as \(k\uparrow \infty \).

Let us use \(\{\zeta _{k}:k=0,1,\ldots \}\) and \(\{\varvec{m} _{k}:k=0,1,\ldots \}\) to denote the corresponding sequences obtained in the previous definition with box \(H_{\varvec{z}}^{\varepsilon }\) replaced by \(Z_{\varvec{z}}^{\varepsilon }\). In other words

$$\begin{aligned} \zeta _{k}(w)=\inf \left\{ t>\zeta _{k-1}(w):w_{t}\in \bigcup \limits _{\varvec{z}\ne \varvec{m}_{k-1}(w)}Z_{\varvec{z} }^{\varepsilon }\right\} \end{aligned}$$

etc.

Let \(M_{H}(w)=\inf \{k:\tau _{k+1}(w)>1\}\) and \(M_{Z}(w)=\inf \{k:\zeta _{k+1}(w)>1\}\). Then both \(M_{H}<\infty \) and \(M_{Z}<\infty \) almost surely. Since a path which hits the box \(H_{\varvec{z}}^{\varepsilon }\) must first hit the larger one \(Z_{\varvec{z}}^{\varepsilon }\) so that \(\zeta _{k}\le \tau _{k}\) for any \(k, \zeta _{k}<\tau _{k}<\zeta _{k+1}\) on \( \{\zeta _{k+1}<\infty \}\), and therefore \(M_{H}\le M_{Z}\). The last inequality says a continuous path at least hit as many larger boxes than smaller ones.

Let us construct \(w(\varepsilon )\) to be the polygon assuming the point \( \varvec{n}_{k}\varepsilon \) at time \(\tau _{k}\), that is,

$$\begin{aligned} w(\varepsilon )_{t}=\varepsilon \varvec{n}_{k-1}(w)+\frac{t-\tau _{k-1}(w)}{\tau _{k}(w)-\tau _{k-1}(w)}\varepsilon \varvec{n}_{k}(w) \text{ if} t\in [\tau _{k-1}(w),\tau _{k}(w)] \end{aligned}$$

for \(l=0,1,\ldots \) We show that \(w(\varepsilon )\) converges to the Brownian curves almost surely as \(\varepsilon \downarrow 0\).

Lemma 3.2

Let \(W=(W_{t})_{t\ge 0}\) be a planar Brownian motion started at some point inside the box \(H_{\varvec{0}}^{\varepsilon }\), and

$$\begin{aligned} \tau =\inf \left\{ t>0:W_{t}\in \bigcup \limits _{\varvec{z}\ne \varvec{0}}H_{\varvec{z}}^{\varepsilon }\right\} . \end{aligned}$$

Then

$$\begin{aligned} P\left\{ \sup _{0\le t\le \tau }|W_{t}|>3\sqrt{2}\varepsilon \right\} \le \left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon }\right] }. \end{aligned}$$

Proof

Let \(\varvec{z}=(z_{1},z_{2})\in \mathbb Z ^{2}\) be the random variable such that \(W_{\tau }\in H_{\varvec{z}}^{\varepsilon }\). If

$$\begin{aligned} \varvec{z}\ne (\pm 1,\pm 1),(\pm 1,0)\quad \text{ or}\quad (0,\pm 1) \end{aligned}$$

or the Brownian motion \(W\) runs out off the square \([-3\varepsilon ,3\varepsilon ]\times [-3\varepsilon ,3\varepsilon ]\), then \(W\) must travel through a narrow strip of wideness \(\varepsilon ^{2}\) and length \( \varepsilon -2\varepsilon ^{\beta }\), so that the probability

$$\begin{aligned} P\left\{ \varvec{z}\ne (\pm 1,\pm 1),(\pm 1,0) \text{ or} (0,\pm 1)\right\} \le \left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon } \right] }. \end{aligned}$$

Therefore

$$\begin{aligned} P\left\{ \sup _{0\le t\le \tau }|W_{t}|>3\sqrt{2}\varepsilon \right\} \le \left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon }\right] }. \end{aligned}$$

\(\square \)

Lemma 3.3

There is a sequence \(\varepsilon _{n}\downarrow 0\), such that

$$\begin{aligned} P\left\{ w:\lim _{n\rightarrow \infty }\inf _{\sigma }\sup _{0\le t\le 1}|w_{t}-w(\varepsilon _{n})_{\sigma (t)}|=0\right\} =1 \end{aligned}$$

where \(\inf _{\sigma }\) takes over all possible parametrization.

Proof

We need to estimate the numbers of the crossings between different \(H_{ \varvec{z}}^{\varepsilon }\) during the time \(0\) to \(1\). Note that

$$\begin{aligned} P\left\{ M_{H}=k \right\}&\le P\left\{ \text{ at} \text{ least} \text{ for} \text{ one} l, \tau _{l+1}-\tau _{l}\le \frac{1}{k}\right\} \\&\le kP\left\{ \sup _{0<t\le \frac{1}{k}}|w_{t}|\ge 2\varepsilon ^{2} \right\} \le 2k\mathbb P \left\{ \sup _{0<t\le \frac{1}{k} }|w_{t}^{1}|\ge \varepsilon ^{2}\right\} \\&\le 2k\exp \left( -\frac{\varepsilon ^{4}}{2}k\right). \end{aligned}$$

Therefore

$$\begin{aligned}&P\left\{ \sup _{l}\sup _{\tau _{l}\le t\le \tau _{l+1}}|w_{t}-\varepsilon \varvec{n}_{l}|>3\sqrt{2}\varepsilon \right\} \\&\quad \le P\left\{ \sup _{l}\sup _{\tau _{l}\le t\le \tau _{l+1}}|w_{t}-\varepsilon \varvec{n}_{l}|>3\sqrt{2}\varepsilon :M\le k\right\} +P\left\{ M_{H}>k\right\} \\&\quad \le k\left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon }\right] }+2\sum _{l>k}l\exp \left( -\frac{\varepsilon ^{4}}{2}l\right) \\&\quad \le k\left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon }\right] }+Ck\exp \left( -\frac{\varepsilon ^{4}}{2}k\right) \frac{1}{\left( 1-\exp \left( -\frac{\varepsilon ^{4}}{2}\right) \right) ^{2}} \\&\quad \le k\left( \frac{1}{3}\right) ^{\left[ \frac{1}{2\varepsilon }\right] }+ \frac{Ck}{\varepsilon ^{8}}\exp \left( -\frac{\varepsilon ^{4}}{2} (k-2)\right) \end{aligned}$$

by choosing \(k-2=\frac{1}{\varepsilon ^{6}}\) to obtain

$$\begin{aligned} P\left\{ \sup _{l}\sup _{\tau _{l}\le t\le \tau _{l+1}}|w_{t}-\varepsilon \varvec{n}_{l}|>3\sqrt{2}\varepsilon \right\} \le \frac{1}{\varepsilon ^{6}}\left( \frac{1}{3}\right) ^{\left[ \frac{1}{ 2\varepsilon }\right] }+\frac{C}{\varepsilon ^{14}}\exp \left( -\frac{1}{ 2\varepsilon ^{2}}\right) \end{aligned}$$

where \(C\) is a constant, so by the Borel-Cantelli lemma, \(w(\varepsilon _{n})\rightarrow w\) almost surely for a properly chosen \(\varepsilon _{n}\) such that

$$\begin{aligned} \sum \frac{1}{\varepsilon _{n}^{6}}\left( \frac{1}{3}\right) ^{\left[ \frac{1 }{2\varepsilon _{n}}\right] }+\sum \frac{1}{\varepsilon _{n}^{14}}\exp \left( -\frac{1}{2\varepsilon _{n}^{2}}\right) <\infty . \end{aligned}$$

\(\square \)

On the other hand the gap between two boxes \(H_{\varvec{z}}^{\varepsilon }\) and \(Z_{\varvec{z}}^{\varepsilon }\) in comparison to the gap between \( Z_{\varvec{z}}^{\varepsilon }\) and \(V_{\varvec{z}}^{\varepsilon }\) is so small, it happens that \(M_{H}=M_{Z}\) and \(\varvec{n}_{k}= \varvec{m}_{k}\) on \(\{k\le M_{H}=M_{Z}\}\) with a large probability, which is the context of the following lemma.

Lemma 3.4

There is \(\varepsilon _{0}\in (0,1/4)\) such that for any \( \varepsilon \in (0,\varepsilon _{0})\) we have

$$\begin{aligned} P\{M_{H}=M_{Z} \text{ and} \varvec{n}_{k}=\varvec{m}_{k} \text{ for} k\le M_{H}\}\ge \beta _{\varepsilon } \end{aligned}$$
(3.2)

where \(\beta _{\varepsilon }=1-2\varepsilon ^{4}-e^{-\frac{1}{\varepsilon ^{2}}}\).

Proof

Let \(A_{k}=\{\omega :\) \(\varvec{n}_{k}=\varvec{m}_{k}\}\) and \( B_{k}=\cap _{l\le k}A_{l}\). Then, as

$$\begin{aligned} \zeta _{k+1}\ge \zeta _{k}+T_{\partial V_{\varvec{m}_{k}}^{\varepsilon }}\circ \theta _{\zeta _{k}}, \end{aligned}$$

by strong Markov property and (3.1), \(P\{B_{k+1}|B_{k}\}\ge 1-\varepsilon ^{10}\). Therefore

$$\begin{aligned} P\{B_{[\varepsilon ^{-6}]}\}\ge (1-\varepsilon ^{10})^{\varepsilon ^{-6}}. \end{aligned}$$

Since \(\varepsilon \in (0,\frac{1}{4})\) and \(\log (1-x)\ge -2x\) for \(x\in (0,\frac{1}{2})\) we therefore have

$$\begin{aligned} \varepsilon ^{-6}\log (1-\varepsilon ^{10})\ge -2\varepsilon ^{4} \end{aligned}$$

so that

$$\begin{aligned} P\{B_{[\varepsilon ^{-6}]}\}\ge e^{-2\varepsilon ^{4}}\ge 1-2\varepsilon ^{4}. \end{aligned}$$

On the other hand, from the proof of Lemma 3.3, there is \( \varepsilon _{0}\in (0,1/4)\) such that for \(\varepsilon \in (0,\varepsilon _{0})\)

$$\begin{aligned} P\{M_{H}>\varepsilon ^{-6}\}\le \frac{C}{\varepsilon ^{14}}e^{-\frac{1}{ 2\varepsilon ^{2}}}\le e^{-\frac{1}{\varepsilon ^{2}}} \end{aligned}$$

for so that

$$\begin{aligned}&P\{M_{H} =M_{Z} \text{ and} \varvec{n}_{k}=\varvec{m}_{k} \text{ for} k\le M_{H}\} \\&\quad \ge P\{B_{[\varepsilon ^{-6}]}\}-P\{M_{H}>\varepsilon ^{-6}\} \\&\quad \ge 1-2\varepsilon ^{4}-e^{-\frac{1}{\varepsilon ^{2}}} \end{aligned}$$

which proves the lemma. \(\square \)

4 Proof of Theorem 1.2: using the signatures

This section is devoted to the proof of Theorem 1.2 by using information of its (extended) Stratonovich signatures. To this end, we need to choose a good version of multiple iterated Stratonovich’s integrals.

Recall that \((\varvec{W},\mathcal B ,P)\) is the classical Wiener space, where \(\varvec{W}\) is the sample space of all continuous paths started at \(0\), on which the coordinate process \((W_{t})_{t\ge 0}\) is Brownian motion under probability measure \(P\). For each path \(w\in \varvec{W}\), and natural number \(n\), we consider its dyadic approximations \(w^{(n)}\in \varvec{W}\) defined to be the polygon assuming the same values as \(w\) at dyadic points \(\frac{j}{2^{n}}\) (for \(j\in \mathbb Z _{+}\)). According to Wong-Zakai [13] and Ikeda-Watanabe [8], there is a subset \(\mathcal N \subset \varvec{W}\) with probability zero, such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{s<t_{1}<\cdots <t_{k}<t}\alpha ^{1}(dw_{t_{1}}^{(n)})\ldots \alpha ^{k}(dw_{t_{k}}^{(n)}) \end{aligned}$$

exists for every \(w\in \varvec{W}\setminus \mathcal N \), for all smooth differential forms \(\alpha ^{j}\) with bounded derivatives and for every pair \(s<t\). The previous limit is denoted by \([\alpha ^{1}\ldots \alpha ^{k}](w)_{s,t}\). We fix such an exceptional set \(\mathcal N \), and assign \( [\alpha ^{1}\ldots \alpha ^{k}](w)\) to be zero for \(w\in \mathcal N \). The important fact is that \([\alpha ^{1}\ldots \alpha ^{k}]_{s,t}\) is a version of Stratonovich’s iterated integral

$$\begin{aligned} \int _{s<t_{1}<\ldots <t_{k}<t}\alpha ^{1}(\circ dW_{t_{1}})\ldots \alpha ^{k}(\circ dW_{t_{k}}). \end{aligned}$$

In Lyons and Qian [10], a specific exceptional set \(\mathcal N \) was constructed by means of the so-called \(p\)-variation metric, which is however not needed in our proof of the main theorem.

In this section \([\alpha ^{1}\ldots \alpha ^{k}]\) denotes the version of Stratonovich’s iterated integral \([\alpha ^{1}\ldots \alpha ^{k}]_{0,1}\) defined as above, so that \([\alpha ^{1}\ldots \alpha ^{k}]=0\) on \(\mathcal N \).

Our goal is to show that \(W_{t}\) for all \(t\le 1\) is \(\mathcal G _{1}\) -measurable. For \(\varepsilon \in (0,1/4)\), and choose \(\alpha \) and \(\beta \) big enough so that the estimates in Lemmata 3.1 and 3.2 hold. Choose a smooth 1-form on \(\mathbb R ^{2}, \phi (x_{1},x_{2})=f(x_{1},x_{2})dx_{1}\), with a compact support in \(Z_{ \varvec{0}}^{\varepsilon }\) such that \(f(x_{1},x_{2})=x_{2}\) on \(K_{ \varvec{0}}^{\varepsilon }\). For each \(\varvec{z}\in \mathbb Z ^{2}\) , let \(\phi ^{\varvec{z}}=\phi (\cdot -\varepsilon \varvec{z})\) (or \( \phi ^{\varvec{z},\varepsilon }\) if we wish to indicate the dependence on \(\varepsilon \)) be the translation of \(\phi \) with compact support in \(Z_{ \varvec{z}}^{\varepsilon }\). Therefore, \(\{\phi ^{\varvec{z}}: \varvec{z}\in \mathbb Z ^{2}\}\) is a countable family of non-trivial differential forms with disjoint compact supports for every fixed \( \varepsilon \). The key idea, as we have explained in the Introduction, is to read out the blocks \(Z_{\varvec{n}_{l}}^{\varepsilon }\)’s which have been visited by the Brownian motion by using the extended Stratonovich’s signatures of form \([\phi ^{\varvec{z}_{1}}\ldots \phi ^{\varvec{z} _{m}}]\).

Let \(m\ge 0\). A finite ordered sequence (or called a word) of length \(m+1, \langle \varvec{z}_{0}\ldots \varvec{z}_{m}\rangle \) (where all \( \varvec{z}\)’s belong to the lattice \(\mathbb Z ^{2}\)), is admissible if \( \varvec{z}_{l}\ne \varvec{z}_{l+1}\) for \(l=0,\ldots ,m-1\). Let \( \mathcal W _{m}\) denote the set of all admissible words of length \(m+1\).

If \(w\in \varvec{W}\),

$$\begin{aligned} \hat{M}(w)=\sup \left\{ m:[\phi ^{\varvec{z}_{0}}\ldots \phi ^{ \varvec{z}_{m}}](w)\ne 0 \text{ for} \text{ some} \langle \varvec{z} _{0}\ldots \varvec{z}_{m}\rangle \in \mathcal W _{m}\right\} \end{aligned}$$

so that \(\hat{M}\) is \(\mathcal G _{1}\)-measurable. For each \(m\in \mathbb N \) and each admissible word \(\langle \varvec{z}_{0}\ldots \varvec{z} _{m}\rangle \in \mathcal W _{m}\) define

$$\begin{aligned} \varvec{A}_{m,\langle \varvec{z}_{0}\ldots \varvec{z}_{m}\rangle }=\{w\in \varvec{W}:\hat{M}(w)=m \text{ and} \left[ \phi ^{\varvec{z} _{0}}\ldots \phi ^{\varvec{z}_{m}}\right] (w)\ne 0\}. \end{aligned}$$
(4.1)

Since \(\phi ^{\varvec{z}}\) have disjoint supports, therefore, if \(\zeta _{m+1}(w)>1\), then \(\hat{M}(w)\) can not be greater than \(m\), so that \(\hat{M} \le M_{Z}\) except on the exceptional set \(\mathcal N \). On the other hand, according to Lemma 2.3 and the strong Markov property, \(\hat{M} \ge M_{H}\) almost surely. Therefore \(M_{H}\le \hat{M}\le M_{Z}\) almost surely.

If \(\hat{M}(w)=m\), there is at most one \(\langle \varvec{z}_{0}\ldots \varvec{z}_{m}\rangle \in \mathcal W _{m}\) such that \(\left[ \phi ^{ \varvec{z}_{0}}\ldots \phi ^{\varvec{z}_{m}}\right] (w)\ne 0\) and all other \([\phi ^{\varvec{z}_{0}^{\prime }}\ldots \phi ^{\varvec{z} _{n}^{\prime }}](w)=0\) for \(\langle \varvec{z}_{0}^{\prime }\ldots \varvec{z}_{n}^{\prime }\rangle \in \mathcal W _{n}\) if \(n>m\) or if \(n=m\) but \(\langle \varvec{z}_{0}^{\prime }\ldots \varvec{z}_{m}^{\prime }\rangle \ne \langle \varvec{z}_{0}\ldots \varvec{z}_{m}\rangle \).

Let

$$\begin{aligned} \varvec{\tilde{W}}_{m,\langle \varvec{z}_{0}\ldots \varvec{z} _{m}\rangle }=\{M_{H}=m,\varvec{n}_{l}=\varvec{z}_{l} \text{ for} l=0,\ldots ,m\}. \end{aligned}$$
(4.2)

for each admissible word \(\langle \varvec{z}_{0}\ldots \varvec{z} _{m}\rangle \in \mathcal W _{m}\), and

$$\begin{aligned} \varvec{\tilde{W}}_{\varepsilon }=\bigcup _{m=0}^{\infty }\bigcup \limits _{\langle \varvec{z}_{0}\ldots \varvec{z}_{m}\rangle \in \mathcal W _{m}}\varvec{\tilde{W}}_{m,\langle \varvec{z} _{0}\ldots \varvec{z}_{m}\rangle }. \end{aligned}$$

Then, according to Lemma 3.4, \(P(\varvec{\tilde{W}} _{\varepsilon })\ge \beta _{\varepsilon }\).

We are now in a position to complete our proof. Set

$$\begin{aligned} \varvec{\tilde{n}}_{l}=\sum _{m=0}^{\infty }\sum _{\langle \varvec{z} _{0}\ldots \varvec{z}_{m}\rangle \in \mathcal W _{m}}\varvec{z} _{l}1_{\varvec{A}_{m,\langle \varvec{z}_{0}\ldots \varvec{z} _{m}\rangle }} \end{aligned}$$

and redefine

$$\begin{aligned} \hat{w}(\varepsilon )_{t}=\varvec{\tilde{n}}_{l}\varepsilon +\frac{ t-\tau _{l}}{\tau _{l+1}-\tau _{l}}\varvec{\tilde{n}}_{l+1}\varepsilon \text{ if} t\in [\tau _{l},\tau _{l+1}] \end{aligned}$$

then, we may choose a sequence \(\varepsilon _{n}\downarrow 0\) so that \( \sum _{n}(1-\beta _{\varepsilon _{n}})<\infty \). Since \(\hat{w}(\varepsilon _{n})=w(\varepsilon _{n})\) almost surely on \(\varvec{\tilde{W}} _{\varepsilon _{n}}\) and \(P(\varvec{\tilde{W}}_{\varepsilon _{n}})\ge \beta _{\varepsilon _{n}}\), it follows from the Borel-Cantelli lemma that \( \sup _{t\in [0,1]}|\hat{w}(\varepsilon _{n})-w(\varepsilon _{n})|\rightarrow 0\) in probability as \(n\rightarrow \infty \), and therefore \(W_{t}\) is measurable with respect to \(\mathcal G _{1}\) for \(t\le 1\).