# Some applications of the Tsirelson spectral measures to noise filtering
problems

Rémi Lassalle  Ceremade, Université Paris 9 (Dauphine), PSL, Place du Maréchal
De Lattre De Tassigny, 75775 Paris Cedex 16, France
<EMAIL_ADDRESS>
###### Abstract.

This paper investigates applications of the Tsirelson spectral measures to
noise filtering problems within the classical Wiener noise framework. We
particularly focus on those among those measures associated to square
integrable Radon-Nikodym derivatives of laws of observation signals absolutely
continuous with respect to the Wiener measure. Explicit formulas are provided
to compute the values of the Tsirelson spectral measure of specific sets of
interests. Then, several applications to filtering problems are considered,
notably to the so-called innovation problem of filtering.

###### Key words and phrases:

Stochastic analysis, Noise filtering, Tsirelson’s spectral measure, Innovation
noise.

Mathematics Subject Classification : 60H30; 60H40; 60H99.

##  1\. Introduction

This paper investigates practical applications of the so-called Tsirelson
spectral measures to address precise noise filtering problems, on the somehow
tangible ground of the classical Wiener noise. Within the range of stochastic
analysis, the corresponding Tsirelson’s spectrum of noise has been recently
qualified to be an important notion within the lines of Itô’s theorems on
chaos expansions and martingale representations theorems in [44], which
motivates the study unrolled in the present paper. Although the whole paper
will be performed within a classical continuous framework, at this stage it
seems to be worth to notice that the discrete counterpart to those tools has
recently received further developments and applications related to other
fields as percolation (for instance, see the Lemma 5.3. of [14]). Recall that
these mathematical objects have been introduced by Tsirelson (p.274 of [37],
Definition 3d2 of [38]) within a systematic study of mathematical models of
what may be interpreted as pure noises : up to a normalization constant, they
provide quantitative instruments which describe accurately the statistical
properties of some random compact subsets, which are associated to square
integrable functionals over the underlying probability space of a noise. Their
behavior is known to yield a distinction between classical noises and non-
classical noises (see [27], [37], [43]), depending on whether all the related
random compact sets are almost surely finite, or not (see [44], and the
references therein). Thus, up to a normalization, the present paper will
essentially focus on the quantitative description of some finite random
compact sets associated to the classical Wiener noise. As it is well known,
still in this case, given $f\in L^{2}(\mu_{W})$, where $\mu_{W}$ denotes the
so-called Wiener measure, the existence of the Tsirelson spectral measure
$\rho_{f}$ of $\mu_{W}$ with respect to $f$ readily follows from the so-called
Itô-Wiener chaos expansion theorem of $L^{2}(\mu_{W})$, as it will be
recalled.

For the sake of clarity, in the whole paper, the time parameter $t$ will run
through the whole closed unit interval $\mathbb{T}:=[0,1]$ (rather than
$\mathbb{R}$), so that the noise will be conveniently described here by a
standard Brownian motion $(B_{t})_{t\in\mathbb{T}}$ defined on a complete
probability space $(\Omega,\mathcal{A},\mathbb{P})$ which is endowed with a
filtration $(\mathcal{A}_{t})_{t\in\mathbb{T}}$ which satisfies to the usual
conditions (it is right continuous and complete, see [33]). Subsequently, we
handle ${\bf B}$ as an (equivalence class of suitably measurable)
application(s) ${\bf B}:\omega\in\Omega\to C([0,1],\mathbb{R})$ which to a
given $\omega\in\Omega$ associates the continuous curve $t\in\mathbb{T}\to
B_{t}(\omega)\in\mathbb{R}$. The noise filter will be described by an
(equivalence class of measurable) application(s) of the form

${\bf F}^{u}:{\bf x}\in C([0,1],\mathbb{R})\to{\bf x}+{\bf u}({\bf x})\in
C([0,1],\mathbb{R}),$

where $t\to{\bf u}_{t}({\bf x})$ is assumed to be absolutely continuous, with
finite energy, and determined at each $t$ by the sole past values $\\{{\bf
x}(s):s\in[0,t]\\}$ of ${\bf x}\in C([0,1],\mathbb{R})$, outside some suitable
negligible sets. This filter turns the input noise $(B_{t})_{t\in\mathbb{T}}$
into an output observation signal which we describe by a stochastic process
$(Y_{t})$, which in functional form reads

${\bf Y}={\bf F}^{u}\circ{\bf B},$

$\circ$ denoting the pullback of equivalence classes of maps (more accurately
of morphisms of probability spaces, see [1]), which will require to take into
account precise aspects of absolute continuity of some deterministic causal
transport plans of $\mu_{W}$, and for short we shall also denote it by ${\bf
F}^{u}({\bf B})$. Similarly, when ${\bf h}\in H^{1}$ is an element of the so-
called Cameron-Martin space (see [28]), so that
$t\in[0,1]\to{h}_{t}\in\mathbb{R}$ is an absolutely continuous function with
square integrable derivative $(\dot{h})$, which describes a deterministic
signal of finite energy, we denote by ${\bf Y}^{h}$ the output which is
obtained by applying the filter to the noisy input signal ${\bf X}:={\bf
B}+{\bf h}$, which with a similar notation reads

${\bf Y}^{h}:={\bf F}^{u}({\bf B}+{\bf h}),$

which is well defined. Under further assumptions, notably that the Radon-
Nikodym derivative $f_{Y}:=\frac{dp_{Y}}{d\mu_{W}}$ of the absolutely
continuous probability $p_{Y}$ with respect to $\mu_{W}$ is square integrable,
this paper investigates applications of the Tsirelson spectral measure
$\rho_{f_{Y}}$ to associated noise filtering problems. More accurately, this
work is directed toward the following practical cases of study :

  1. (1)

The question of the existence of a realizable inverse filter described by some
${\bf H}^{u}$ to the real-time filter described by ${\bf F}^{u}$, and its
application to the statistical estimation of a deterministic input signal
$(h_{t})$ by a robust estimator $(\hat{h}_{t})$, for $t\in\mathbb{T}$.

  2. (2)

The so-called innovation problem (see [2], [29]). In a nutshell, this problem,
which will be recalled accurately below, is to decide whether the data of a
specific observable noise, which is called the innovation process, provides a
sufficient summary to access in real-time to the whole information on past
values of the observation process $(Y_{t})$.

As far as $(1)$ is concerned, we obtain a necessary and sufficient condition
which is based on Tsirelson’s spectral measures in Theorem 4.1, for the
existence of a realizable inverse filter. As an application, we establish a
sufficient condition on $\rho_{f_{\bf Y}}$ for the existence of the real-time
unbiased robust statistical estimator $\widehat{h}_{t}:={H}^{\bf u}_{t}({\bf
Y^{h}})$ of the value at $t$ of the deterministic signal described by
$h_{t}\in\mathbb{R}$. Here, we emphasize that the problematic of existence of
such an inverse filter is a classical problem of the spectral analysis of time
series. As a classical reference, we refer here to [21] p.105 where similar
problems are stated with linear filters and stationary random inputs : within
the framework and hypothesis of [21] the linearity and the stationarity allow
to use classical spectral measures. However, within our framework, the noise
is Wiener and the filter is not necessarily linear. Still, Theorem 4.1 shows
that within the specific circumstances which we encounter within this context,
the Tsirelson spectral measure $\rho_{f_{Y}}$ of $\mu_{W}$ with respect to
$f_{Y}$ can be used as a somehow analogous substitute to the usual spectral
measure to provide a criterion for the existence for the inverse filter ${\bf
H}^{u}$. The proof of the latter takes advantage of recent advances in
stochastic analysis from works of D.Feyel, A.S. Üstünel, M.Zakai and their
collaborators (for instance, see [11], [12], [39], [41]) on distinct but
related problems which, as a by product, can be used to study the
invertibility of filters in particular cases. More accurately, within the
proof a key role will be played by a causal Monge-Ampère equation, while some
values of $\rho_{f_{Y}}$ plays a role somehow analogous to the relative
entropy in [41]. Turning now to the case $(2)$, by following a parallel
strategy, we show that the so-called innovation problem of filtering can be
also conveniently addressed by using the Tsirelson spectral measure of
$f_{Y}$.

The structure of this paper is the following. In Section 2 we fix the notation
for the whole paper. In particular, the definition of the Tsirelson spectral
measures is recalled within the particular framework of the Wiener noise,
under the specific hypothesis which we adopt here. Then, in Section 3, under
precise hypothesis, we obtain explicit formulas for the values of the
Tsirelson spectral measure $\rho_{f_{Y}}(A)$, for specific Borel subsets $A$
of the set $Comp(\mathbb{T})$ of compact subsets of $\mathbb{T}$, which is
endowed with the corresponding Hausdorff distance. Those formulas are then
applied in Section 4 to state Theorem 4.1 which corresponds to the case $(1)$
above. Then, Corollary 5.1 of Section 5 provides a precise result toward the
application $(2)$ to the innovation problem. Finally, in Section 6 which also
provides further perspectives, a probabilistic normalization is performed, so
that the results of Section 4 first yield an explicit lower bond for the
probability of a precise random set associated to $(Y_{t})$ to be empty, and
then provide a criterion for $(1$) which is founded on the law of this
specific spectral random set.

##  2\. Preliminaries and notation

###  2.1. Notation for stochastic processes

Within the filtering model which we consider here, any realization of random
signals on the interval of time $[0,1]$ will be described by a continuous
function ${\bf\omega}:t\in[0,1]\to\omega(t)\in\mathbb{R}$. Therefore, we
denote by $C([0,1],\mathbb{R})$ the space of continuous real valued functions
defined on $[0,1]$, which we endow with the norm $\|.\|_{\infty}$ of uniform
convergence, so that it is turned into a separable Banach space, whose
associated Borel sigma-field will be denoted by
$\mathcal{B}(C([0,1],\mathbb{R}))$. Within this point of view, the law of a
random continuous signal is described by an element $\nu$ of the set ${\bf
M}_{1}(C([0,1],\mathbb{R}))$ of Borel probability measures on
$C([0,1],\mathbb{R})$. Among those laws, the standard Wiener measure
$\mu_{W}\in{\bf M}_{1}(C([0,1],\mathbb{R}))$ will play a key role within this
model, as it will be used to model a white noise, since the latter is well
known to be conveniently interpreted as some generalized derivative of the
Brownian motion. Given a standard Brownian motion $(B_{t})_{t\in[0,1]}$, with
$B_{0}=0$, $\mathds{P}-a.s.$, which is defined on a complete probability space
$(\Omega,\mathcal{A},\mathds{P})$, and denoting by ${\bf B}:\Omega\to
C([0,1],\mathbb{R})$ the application which to $\omega\in\Omega$ associates the
curve $t\to B_{t}(\omega)$ of a continuous modification, outside some
$\mathbb{P}-$ null set, recall that we have $\mu_{W}=p_{B}$, where
$p_{B}:={\bf B}_{\star}\mathds{P}$ denotes the pushforward of the probability
measure $\mathds{P}$ by the $\mathcal{A}/\mathcal{B}(C([0,1],\mathbb{R}))-$
measurable function ${\bf B}$, that is,
$p_{B}(A):=\mathbb{P}(\\{\omega\in\Omega:{\bf B(\omega)}\in A\\})$, $\forall
A\in\mathcal{B}(C([0,1],\mathbb{R}))$. Realizations of signals of finite
energy will be described as elements of the classical Cameron-Martin space
$H^{1}$ (see [28]), which is the linear subspace of $C([0,1],\mathbb{R})$
defined by

$H^{1}=\left\\{{\bf h}:[0,1]\to\mathbb{R}^{d}:\ {\bf
h}=\int_{0}^{.}{\bf\dot{h}_{s}}ds\ ,\
\int_{0}^{1}\|{\bf\dot{h}_{s}}\|^{2}_{\mathbb{R}^{d}}ds<+\infty\ \right\\};$

recall that the latter takes its name from a celebrated sequence of papers of
R.H. Cameron, and W.T. Martin (for instance see [5], [6]).

When it is endowed with the scalar product $<.,.>_{H^{1}}:(h,k)\in H^{1}\times
H^{1}\to\int_{0}^{1}\dot{h}_{s}\dot{k}_{s}ds\in\mathbb{R}$, $H^{1}$ is a
separable Hilbert space. Subsequently, to handle random signal models as time
evolves together with martingales, and to overcome technical aspects, we will
require to work on a complete stochastic basis
$(\Omega,\mathcal{A},(\mathcal{A}_{t})_{t\in[0,1]},\mathds{P})$, where
$(\Omega,\mathcal{A},\mathds{P})$ denotes a $\mathbb{P}-$ complete probability
space, and $(\mathcal{A}_{t})_{t\in[0,1]}$ a filtration which satisfies to the
usual conditions. Furthermore, when $X:\Omega\to\mathbb{R}$ is an
$\mathcal{A}/\mathcal{B}(\mathbb{R})-$ measurable function, where
$\mathcal{B}(\mathbb{R})$ denotes the Borel sigma field on $\mathbb{R}$, we
use the standard notation of the mathematical expectation
$\mathbb{E}_{\mathds{P}}\left[X\right]:=\int_{\Omega}X(\omega)\mathds{P}(d\omega)$,
if $E_{\mathbb{P}}[|X|]<+\infty$, in which case
$E_{\mathbb{P}}[X]\in\mathbb{R}$ and $X$ is said to be
$\mathbb{P}$-integrable, or if $X\geq 0$, $\mathbb{P}-a.s.$, in which case
$E_{\mathbb{P}}[X]\in[0,+\infty]$. At this stage it seems to be worth to
mention that within the whole paper, we adopt the usual convention according
to which the $\mathbb{P}-$ completion $\mathcal{G}^{\mathbb{P}}$ of a sub-
sigma field $\mathcal{G}$ of $\mathcal{A}$ is defined to be the smallest
sigma-field on $\Omega$ among those which contain both all the elements of
$\mathcal{G}$ and all the elements of
$\mathcal{N}^{\mathds{P}}:=\\{N\subset\Omega:\exists A\in\mathcal{A},N\subset
A,\mathbb{P}(A)=0\\}$. To handle random processes of finite energy, we define
the Hilbert space $L^{2}(\mathbb{P},H^{1})$ to be the set which is constituted
of ($\mathds{P}$\- equivalence classes) of
$\mathcal{A}/\mathcal{B}(C([0,1],\mathbb{R}))$\- measurable functions
$f:\Omega\to C([0,1],\mathbb{R})$ such that
$E_{\mathbb{P}}[\|f\|_{H^{1}}^{2}]<+\infty$. Furthermore, in order to take
into account the information flows described by
$(\mathcal{A}_{t})_{t\in[0,1]}$ (see [7], Definition 2.11, p. 39), we will
further consider the subset $L^{2}_{a}(\mathbb{P},H^{1})$ of the ${\bf u}\in
L^{2}(\mathbb{P},H^{1})$, such that the random variable
$f_{t}:\omega\in\Omega\to f_{t}(\omega)\in\mathbb{R}$ is $\mathcal{A}_{t}-$
measurable, $\forall t\in[0,1]$, for any (and then all)
$\mathcal{A}/\mathcal{B}(C([0,1],\mathbb{R})-$measurable function $f:\Omega\to
C([0,1],\mathbb{R})$ whose equivalence classes is ${\bf u}$, $f_{t}(\omega)$
denoting the value of the continuous function $f(\omega)\in
C([0,1],\mathbb{R})$ at point $t\in[0,1]$, $\forall\omega\in\Omega$. Then,
given $t\in[0,1]$, to shorten subsequent statements and proofs, it is
convenient to define the operator $\pi_{t}:u\in
L^{2}_{a}(\mathbb{P},H^{1})\to\int_{0}^{.}1_{[0,t]}(\sigma)\dot{u}_{\sigma}d\sigma\in
L^{2}_{a}(\mathbb{P},H^{1})$.

In the particular case where $(\Omega,\mathcal{A},\mathds{P})$ coincides with
the canonical probability space $(C([0,1],\mathbb{R}),$ $\mathcal{B}(C([0,1],$
$\mathbb{R})),\nu)$, for some $\nu\in{\bf M}_{1}(C([0,1],\mathbb{R}))$, in
this paper, we systematically endow it with the usual augmentation
$(\mathcal{F}_{t}^{\nu})$ of the filtration generated by the evaluation
process $(W_{t})_{t\in[0,1]}$, where $W_{t}:\omega\in
C([0,1],\mathbb{R})\to\omega(t)\in\mathbb{R}$ denotes the function which to
any $\omega$ associates its value $W_{t}(\omega)=\omega(t)$ at $t\in[0,1]$ ;
recall that $(\mathcal{F}_{t}^{\nu})$ is right-continuous and $\nu-$ complete.
In this latter case, we simply denote by $L^{2}_{a}(\nu,H^{1})$ for the
corresponding $L^{2}_{a}(\mathds{P},H^{1})$ space, which doesn’t seem to yield
any confusion below. Aside filtrations, to perform computations, given a
stochastic process $(Y_{t})_{t\in[0,1]}$ defined on a complete probability
space $(\Omega,\mathcal{A},\mathds{P})$, and given $s,t\in[0,1]$ such that
$s\leq t$, we shall denote by $(\mathcal{G}_{[s,t]}^{Y})_{t\in[s,1]}$ the
usual augmentation of the filtration $(\sigma(Y_{v}-Y_{u}:s\leq u\leq v\leq
t))_{t\in[s,1]}$, and whenever $Y_{t}=W_{t}$, $\forall t\in[0,1]$, and the
underlying probability space is
$(C([0,1],\mathbb{R}),\mathcal{B}(C([0,1],\mathbb{R})),\nu)$, for some
$\nu\in{\bf M}_{1}(C([0,1],\mathbb{R}))$, we use the specific notation
$\mathcal{F}_{[s,t]}^{\nu}$ to denote $\mathcal{G}_{[s,t]}^{Y}$, so that
$\mathcal{F}_{t}^{\nu}=\mathcal{F}_{[0,t]}^{\nu}$, $\forall t\in[0,1]$. In
particular, with this notation, the independency of increments of the Brownian
motion yields the continuous product structure, with in particular

$\mathcal{F}_{1}^{\mu_{W}}=\mathcal{F}_{[0,s]}^{\mu_{W}}\otimes\mathcal{F}_{[s,1]}^{\mu_{W}},\forall
s\in[0,1]$

where the sigma-field in the r.h.t. denotes the smallest sigma-field which
contains both all elements of $\mathcal{F}_{[0,s]}^{\mu_{W}}$ and elements of
the $\mu_{W}-$ independent sigma-field $\mathcal{F}_{[s,1]}^{\mu_{W}}$.

###  2.2. The Tsirelson spectral measures

We denote by $Comp([0,1])$ the set which is constituted by the compact subsets
of $[0,1]$, including the empty-set $\emptyset$, which is endowed with the so-
called Hausdorff distance $d_{Haus}:Comp([0,1])\times
Comp([0,1])\to[0,+\infty]$, defined by

$d_{Haus}(K,\widetilde{K}):=\inf\left(\left\\{\epsilon\in(0,+\infty):K\subset\widetilde{K}^{\epsilon}\
\text{and}\ \widetilde{K}\subset K^{\epsilon}\right\\}\right),$

for any $K,\widetilde{K}\in Comp([0,1])$, where for any subset $D$ contained
in $[0,1]$, and $\epsilon\in(0,+\infty)$, we use the notation
$D^{\epsilon}:=\\{t\in[0,1]:\exists s\in D,\ |s-t|<\epsilon\\}$. We denote by
$\mathcal{B}(Comp([0,1]))$ the associated Borel sigma-field on $Comp([0,1])$.
Within our specific framework, we recall the definition of the Tsirelson
spectral measures, in particular see section 3d4 of [38]. Although this
definition usually holds for much general noise models with rather elaborated
proofs, by following [38], we emphasize that within our specific framework,
its existence follows from the so-called Itô-Wiener chaotic representation of
$L^{2}(\mu_{W})$, as it will be recalled in this section. For the sake of
clarity denote by $\mathcal{E}:=\\{\cup_{i=1}^{n}[s_{i},t_{i}]:0\leq s_{1}\leq
t_{1}\leq...\leq s_{s}\leq t_{n},\ n\in\mathbb{N}\\}$, and for
$E=\cup_{i=1}^{n}[s_{i},t_{i}]\in\mathcal{E}$, $\mathcal{F}_{E}^{\mu_{W}}$
denotes the sigma-field generated by the
$\mathcal{F}_{[s_{i},t_{i}]}^{\mu_{W}}$, $\forall i\in\\{1,...,n\\}$.

######  Definition 2.1.

(Tsirelson’s spectral measures within this specific framework.) Given $f\in
L^{2}(\mu_{W})$ there exists a unique finite Borel measure $\rho_{f}$ on
$\text{Comp}([0,1])$ such that

$\rho_{f}(\\{K\in\text{Comp}([0,1]):K\subset
E\\})=E_{\mu_{W}}\left[\left(E_{\mu_{W}}[f|\mathcal{F}_{E}\right])^{2}\right],$

holds, $\forall E\in\mathcal{E}$. $\rho_{f}$ is called the Tsirelson spectral
measure of $\mu_{W}$ with respect to $f$.

As far as the existence is concerned, for the reader’s convenience, we now
recall how this measure can be explicitly defined for $f\in L^{2}(\mu_{W})$ by
setting

$\rho_{f}(\mathcal{K}):=|f_{0}|^{2}\mathds{1}_{\mathcal{K}}(\emptyset)+\sum_{n=1}^{+\infty}n!\int_{[0,1]^{n}}|f_{n}(t_{1},...,t_{n})|^{2}\mathds{1}_{\mathcal{K}}(\\{t_{1},...,t_{n}\\})\lambda_{\mathbb{R}^{n}}(dt_{1},...,dt_{n}),$
(2.1)

for any $\mathcal{K}\in\mathcal{B}(Comp([0,1]))$, where
$\lambda_{\mathbb{R}^{n}}$ denotes the Lebesgue measure on $\mathbb{R}^{n}$,
while

$f=\sum_{n=0}^{+\infty}I_{n}(f_{n}),\ \mu_{W}-a.s.$

denotes the Itô-Wiener chaos expansion (section XXI of [9], [16], [28], p. 68
of [30], [31], [32], Theorem 3.7 p.202 of [34]) of $f$ (the limit is in
$L^{2}(\mu_{W})$), with $f_{0}=E_{\mu_{W}}[f]\in\mathbb{R}$, with $f_{n}\in
L^{2}([0,1]^{n})$ which is a symmetric square integrable real-valued function
defined on $[0,1]^{n}$ which is endowed with the restriction of the measure
$\lambda_{\mathbb{R}^{n}}$ to the Borel $\sigma-$ field
$\mathcal{B}([0,1]^{n})$, for any $n\in\mathbb{N}$, and where

$I_{n}(f_{n}):=\int_{[0,1]^{n}}f_{n}dW^{\otimes
n}:=n!\int_{0}^{1}(\int_{0}^{s_{1}}...(\int_{0}^{s_{n-1}}f_{n}(s_{1},...,s_{n})dW_{s_{n}})...)dW_{s_{1}},$

denotes iterated Itô’s integrals, with $(n+1)!=(n+1)n!$ and $0!=1$, while
$I_{0}(f_{0})(\omega):=f_{0}$, $\mu_{W}-$ a.s. coincides with
$f_{0}\in\mathbb{R}$ outside some $\mu_{W}-$ negligible set ; notice that,
since the function

$(t_{1},...,t_{n})\in[0,1]^{n}\to\\{t_{1},...,t_{n}\\}\in Comp([0,1])$

is continuous and therefore Borel measurable, for any $n\in\mathbb{N}$, the
r.h.t. of (2.1) is well defined. Then, it follows that $\rho_{f}$ meets the
hypothesis of Definition 2.1, while the uniqueness is clear.

######  Example 2.1.

For $f=\frac{W_{1}^{4}}{\sqrt{E_{\mu_{W}}[W_{1}^{8}]}}$, $\rho_{f}$ is a
probability measure. In this particular case, $\rho_{f}$ is the law of a
random set, which is obtained from an i.i.d. sequence of random variables
$(\xi_{n})_{n\in\mathbb{N}}$ of uniform law on $]0,1[$
($\xi_{1}\sim\mathcal{U}(]0,1[)$), which is defined on some probability space
$(\Omega,\mathcal{A},\mathbb{P})$, where is also defined an independent
integer-valued random variable $N:\Omega\to\mathbb{R}$, which is distributed
according to the law $p_{N}:=\sum_{n=0}^{+\infty}p_{N}(\\{n\\})\delta_{n}$,
where $\delta_{n}$ denotes the Dirac mass concentrated at
$n\in\\{0\\}\cup\mathbb{N}$, and where

$p_{N}(\\{n\\}):=\rho_{f}(\\{K\in Comp([0,1]):Card(K)=n\\}),$

$\forall n\in\\{0\\}\cup\mathbb{N}.$ Then $\rho_{f}$ is the law of the random
set

$C:\omega\in\Omega\to\begin{cases}\\{\xi_{1}(\omega),...,\xi_{N(\omega)}(\omega)\\}\in
Comp([0,1])&\text{if }\ N(\omega)\geq 1\\\ \emptyset\in Comp([0,1])&\text{if}\
N(\omega)=0\end{cases}.$

Table 1. Numerical table : numerical approximations with $10^{-5}$ accuracy of values of the function $n\to p_{N}(\\{n\\})$ of Example 2.1. $n$ | 0 | 1 | 2 | 3 | 4 |  $>$4
---|---|---|---|---|---|---
$p_{N}(\\{n\\})$ | 0.08571 | 0 | 0.68571 | 0 | 0.22857 | 0

######  Example 2.2.

For $\beta\in(0,+\infty)$, set $f:=\exp(\sqrt{\beta}W_{1}-\beta)$, so that
once again $\rho_{f}$ is a probability since $E_{\mu_{W}}[f^{2}]=1$. Then, the
probability law $p_{N}$ defined by $p_{N}(\\{n\\}):=\rho_{f}(\\{K\in
Comp([0,1]):Card(K)=n\\})$, $\forall n\in\mathbb{N}$, is now a Poisson
distribution of parameter $\beta$.

Figure 2.1. The figure on the left (resp. on the right) illustrates Example
2.1 (resp. Example 2.2). It represents a restriction of the graph of the
Tsirelson spectral mass function $n\in\\{0\\}\cup\mathbb{N}\to
p_{N}(\\{n\\})\in[0,1]$, with the data of Table 1 (resp. by taking $\beta=4$
in Example 2.2).

###  2.3. The noise filter

In the whole paper, $(B_{t})_{t\in[0,1]}$ will denote a standard brownian
motion defined on a complete stochastic basis
$(\Omega,\mathcal{A},(\mathcal{A}_{t})_{t\in[0,1]},\mathds{P})$, where
$(\mathcal{A}_{t})$ satisfies to the usual conditions. Given ${\bf h}\in
H^{1}$, we may consider noisy input signals of the form

$X_{t}={B}_{t}+h_{t},\ \forall t\in[0,1],\mathbb{P}-a.s..$

It is a well known consequence of the so-called Cameron-Martin-Segal theorem
(see [28]) that the law $p_{X}$ is actually equivalent to the Wiener measure
$p_{B}=\mu_{W}$, which we denote by $p_{X}\sim\mu_{W}$. We further assume that
this noisy input signal is then turned into the observation signal $(Y_{t})$
by a not necessarily linear filter ${\bf F}^{\bf u}$ which is determined by a
given ${\bf u}\in L^{2}_{a}(\mu_{W},H^{1})$, from the formula

${\bf F}^{\bf u}:=\mathds{I}_{C([0,1],\mathbb{R})}+{\bf u},$ (2.2)

so that we define

${\bf Y^{h}}={\bf F}^{u}\circ{\bf X},$ (2.3)

$\circ$ denoting the usual pullback of (equivalence classes) of measurable
functions which is well defined here, since $p_{X}\sim\mu_{W}$ (i.e.
equivalent), and for short we may also denote it by ${\bf F}^{u}({\bf X})$.
The latter means that

$Y_{t}^{\bf h}={F}^{\bf u}_{t}\circ{\bf X},\ \forall
t\in[0,1],\mathds{P}-a.s.,$

where ${F}^{\bf u}_{t}=W_{t}\circ{\bf F}^{\bf u}$, and subsequently, to
simplify the notation, we may also denote it by ${Y_{t}^{\bf h}}={F}^{\bf
u}_{t}({\bf X})$, $\forall t\in[0,1]$, while in the particular case where
${\bf h}={\bf 0_{H^{1}}}$ we shall denote ${\bf Y^{0_{H^{1}}}}$ by ${\bf Y}$.
Notice here that, for given $t\in[0,1]$, the value of ${F}^{\bf u}_{t}({\bf
X}):={F}^{\bf u}_{t}\circ{\bf X}$ may be determined by the whole past values
$\\{X_{s},s\in[0,t]\\}$ of the input $(X_{s})_{s\in[0,1]}$, in a not
necessarily linear fashion. Although filters of the form (2.3) may seem to be
quite restrictive with respect to some of the forthcoming computations, it
turns out to be taylor made to obtain a compact statement of our main results
Theorem 4.1, and Corollary 5.1 on the innovation conjecture, while it
emphasizes the clarity of the method. Sunsequently, $p_{\bf Y}$ (resp.
$p_{{\bf X}}$) denotes the law of ${\bf Y}$ (resp. ${\bf X}$) that is, the
Borel probability measure on $C([0,1],\mathbb{R})$ which is the pushforward of
$\mathbb{P}$ by the $C([0,1],\mathbb{R})-$ valued (equivalence class of)
measurable function(s) ${\bf Y}$ (resp. ${\bf X}$) which is defined on
$(\Omega,\mathcal{A},\mathbb{P})$. In the particular case where ${\bf X}={\bf
B}$, notice again that we have $p_{\bf B}=\mu_{W}$.

###  2.4. The innovation conjecture

In this whole subsection we focus on the case where $h_{t}=0$, $\forall
t\in[0,1]$, so that the input signal $(X_{t})$ coincides with the noise
$(B_{t})$ ; as it will be clear below, within our specific purposes, the case
where ${\bf h}\neq{\bf 0_{H^{1}}}$ can actually be reduced to this case, by
considering the composition of the filter described by ${\bf F}^{u}$ with the
filter described by the function ${\bf
T}_{h}:=\mathds{I}_{C([0,1],\mathbb{R})}+{\bf h}$. Hence, in this section,
given ${\bf u}\in L^{2}_{a}(\mathbb{P},H^{1})$ we still define ${\bf Y}:={\bf
Y^{0_{H^{1}}}}$ by (2.4) with ${\bf F}^{u}$ given by (2.2) with now ${\bf
X}={\bf B}$, so that we have

${\bf Y}={\bf F}^{u}\circ{\bf B}={\bf B}+{\bf\widetilde{u}},$ (2.4)

where ${\bf\widetilde{u}}\in L^{2}_{a}(\mathbb{P},H^{1})$ is defined by

${\bf\widetilde{u}}:={\bf u}\circ{\bf B}={\bf Y}-{\bf B}.$

Recall that, in this case, the innovation process of $(Y_{t})$ which we denote
here by $(B_{t}^{\bf\widetilde{u}})$ is the $(\mathcal{G}_{[0,t]}^{Y})-$
brownian motion on $(\Omega,\mathcal{A},\mathds{P})$ (see [3], [18], [29])
which is defined by

$B_{t}^{\bf\widetilde{u}}:=B_{t}+\int_{0}^{t}(\dot{\widetilde{u}}_{s}-E_{\mathds{P}}\left[\dot{\widetilde{u}}_{s}|\mathcal{G}_{[0,s]}^{Y}\right])ds,\
\forall t\in[0,1],\mathds{P}-a.s.,$

so that it satisfies to

$Y_{t}=B_{t}^{\bf\widetilde{u}}+\int_{0}^{t}E_{\mathds{P}}\left[\dot{\widetilde{u}}_{s}|\mathcal{G}_{[0,s]}^{Y}\right]ds,\
\forall t\in[0,1],\mathds{P}-a.s..$

At this stage, it seems to be worth to notice that the small increments of the
so-called innovation noise process $(B^{\bf\widetilde{u}}_{t})$ may be
interpreted as the part of increments of $(Y_{t})$ which are not determined by
the past of $(Y_{t})$ and are independent to it under $\mathbb{P}$. A natural
question is then to determine whether $(Y_{t})$ is totally determined by its
innovation $(B_{t}^{\bf\widetilde{u}})$, which is the so-called innovation
problem of Frost and Kailath (see [2], [29] and the references therein).
Subsequently, given $t\in[0,1]$, $(Y_{s})$ is said to satisfy to the
innovation conjecture of filtering on $[0,t]$, if and only if, the answer to
the previous question is affirmative, that is, if and only if, the following
coincidence of filtrations

$(\mathcal{G}_{[0,s]}^{{\bf
B^{\bf\widetilde{u}}}})_{s\in[0,t]}=(\mathcal{G}_{[0,s]}^{\bf Y})_{s\in[0,t]}\
(\text{innovation conjecture on [0,t]}),$

holds. In cases, this problems may involves sharp aspects of filtrations
theory (for instance, see [36]).

Denoting by $p_{Y}:={\bf Y}_{\star}\mathds{P}\in{\bf
M}_{1}(C([0,1],\mathbb{R}))$ the law of $({Y}_{t})$, it is known (see [19]) to
be absolutely continuous to the Wiener measure $\mu_{W}=p_{B}$ ([45], see also
[22]), so that the corresponding Radon-Nikodym derivative
$\frac{dp_{Y}}{dp_{B}}\in L^{1}(p_{B})$ is well defined. Under the further
assumption that $\frac{dp_{Y}}{dp_{B}}\in L^{2}(p_{B})$, we can consider the
Tsirelson spectral measure $\rho_{f_{Y}}$ with $f_{Y}:=\frac{dp_{Y}}{dp_{B}}$.
Subsequently, it is shown that under suitable hypothesis, such $(Y_{t})$ which
satisfy to the innovation conjecture can be characterized through
$\rho_{f_{Y}}$. Finally, given ${\bf{\widetilde{u}}}\in
L^{2}_{a}(\mathds{P},H^{1})$,
$\delta^{B}{\bf\widetilde{u}}:=\int_{0}^{1}\dot{\widetilde{u}}_{s}dB_{s}$
denotes the Itô stochastic integral ([17]), while

$\mathcal{E}\left(-\delta^{B}{\bf\widetilde{u}}\right):=\exp\left(-\delta^{B}{\bf\widetilde{u}}-\frac{\|{\bf\widetilde{u}}\|_{H^{1}}^{2}}{2}\right)$
(2.5)

denotes the so-called Doléans-Dade exponential (see [33]).

##  3\. Tsirelson’s spectral measures associated to the Radon-Nikodym
derivatives of $p_{Y}$ with respect to the law of the Wiener noise

######  Proposition 3.1.

Let $s,t\in[0,1]$ be such that $s\leq t$, and let ${\bf u}\in
L^{2}_{a}(\mu_{W},H^{1})$ which satisfies to the hypothesis

$E_{\mu_{W}}\left[\exp\left(-\int_{0}^{t}\dot{u}_{s}dW_{s}-\frac{1}{2}\int_{0}^{t}\dot{u}_{s}^{2}ds\right)\right]=1.$
(3.6)

Further define ${\bf Y}$ by (2.4), with ${\bf F}^{\bf u}$ given by (2.2), and
set ${\bf\widetilde{u}}:={\bf Y}-{\bf B}\in L^{2}_{a}(\mathds{P},H^{1})$ so
that ${\bf Y}={\bf B}+{\bf\widetilde{u}}$. Then, we have

$E_{\mathds{P}}\left[\mathcal{E}\left(-\delta^{B}\pi_{t}{\bf\widetilde{u}}\right)\middle|\mathcal{G}_{[s,t]}^{\bf
Y}\right]=\exp\left(-\int_{s}^{t}E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]dY_{\sigma}+\frac{1}{2}\int_{s}^{t}\left(E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)^{2}d\sigma\right),$

where the notation of the l.h.t is given by (2.5), while the integral in the
r.h.t. denotes the Itô stochastic integral with respect to the semi-martingale
$(Y_{\widetilde{s}})_{\widetilde{s}\in[0,1]}$.

Proof: Let $s,t\in[0,1]$ be such that $s\leq t$. For $n\in\mathbb{N}$, first
define the $(\mathcal{G}_{[s,\sigma]}^{\bf Y})_{\sigma\in[s,t]}-$ stopping
time

$\tau_{n}:=\inf\left(\left\\{\widetilde{s}\in[s,t]:\int_{s}^{\widetilde{s}}\left(E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}\middle|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)^{2}d\sigma>n\right\\}\right)\wedge t,$

with the convention $\inf(\emptyset)=+\infty$, and further notice that since
$\tau_{n}$ is $\mathcal{G}_{[s,t]}^{\bf Y}-$ measurable, it can be written of
the form $\tau_{n}=\sigma_{n}\circ{\bf Y}$, $\mathbb{P}-a.s.$, where
$\sigma_{n}$ is a $\left(\mathcal{F}_{[s,\widetilde{s}]}^{p_{\bf
Y}}\right)_{\widetilde{s}\in[s,t]}-$ stopping time on the probability space
($C([0,1],\mathbb{R})$, $\mathcal{B}(C([0,1],\mathbb{R}))^{p_{\bf Y}}$,
$p_{\bf Y}$). Since ${\bf\widetilde{u}}\in L^{2}(\mathbb{P},H^{1})$, the
conditional Jensen inequality yields $\lim_{n\to+\infty}\tau_{n}=t$,
$\mathbb{P}-a.s.$. Let $\theta_{s,t}$ be a continuous and bounded
$\mathcal{F}_{[s,t]}^{p_{\bf Y}}-$ measurable real-valued function which is
defined on $C([0,1],\mathbb{R})$, and define
$\theta^{n}_{s,t}:=\theta_{s,t}\circ\rho_{n}$ where $\rho_{n}:\omega\in
C([0,1],\mathbb{R})\to\omega_{.\wedge\sigma_{n}(\omega)}\in
C([0,1],\mathbb{R})$. First notice that due to (3.6) the Cameron-Martin-
Girsanov theorem ([15]) ensures that
$(Y_{\widetilde{s}}-Y_{s})_{\widetilde{s}\in[s,t]}$ is an
$(\mathcal{A}_{\widetilde{s}})_{\widetilde{s}\in[s,t]}-$ Brownian motion
starting from $0$ at $s$ on the probability space
$(\Omega,\mathcal{A},\mathds{P}_{u})$, where $\mathds{P}_{u}$ denotes the
probability absolutely continuous (and also due to (3.6) equivalent) with
respect to $\mathbb{P}$ whose Radon-Nikodym derivative is defined by

$\frac{d\mathds{P}_{u}}{d\mathds{P}}=\mathcal{E}\left(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}}\right),\ \mathbb{P}-a.s..$

Therefore, we first obtain

$E_{\mathds{P}}\left[\theta_{s,t}^{n}\circ{\bf Y}\mathcal{E}\left(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}}\right)\right]=E_{\mathds{P}}\left[\theta_{s,t}^{n}\circ{\bf
B}\right].$ (3.7)

On the other hand, notice that

$Y_{\widetilde{s}}-Y_{s}=N_{\widetilde{s}}+\int_{s}^{\widetilde{s}}E_{\mathds{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]d\sigma,\ \forall\widetilde{s}\in[s,t],\mathbb{P}-a.s.,$ (3.8)

where we have defined

$N_{\widetilde{s}}:=B_{\widetilde{s}}-B_{s}+\int_{s}^{\widetilde{s}}\left(\dot{\widetilde{u}}_{\sigma}-E_{\mathds{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)d\sigma,\ \forall\widetilde{s}\in[s,t],\ \mathbb{P}-a.s..$
(3.9)

While (3.8) ensures that $(N_{\widetilde{s}})_{\widetilde{s}\in[s,t]}$ is
$(\mathcal{G}_{[s,\widetilde{s}]}^{\bf Y})_{\widetilde{s}\in[s,t]}-$ adapted
with $N_{s}=0$, $\mathbb{P}-a.s.$, since for $\widetilde{s}\in[s,t]$, we have
$\mathcal{G}_{[s,\widetilde{s}]}^{\bf Y}\subset\mathcal{A}_{\widetilde{s}}$,
from (3.9), the tower property yields

$E_{\mathbb{P}}\left[N_{\widetilde{t}}-N_{\widetilde{s}}|\mathcal{G}_{[s,\widetilde{s}]}^{\bf
Y}\right]=0,\ \mathbb{P}-a.s.,$

so that from Paul Lévy’s criteria we conclude that
$(N_{\widetilde{s}})_{\widetilde{s}\in[s,t]}$ is a
$(\mathcal{G}_{[s,\widetilde{s}]}^{Y})_{\widetilde{s}\in[s,t]}$ Brownian
motion starting from $0$ at time $s$ on $(\Omega,\mathcal{A},\mathbb{P})$.
Henceforth, and until the end of the proof, for $n\in\mathbb{N}$, let
$(M^{n}_{\widetilde{s}})_{\widetilde{s}\in[s,t]}$ be the stochastic process
defined on $(\Omega,\mathcal{A},\mathbb{P})$ by

$M_{\widetilde{s}}^{n}:=\exp\left(-\int_{s}^{\widetilde{s}}1_{[0,\tau_{n}]}(\sigma)E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]dN_{\sigma}-\frac{1}{2}\int_{s}^{\widetilde{s}}1_{[0,\tau_{n}]}(\sigma)\left(E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}\middle|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)^{2}d\sigma\right),$

$\forall{\widetilde{s}}\in[s,t],\mathbb{P}-a.s.,$ which is a non-negative
$(\mathcal{G}_{[s,\widetilde{s}]}^{\bf Y})_{\widetilde{s}\in[s,t]}-$
martingale due to the Novikov criterion. For $\theta_{s,t}^{n}$ as above, from
the Cameron-Martin-Girsanov theorem, now applied to the
$(\mathcal{G}_{[s,\widetilde{s}]})_{\widetilde{s}\in[s,t]}-$ Brownian motion
$(N_{\widetilde{s}})_{\widetilde{s}\in[s,t]}$, it therefore follows that

$E_{\mathbb{P}}[M_{t}^{n}\theta_{s,t}^{n}\circ{\bf
Y}]=E_{\mathbb{P}}[\theta_{s,t}^{n}\circ{\bf B}],$

which together with (3.7) yields

$E_{\mathbb{P}}[M_{t}^{n}\theta_{s,t}^{n}\circ{\bf
Y}]=E_{\mathds{P}}\left[\theta_{s,t}^{n}\circ{\bf
Y}\mathcal{E}\left(-\delta^{\bf B}{\bf\widetilde{u}}\right)\right].$ (3.10)

Now, notice that the optional sampling theorem (see [16], Theorem 6.11, p.34)
ensures that $(M_{t}^{n})_{n\in\mathbb{N}}$ is a uniformly integrable
$(\mathcal{G}_{[s,\tau_{n}]}^{Y})_{n\in\mathbb{N}}-$ martingale with respect
to the probability $\mathbb{P}$, $\mathcal{G}_{[s,\tau_{n}]}^{Y}$ denoting the
corresponding $\sigma-$ field of the $(\mathcal{G}_{[s,\widetilde{s}]}^{\bf
Y})_{\widetilde{s}\in[s,t]}-$ stopping time $\tau_{n}$ (see Definition 5.3. of
[16], p.22), $\forall n\in\mathbb{N}$, and that it is closed by the
$\mathbb{P}-$ integrable random variable

$M_{t}:=\exp\left(-\int_{s}^{t}E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]dN_{\sigma}-\frac{1}{2}\int_{s}^{t}\left(E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}\middle|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)^{2}d\sigma\right).$

Hence, by noticing that $t\in[0,1]$ is fixed, the martingale convergence
theorem (see Theorem 10 and Theorem 13 p.8-9 of [33]) ensures that the
discrete time martingale $(M_{t}^{n})_{n\in\mathbb{N}}$ converges
$\mathbb{P}-$ almost surely to $M_{t}\in L^{1}(\mathbb{P})$, while we have
$\rho_{n}({\bf Y})=Y_{.\wedge\tau_{n}}$, $\forall n\in\mathbb{N}$, where
$\rho_{n}({\bf Y}):=\rho_{n}\circ{\bf Y}$, $\forall n\in\mathbb{N}$. The
latter entails that $(\rho_{n}({\bf Y}))_{n\in\mathbb{N}}$ converges
$\mathbb{P}-a.s.$ to ${\bf Y}$, and then, by continuity, that
$(\theta_{s,t}^{n}({\bf Y}))_{n\in\mathbb{N}}$ converges almost surely to
$\theta_{s,t}({\bf Y})$, where $\theta_{s,t}^{n}({\bf
Y}):=\theta_{s,t}^{n}\circ{\bf Y}$, $\forall n\in\mathbb{N}$ (respectively
$\theta_{s,t}({\bf Y}):=\theta_{s,t}\circ{\bf Y}$). Since $M_{t}\in
L^{1}(\mathbb{P})$, and since $(\theta_{s,t}^{n})$ is uniformly bounded, the
Lebesgue convergence theorem (see [8]) yields

$\lim_{n\to+\infty}E_{\mathbb{P}}[M_{t}^{n}\theta_{s,t}^{n}\circ{\bf
Y}]=E_{\mathbb{P}}\left[M_{t}\theta_{s,t}\circ{\bf Y}\right],$ (3.11)

while

$\lim_{n\to+\infty}E_{\mathds{P}}\left[\theta_{s,t}^{n}\circ{\bf
Y}\mathcal{E}\left(-\delta^{\bf
B}{\bf\pi_{t}\widetilde{u}}\right)\right]=E_{\mathbb{P}}\left[\theta_{s,t}\circ{\bf
Y}\mathcal{E}\left(-\delta^{\bf B}{\bf\widetilde{u}}\right)\right]$ (3.12)

follows similarly.

Then, taking the limit in (3.10), from (3.11) and (3.12), it follows that

$E_{\mathbb{P}}\left[\theta_{s,t}\circ{\bf Y}\mathcal{E}\left(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}}\right)\right]=E_{\mathbb{P}}\left[M_{t}\theta_{s,t}\circ{\bf
Y}\right],$

so that, taking into account that $M_{t}$ is $\mathcal{G}_{[s,t]}^{\bf Y}-$
measurable, we get the result.

Figure 3.2. This figure illustrates Lemma 3.1. It represents the graph of the
function
$(s,t)\in[0,1]\times[0,1]\to\rho_{f_{Y}}(\\{K\in\text{Comp}([0,1]):K\subset[s\wedge
t,s\vee t]\\})\in\mathbb{R}_{+},$ for the particular case where
$u=\int_{0}^{.}\sin(s)ds$, $\mu_{W}-a.s.$ ; $s\wedge t$ (resp. $s\vee t$)
denotes $\min(s,t)$ (resp. $\max(s,t)$), $\forall s,t\in[0,1]$.

######  Lemma 3.1.

Given ${\bf u}\in L^{2}_{a}(\mu_{W},H)$, further define ${\bf Y}$ by (2.4),
where ${\bf F}^{\bf u}$ is given by (2.2), and set ${\bf\widetilde{u}}:={\bf
Y}-{\bf B}\in L^{2}_{a}(\mathds{P},H^{1})$. Further assume that $f_{\bf Y}\in
L^{2}(\mu_{W})$, where $f_{\bf Y}=\frac{dp_{\bf Y}}{dp_{\bf B}}$ denotes the
Radon-Nikodym derivative of the absolutely continuous probability $p_{\bf Y}$
with respect to $p_{\bf B}=\mu_{W}$. Then, for any $0\leq s\leq t\leq 1$ such
that (3.6) holds with $t$, we have

$\rho_{f_{Y}}(J([s,t]))=E_{\mathbb{P}}\left[\exp\left(\int_{s}^{t}E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{Y}\right]dY_{\sigma}-\frac{1}{2}\int_{s}^{t}\left(E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{\sigma}|\mathcal{G}_{[s,\sigma]}^{\bf
Y}\right]\right)^{2}d\sigma\right)\right],$

where

$J([s,t]):=\\{K\in\text{Comp}([0,1]):K\subset[s,t]\\}\in\mathcal{B}(\text{Comp}([0,1])),$
(3.13)

and where $\rho_{f_{\bf Y}}$ denotes the Tsirelson spectral measure (see
Definition 2.1) of $\mu_{W}$ with respect to $f_{\bf Y}$.

Proof: From the definitions, we obtain

$\rho_{f_{\bf Y}}(J([s,t]))=E_{\mathbb{P}}\left[E_{{p_{\bf
B}}}\left[\frac{dp_{\bf Y}}{dp_{\bf B}}\middle|\mathcal{F}_{[s,t]}^{{p_{\bf
B}}}\right]\circ{\bf Y}\right].$ (3.14)

Since for any real valued functions $\theta_{s,t}$ which is bounded and
$\mathcal{F}_{[s,t]}^{p_{\bf Y}}-$ measurable, the Cameron-Martin-Girsanov
theorem, whose application is ensured by (3.6), yields

$E_{\mathbb{P}}\left[\theta_{s,t}\circ{\bf Y}E_{{p_{\bf
B}}}\left[\frac{dp_{\bf Y}}{dp_{\bf B}}\middle|\mathcal{F}_{s,t}^{{p_{\bf
B}}}\right]\circ{\bf Y}\mathcal{E}(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}})\right]=E_{\mathbb{P}}\left[\theta_{s,t}\circ{\bf
Y}\right],$

we obtain

$E_{{p_{\bf B}}}\left[\frac{dp_{\bf Y}}{dp_{\bf
B}}\middle|\mathcal{F}_{[s,t]}^{{p_{\bf B}}}\right]\circ{\bf
Y}E_{\mathbb{P}}\left[\mathcal{E}(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}})\middle|\mathcal{G}_{[s,t]}^{\bf Y}\right]=1,\
\mathbb{P}-a.s..$ (3.15)

Substituting (3.15) into (3.14), the result follows from Proposition 3.1.

Remark:  By taking $s=0$, (3.15) boils down to the causal Monge-Ampère
equation as Proposition 3.4. of [12].

##  4\. First application : realizable inverse of a real-time noise filter

######  Theorem 4.1.

Let $t\in[0,1]$ and let ${\bf u}\in L^{2}_{a}(\mu_{W},H^{1})$ be such that
(3.6) holds. Further define ${\bf Y}$ by (2.4), set ${\bf\widetilde{u}}:={\bf
Y}-{\bf B}$, and denote by $f_{\bf Y}:=\frac{dp_{\bf Y}}{d\mu_{W}}$ the Radon-
Nikodym of the absolutely continuous probability $p_{\bf Y}$ with respect to
the Wiener measure. Moreover, assume that $f_{\bf Y}\in L^{2}(\mu_{W})$. Then,
we have

$\rho_{f_{\bf Y}}(J([0,t]))\leq
E_{\mathds{P}}\left[\exp\left(\int_{0}^{t}\dot{\widetilde{u}}_{s}dY_{s}-\frac{1}{2}\int_{0}^{t}\dot{\widetilde{u}}_{s}^{2}ds\right)\right],$
(4.16)

with equality, if and only if, there exists a
$\mathcal{B}(C([0,1],\mathbb{R}))^{p_{\bf Y}}/\mathcal{B}(C[0,1],\mathbb{R})-$
measurable function

${\bf H}^{u}:C([0,1],\mathbb{R})\to C([0,1],\mathbb{R})$

such that ${\bf H}^{u}_{s}$ is $\mathcal{F}_{s}^{p_{\bf Y}}-$ measurable,
where ${\bf H}^{u}_{s}:=W_{s}\circ{\bf H}^{u}$, $\forall s\in[0,t]$, and

$B_{s}={H}^{\bf u}_{s}({\bf Y}),\ \forall s\in[0,t],\mathbb{P}-a.s..$ (4.17)

Moreover, still in this case, for any ${\bf h}\in H^{1}$, and $t\in[0,1]$,
$\widehat{h}_{s}:=H_{s}^{u}({\bf Y^{h}})$, $\mathbb{P}-a.s.$ provides an
unbiased estimator for $h_{s}$ for any $s\in[0,t]$, that is,

$h_{s}=E_{\mathbb{P}}\left[\widehat{h}_{s}\right],\ \forall s\in[0,t],$ (4.18)

where ${\bf Y^{h}}$ is given by (2.3), with ${\bf X}:={\bf B}+{\bf h}$.

Proof: Lemma 3.1 and Proposition 3.1 yield

$\rho_{f_{\bf
Y}}(J([0,t]))=E_{\mathbb{P}}\left[\frac{1}{E_{\mathds{P}}\left[\mathcal{E}(-\delta^{\bf
B}\pi_{t}\widetilde{u})\middle|\mathcal{G}_{[0,t]}^{Y}\right]}\right],$

so that (4.16) follows from the conditional Jensen inequality, by taking into
account that $Y_{s}=B_{s}+\widetilde{u}_{s}$, $\forall s\in[0,t]$, outside
some $\mathbb{P}-$ negligible set. Since the inverse function is strictly
convex on $(0,+\infty)$, and the $\sigma-$ field $\mathcal{G}_{[0,t]}^{Y}$ is
$\mathbb{P}-$ complete, the equality occurs in (4.16), if and only if,
$\mathcal{E}(-\delta^{\bf B}\pi_{t}\widetilde{u})$ is
$\mathcal{G}_{[0,t]}^{Y}-$ measurable. From the proof of Lemma 3.1 (see
(3.15)), the latter condition is also equivalent to the following causal
Monge-Ampère equation

$E_{p_{\bf B}}\left[\frac{dp_{\bf Y}}{dp_{\bf
B}}\middle|\mathcal{F}_{t}^{\nu}\right]\circ{\bf Y}\mathcal{E}(-\delta^{\bf
B}\pi_{t}\widetilde{u})=1,\ \mathbb{P}-a.s..$ (4.19)

Then, it is routine to check that the latter is indeed equivalent to the
existence of ${\bf H}^{{\textbf{u}}}$, which satisfies the assertions of the
claim, by applying same methods as Proposition 3.1. of [12]. However, since
the equality is assumed here at time $t$, for the reader’s convenience, and
for the paper to be enough self-contained, we fix details of the proof for the
sake of completeness, and recall why (4.19) is indeed equivalent to the
existence of ${\bf H^{u}}$ as stated, which in particular satisfies to (4.17).
First assuming the existence of ${\bf H}^{u}$ as stated, since ${\bf Y}={\bf
F}^{u}\circ{\bf B}$ entails $\mathcal{G}_{[0,s]}^{\bf
Y}\subset\mathcal{G}_{[0,s]}^{\bf B}$, we first obtain
$\mathcal{G}_{[0,s]}^{{\bf B}}=\mathcal{G}_{[0,s]}^{\bf Y}$, $\forall
s\in[0,t]$. As a consequence, since $(\widetilde{u}_{s})$ is
$(\mathcal{G}_{[0,s]}^{\bf B})-$ adapted, so that together with (3.6), Itô’s
Lemma ensures that $(\mathcal{E}(-\delta^{\bf
B}{\bf\pi_{s}\widetilde{u}}))_{s\in[0,t]}$ is a $(\mathcal{G}_{[0,s]}^{\bf
B})_{s\in[0,t]}-$ martingale on $(\Omega,\mathcal{A},\mathbb{P})$, we get

$E_{\mathbb{P}}[\mathcal{E}(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u}})|\mathcal{G}_{[0,t]}^{\bf
Y}]=E_{\mathbb{P}}[\mathcal{E}(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{u})}|\mathcal{G}_{[0,t]}^{\bf
B}]=\mathcal{E}(-\delta^{\bf B}\pi_{t}{\bf\widetilde{u}}),\ \mathbb{P}-a.s.,$
(4.20)

while similarly as above (see Remark 3.1), the Cameron-Martin-Girsanov theorem
yields

$E_{p_{\bf B}}\left[\frac{dp_{\bf Y}}{dp_{\bf
B}}\middle|\mathcal{F}_{t}^{p_{\bf B}}\right]\circ{\bf
Y}E_{\mathbb{P}}[\mathcal{E}(-\delta^{\bf
B}{\bf\widetilde{u}})|\mathcal{G}_{[0,t]}^{\bf Y}]=1,\ \mathbb{P}-a.s..$
(4.21)

Substituting (4.20) into (4.21), we obtain (4.19).

On the converse, assume now that (4.19) holds, and recall that since the
Cameron-Martin-Girsanov theorem ensures that $p_{\bf Y}$ is absolutely
continuous with respect to $p_{\bf B}$, it is known (see [20], see also [10],
or [13]) that there exists a unique

$b^{p_{\bf Y}}:=\int_{0}^{.}v^{p_{\bf Y}}ds\in L^{2}_{a}(p_{\bf Y},H^{1}),$
(4.22)

which belongs to $L^{2}_{a}(p_{\bf Y},H^{1})$ since ${\bf\widetilde{u}}\in
L^{2}_{a}(\mathbb{P},H^{1})$ (see [4] formula $(75)$ and $(76)$ and see also
[13]), such that setting ${\bf
H}^{u}:=\mathds{I}_{C([0,1],\mathbb{R})}-b^{p_{\bf Y}}$, the stochastic
process $(H^{\bf u}_{s})_{s\in[0,1]}$ is a $(\mathcal{F}_{s}^{p_{\bf Y}})-$
brownian motion on $(C([0,1],\mathbb{R}),$ $\mathcal{B}(C([0,1],\mathbb{R})),$
$p_{\bf Y})$, where $H^{\bf u}_{s}:=W_{s}\circ{\bf H^{u}}$, $\forall
s\in[0,1]$, and

$E_{p_{\bf B}}\left[\frac{dp_{\bf Y}}{dp_{\bf
B}}\middle|\mathcal{F}_{\sigma}^{p_{\bf B}}\right]=\frac{dp_{\bf
Y}|_{\mathcal{F}_{\sigma}^{p_{\bf B}}}}{dp_{\bf
B}|_{\mathcal{F}_{\sigma}^{p_{B}}}}=\exp\left(\int_{0}^{\sigma}v_{s}^{p_{\bf
Y}}dW_{s}-\frac{1}{2}\int_{0}^{\sigma}(v_{s}^{p_{\bf Y}})^{2}ds\right),\
p_{\bf Y}-a.s.,$ (4.23)

$\forall\sigma\in[0,1]$. With this notation, the definitions easily yield

$b^{p_{\bf Y}}\circ{\bf
Y}=\int_{0}^{.}E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{s}|\mathcal{G}_{[0,s]}^{\bf
Y}\right]ds,\ \mathbb{P}-a.s.,$

where the r.h.t denotes the dual predictable projection of
$(\widetilde{u}_{s})$ on $(\mathcal{G}_{[0,s]}^{\bf Y})_{s\in[0,t]}$, as it is
well known (for instance see Proposition 3.1. of [23] for a direct proof).
Hence, taking the logarithm and then the expectation under $\mathbb{P}$ in
both terms of (4.19) we get

$(\pi_{t}b^{p_{\bf Y}})\circ{\bf
Y}=\int_{0}^{.}1_{[0,t]}(s)E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{s}|\mathcal{G}_{[0,s]}^{\bf
Y}\right]ds=\pi_{t}\widetilde{u},\ \mathbb{P}-a.s.,$

where we used that since the $L^{2}(\mathbb{P},H^{1})$ norm of $\pi_{t}u$
coincides with the norm of its projection
$\int_{0}^{.}1_{[0,t]}(s)E_{\mathbb{P}}\left[\dot{\widetilde{u}}_{s}|\mathcal{G}_{[0,s]}^{\bf
Y}\right]ds$ on the closed linear subspace of the ${\bf
k}:=\int_{0}^{.}k_{\widetilde{s}}d\widetilde{s}\in
L^{2}_{a}(\mathbb{P},H^{1})$ such that $(k_{\widetilde{s}})$ is
$(\mathcal{G}_{[0,\widetilde{s}]}^{\bf Y})_{\widetilde{s}\in[0,1]}-$ adapted,
both elements of $L^{2}_{a}(\mathbb{P},H^{1})$ necessarily coincide. Thus, we
get

${\bf H}_{s}^{u}\circ{\bf F}^{u}\circ{\bf B}=B_{s},\ \forall s\leq t,\
\mathbb{P}-a.s.,$ (4.24)

so that, taking into account once again that ${\bf Y}={\bf F^{u}}\circ{\bf
B}$, $\mathbb{P}-a.s.$, we obtain (4.17).

We now turn to the proof of (4.18). Thus, assume that we are in the equality
case in the above inequality, and still denote by ${\bf H}^{u}$ the previous
map. We then take ${\bf h}\in H^{1}$, and denote by ${\bf Y}^{h}:={\bf
F}^{u}({\bf B}+{\bf h})$. First observe that

${\bf Y}^{h}={\bf F}^{u}\circ({\bf B}+{\bf h})={\bf F}^{u}\circ{\bf
T_{h}}\circ{\bf B},$

where ${\bf T_{h}}(\omega):={\bf\omega}+{\bf h}\in C([0,1],\mathbb{R})$,
$\forall\omega\in C([0,1],\mathbb{R})$. Therefore, taking into account that
$p_{T_{h}({\bf B})}\sim\mu_{W}$, which follows from the Cameron-Martin-Segal
theorem (see [28], [35], Theorem 2.2. p. 339 of [34]), from (4.24) we get

$H_{s}^{\bf u}\circ{\bf F^{u}}=W_{s},\forall s\leq t,p_{\bf T_{h}(B)}-a.s.,$

from which we deduce that

$H_{s}^{u}\circ{\bf Y}^{\bf h}=H_{s}^{\bf u}\circ{\bf F^{u}}\circ{\bf
T_{h}}\circ{\bf B}=B_{s}+h_{s},\forall s\leq t,\mu_{W}-a.s..$

By taking the expectation under $\mathbb{P}$, and since $B_{0}=0$, (4.18)
follows.

##  5\. Second application : the innovation problem of filtering

The following result may be seen as complementary to the approaches unrolled
in [2], [39], [41], [42] and [26] ; for related topics, see also [24] or [25].
It yields a criterion for the innovation conjecture to hold which is based on
the Tsirelon spectral measure.

######  Corollary 5.1.

Let $t\in[0,1]$ and let ${\bf u}\in L^{2}_{a}(\mu_{W},H^{1})$ be such that
(3.6) holds. Further define ${\bf Y}$ by (2.4), set ${\bf\widetilde{u}}:={\bf
Y}-{\bf B}$, denote by $f_{\bf Y}:=\frac{dp_{\bf Y}}{d\mu_{W}}$ the Radon-
Nikodym of the absolutely continuous probability $p_{\bf Y}$ with respect to
the Wiener measure $\mu_{W}$, and assume that $f_{\bf Y}\in L^{2}(\mu_{W})$.
Then, the innovation conjecture is satisfied by $(Y_{s})$ until time
$t\in[0,1]$, that is,

$(\mathcal{G}_{[0,s]}^{{\bf
B^{\widetilde{u}}}})_{s\in[0,t]}=(\mathcal{G}_{[0,s]}^{\bf Y})_{s\in[0,t]},$
(5.25)

if and only if, there exists a ${\bf\xi}\in L^{2}_{a}(\mu_{W},H^{1})$ which
meets the following assumptions :

  1. (i)

$p_{\bf Y}|_{\mathcal{F}_{t}^{\mu_{W}}}=p_{\bf
Z}|_{\mathcal{F}_{t}^{\mu_{W}}},$ where ${\bf Z}:={\bf F}^{\bf\xi}({\bf B})$,
with ${\bf F}^{\bf\xi}:=\mathds{I}_{C([0,1],\mathbb{R})}+{\bf\xi}$.

  2. (ii)

$E_{\mathbb{P}}\left[\mathcal{E}(-\delta^{\bf
B}\pi_{t}{\bf\widetilde{\xi}})\right]=1$, where
${\bf\widetilde{\xi}}:={\bf\xi}\circ{\bf B}={\bf Z}-{\bf B}\in
L^{2}_{a}(\mathds{P},H^{1})$.

  3. (iii)

The following formula holds :

$\rho_{f_{\bf
Y}}(J([0,t]))=E_{\mathds{P}}\left[\exp\left(\int_{0}^{t}\dot{\widetilde{\xi}}_{s}dZ_{s}-\frac{1}{2}\int_{0}^{t}\dot{\widetilde{\xi}}_{s}^{2}ds\right)\right],$
(5.26)

where

$J([0,t]):=\\{K\in Comp([0,1]):K\subset[0,t]\\}.$

Proof: First assuming the existence of a ${\bf\xi}\in
L^{2}_{a}(\mu_{W},H^{1})$ as above, notice that $(i)$ implies

$E_{\mu_{W}}\left[\frac{dp_{\bf
Y}}{d\mu_{W}}\middle|\mathcal{F}_{[0,t]}^{\mu_{W}}\right]=E_{\mu_{W}}\left[\frac{dp_{\bf
Z}}{d\mu_{W}}\middle|\mathcal{F}_{[0,t]}^{\mu_{W}}\right],\ \mu_{W}-a.s.,$

so that

$\rho_{f_{\bf Y}}(J([0,t]))=\rho_{f_{\bf Z}}(J([0,t])),$

where $f_{\bf Z}:=\frac{dp_{\bf Z}}{d\mu_{W}}$. Thus, by $(ii)$ and $(iii)$,
Theorem 4.1 applied to ${\bf Z}$ yields the existence of ${\bf H}^{\bf\xi}$
such that ${\bf H}^{\xi}_{s}\circ{\bf Z}=B_{s}$, $\forall s\leq t$, outside
some $\mathbb{P}-$ null set. Since $Z_{s}={\bf F}^{\xi}_{s}\circ{\bf B}$, for
all $s\leq t$, $\mathbb{P}-a.s.$, it entails

$H^{\bf\xi}_{s}\circ{\bf F}^{\xi}=W_{s},\forall s\leq t,\ \mu_{W}-a.s.,$
(5.27)

which notably implies that $(H^{\bf\xi}_{s})_{s\in[0,t]}$ is a canonical
Brownian motion until time $t$, on the canonical probability space endowed
with the probability $p_{Z}$.

Let $A:=\\{\omega\in C([0,1],\mathbb{R}):H^{\bf\xi}_{s}\circ{\bf
F}^{\xi}=W_{s},\forall s\leq t\\},$ $\widetilde{A}:=\\{\omega\in
C([0,1],\mathbb{R}):F^{\bf\xi}_{s}\circ{\bf H}^{\bf\xi}\circ{\bf
F}^{\bf\xi}=F^{\bf\xi}_{s},\forall s\leq t\\},$ and $C=\\{\omega\in
C([0,1],\mathbb{R}):F^{\bf\xi}_{s}\circ{\bf H}^{\bf\xi}=W_{s},\forall s\leq
t\\}$. Since $A\subset\widetilde{A}$ and $\mu_{W}(A)=1$, by noticing that
$C\in\mathcal{F}_{t}^{p_{\bf Z}}$, we get

$1\geq p_{\bf Z}(C)={\bf
F^{\xi}}_{\star}\mu_{W}(C)=\mu_{W}(\widetilde{A})\geq\mu_{W}(A)=1,$

${\bf F^{\xi}}_{\star}{\mu_{W}}$ denoting the pushforward of the probability
$\mu_{W}$ by ${\bf F^{\xi}}$, so that

$F_{s}^{{\bf\xi}}\circ{\bf H}^{\bf\xi}=W_{s},\ \forall s\in[0,t],\ p_{\bf
Z}-a.s..$ (5.28)

From this, we deduce that $H_{s}^{\bf\xi}=W_{s}-\xi_{s}\circ{\bf H^{\xi}}$,
$\forall s\leq t$, $p_{Z}|_{\mathcal{F}_{t}}-a.s.$. Since $p_{\bf
Y}|_{\mathcal{F}_{t}^{\mu_{W}}}=p_{\bf Z}|_{\mathcal{F}_{t}^{\mu_{W}}}$ and
$p_{\bf Y}\sim\mu_{W}$ (equivalent), the uniqueness of the semi-martingale
decomposition of $(W_{s})_{s\in[0,t]}$ entails, $\xi_{s}\circ{\bf
H^{\xi}}=b_{s}^{p_{\bf Y}}$, $\forall s\leq t$ $p_{\bf Y}-a.s.$ (and then
$p_{\bf B}-a.s.$), where ${\bf b^{p_{\bf Y}}}:=\int_{0}^{.}v_{s}^{p_{\bf
Y}}ds$ is as in (4.22), so that in particular $(v_{s}^{p_{\bf Y}})$ satisfies
to (4.23) (finite variation part), and
$B^{\widetilde{u}}_{s}=H_{s}^{\bf\xi}\circ{\bf Y}$, $\forall s\leq t$,
$\mathbb{P}-a.s.$, which easily follows from the identification of the
martingale parts together with the definition of the innovation process
$(B^{\widetilde{u}}_{s})$. Hence (5.28) yields $F^{\xi}_{s}\circ{\bf
B^{\widetilde{u}}}=Y_{s},$ $\forall s\in[0,t]$, $\mathbb{P}-a.s.$, and
therefore $\mathcal{G}_{[0,s]}^{\bf Y}\subset\mathcal{G}_{[0,s]}^{{\bf
B^{\widetilde{u}}}}$, $\forall s\in[0,t]$. Since $(B^{\widetilde{u}}_{s})$ is
known to be a $(\mathcal{G}_{[0,s]}^{\bf Y})_{s\in[0,1]}-$ brownian motion,
this proves (5.25).

Conversely, we now assume that (5.25) holds, and define the continuous
stochastic processes $(\widetilde{Y}_{s})$ and $(\widetilde{X}_{s})$ by

$\widetilde{Y}_{s}=1_{[0,t)}(s)Y_{s}+1_{[t,1]}(s)(Y_{t}+B_{s}-B_{t}),\ \forall
s\in[0,1],\mathbb{P}-a.s.,$

and

$\widetilde{X}_{s}=1_{[0,t)}(s)B_{s}^{\bf\widetilde{u}}+1_{[t,1]}(s)(B_{t}^{\bf\widetilde{u}}+B_{s}-B_{t})\
\forall s\in[0,1],\mathbb{P}-a.s.$

Then, (5.25) implies

$(\mathcal{G}_{[0,s]}^{{\bf\widetilde{Y}}})_{s\in[0,1]}=(\mathcal{G}_{[0,s]}^{\widetilde{X}})_{s\in[0,1]}.$

This ensures the existence of two maps
${\bf\widetilde{I}},{\bf\widetilde{J}}:C([0,1],\mathbb{R})\to
C([0,1],\mathbb{R})$ such that ${\bf\widetilde{I}}$ is
$\mathcal{B}^{p_{\widetilde{X}}}(C([0,1],\mathbb{R}))/\mathcal{B}(C([0,1],\mathbb{R}))$
\- measurable, while ${\bf\widetilde{J}}$ is
$\mathcal{B}^{p_{{\bf\widetilde{Y}}}}(C([0,1],\mathbb{R}))/$
$\mathcal{B}(C([0,1],\mathbb{R}))$ \- measurable,
${\bf\widetilde{J}}\circ{\bf\widetilde{I}}=\mathds{I}_{C([0,1],\mathbb{R})}$,
$p_{\widetilde{X}}-a.s.$,
${\bf\widetilde{I}}\circ{\bf\widetilde{J}}=\mathds{I}_{C([0,1],\mathbb{R})}$,
$p_{{\bf\widetilde{Y}}}-a.s.$, ${\bf\widetilde{I}}$ (resp.
${\bf\widetilde{J}}$) is $(\mathcal{F}_{t}^{p_{\widetilde{X}}})-$ (resp.
$(\mathcal{F}_{t}^{p_{{\bf\widetilde{Y}}}})-$) adapted, and
${\bf\widetilde{Y}}={\bf\widetilde{I}}\circ{\bf\widetilde{X}}$ (resp.
${\bf\widetilde{X}}={\bf\widetilde{J}}\circ{\bf\widetilde{Y}}$),
$\mathbb{P}-a.s.$ (for instance, see Proposition 1.3. of [23]). Hence, we
obtain

$Y_{s}=\widetilde{Y}_{s}=\widetilde{I}_{s}\circ{\bf\widetilde{X}}=\widetilde{I}_{s}\circ{\bf
B^{\widetilde{u}}},\ \forall s\leq t,\ \mathbb{P}-a.s.,$ (5.29)

and
$B_{s}^{\widetilde{u}}=\widetilde{X}_{s}=\widetilde{J}_{s}\circ{\bf\widetilde{Y}}=\widetilde{J}_{s}\circ{\bf
Y}$, $\forall s\leq t$, $\mathbb{P}-a.s.$. On the other hand, since
$B^{\widetilde{u}}=(\mathds{I}_{C([0,1],\mathbb{R})}-b^{p_{\bf Y}})\circ{\bf
Y}$, where $b^{p_{\bf Y}}$ is defined by (4.22), we get
$\widetilde{J}_{s}\circ{\bf Y}=Y_{s}-b_{s}^{p_{\bf Y}}\circ{\bf Y}$, $\forall
s\in[0,t]$, $\mathbb{P}-a.s.$, so that $\widetilde{J}_{s}=W_{s}-b_{s}^{p_{\bf
Y}}$, $\forall s\in[0,t]$, $p_{\bf Y}|_{\mathcal{F}_{t}^{p_{\bf Y}}}-$ a.s.,
which implies that
$W_{s}=\widetilde{J}_{s}\circ{\bf\widetilde{I}}=\widetilde{I}_{s}-\xi_{s}$,
$\forall s\leq t$, $\mu_{W}-$a.s., where we have defined
$\xi:=(\pi_{t}b^{p_{\bf
Y}})\circ{\bf\widetilde{I}}=\int_{0}^{.}1_{[0,t]}(s)v_{s}^{p_{\bf
Y}}\circ{\bf\widetilde{I}}ds$. Further notice that
$p_{\bf\widetilde{X}}=\mu_{W}$, so that ${\bf\xi}\in L^{2}_{a}(\mu_{W},H^{1})$
follows from $b^{p_{\bf Y}}\in L^{2}_{a}(p_{\bf Y},H^{1})$. Then, we define
${\bf F^{\xi}}:=\mathds{I}_{C([0,1],\mathbb{R})}+{\bf\xi}$,
${\bf\widetilde{\xi}}:={\bf\xi}\circ{\bf B}$, and ${\bf Z}:={\bf F^{\xi}}({\bf
B})={\bf B}+{\bf\widetilde{\xi}}$. By definition, we obtain
$Z_{s}=\widetilde{I}_{s}({\bf B})$, $\forall s\leq t$, $\mathbb{P}-a.s.$, so
that in particular, $\widetilde{\xi}_{s}=b_{s}^{p_{\bf Y}}\circ{\bf Z}$,
$\forall s\leq t$, $\mathbb{P}-a.s.$, while we obtain
$\widetilde{I}_{s}=W_{s}+\xi_{s}$, $\forall s\leq t$, $\mu_{W}-a.s.$. Thus,
taking into account that $p_{\bf B^{{\widetilde{u}}}}=p_{\bf B}$, (5.29)
entails that $p_{\bf Z}|_{\mathcal{F}_{t}^{\mu_{W}}}=p_{{\bf
Y}}|_{\mathcal{F}_{t}^{\mu_{W}}}$, while $(\mathcal{G}_{[0,s]}^{\bf
Z})_{s\in[0,t]}\subset(\mathcal{G}_{[0,s]}^{\bf B})_{s\in[0,t]}$. On the other
hand, due to the Camron-Martin-Girsanov theorem, (3.6) entails $p_{\bf
Y}|_{\mathcal{F}_{t}^{\mu_{W}}}\sim\mu_{W}|_{\mathcal{F}_{t}^{\mu_{W}}}$, so
that (4.23) yields

$\frac{{dp_{\bf
Y}}|_{\mathcal{F}_{t}^{\mu_{W}}}}{d{p_{B}}|_{\mathcal{F}_{t}^{\mu_{W}}}}=\exp\left(\int_{0}^{t}v_{s}^{p_{\bf
Y}}dW_{s}-\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf Y}})^{2}ds\right),\
{\mu_{W}}|_{\mathcal{F}_{t}^{\mu_{W}}}-a.s..$ (5.30)

Therefore, we get

$\displaystyle
E_{\mathbb{P}}\left[\mathcal{E}(-\delta^{B}\pi_{t}\widetilde{\xi})\right]$
$\displaystyle=$ $\displaystyle
E_{\mathbb{P}}\left[\exp\left(-\int_{0}^{t}v_{s}^{p_{\bf Y}}\circ{\bf
Z}dB_{s}-\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf Y}}\circ{\bf
Z})^{2}ds\right)\right]$ $\displaystyle=$ $\displaystyle
E_{\mu_{W}}\left[\exp\left(-\int_{0}^{t}v_{s}^{p_{\bf
Y}}\circ\widetilde{I}_{s}d\widetilde{I}_{s}+\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf
Y}}\circ\widetilde{I}_{s})^{2}ds\right)\right]$ $\displaystyle=$
$\displaystyle E_{p_{\bf Z}}\left[\exp\left(-\int_{0}^{t}v_{s}^{p_{\bf
Y}}dW_{s}+\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf Y}})^{2}ds\right)\right]$
$\displaystyle=$ $\displaystyle E_{p_{\bf
Y}}\left[\exp\left(-\int_{0}^{t}v_{s}^{p_{\bf
Y}}dW_{s}+\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf Y}})^{2}ds\right)\right]$
$\displaystyle=$ $\displaystyle
E_{\mu_{W}|_{\mathcal{F}_{t}^{\mu_{W}}}}\left[\frac{dp_{\bf
Y}|_{\mathcal{F}_{t}^{\mu_{W}}}}{d\mu_{W}|_{\mathcal{F}_{t}^{\mu_{W}}}}\exp\left(-\int_{0}^{t}v_{s}^{p_{\bf
Y}}dW_{s}+\frac{1}{2}\int_{0}^{t}(v_{s}^{p_{\bf Y}})^{2}ds\right)\right]$
$\displaystyle=$ $\displaystyle 1,$

where the last inequality follows from (5.30). Thus, ${\bf\xi}$ satisfies to
$(i)$ and $(ii)$. Finally, notice that for $s\leq t$, we have
$\widetilde{J}_{s}\circ{\bf
Z}=\widetilde{J}_{s}\circ{\bf\widetilde{I}}\circ{\bf B}=W_{s}\circ{\bf
B}=B_{s}$, $\forall s\leq t$, $\mathbb{P}-a.s.$, so that
$(\mathcal{G}_{[0,s]}^{\bf B})_{s\in[0,t]}=(\mathcal{G}_{[0,s]}^{\bf
Z})_{s\in[0,t]}$. As a consequence, we are in the equality case of Theorem 4.1
applied to ${\bf\xi}$, from which we obtain that the latter satisfies to
$(iii)$. This achieves the proof.

##  6\. Probabilistic perspective by normalization, spectral random sets, and
further developments

To clarify the statements in view of applications to the above problems of
filtering, we did not require the Tsirelson spectral measures to be
necessarily normalized. However, taking a finite measure of finite mass one,
as it is usual in stochastic analysis, notably increases the insight on this
kind of studies, as it is the case in Example 2.1 and Example 2.2 above.
Moreover, we notice that the spectral measures investigated above were
determined by the laws of the associated processes. This motivates to
introduce the following notation to shorten the statement of Corollary 6.1
below :

######  Definition 6.1.

Let $\nu\in{\bf M}_{1}(C([0,1],\mathbb{R}))$ be a Borel probability measure
which is absolutely continuous with respect to the Wiener measure
($\nu<<\mu_{W}$), whose Radon-Nikodym derivative $f_{\bf
Y}:=\frac{d\nu}{d\mu_{W}}$ is square-integrable (i.e.
$E_{\mu_{W}}\left[(\frac{d\nu}{d\mu_{W}})^{2}\right]<+\infty$). Then, we
denote by $\mathbb{P}_{\nu}$ the normalized Tsirelson spectral measure
associated with the probability $\nu$, which is the Borel probability measure
$\mathbb{P}_{\nu}$ on $Comp([0,1])$ given by

$\mathbb{P}_{\nu}(\mathcal{K}):=\frac{\rho_{\frac{d\nu}{d\mu_{W}}}(\mathcal{K})}{\rho_{{\frac{d\nu}{d\mu_{W}}}}(\text{Comp}([0,1]))},\
\forall\mathcal{K}\in\mathcal{B}(\text{Comp}([0,1])),$ (6.31)

i.e.,

$\mathbb{P}_{\nu}:=\rho_{g_{\nu}}\in{\bf M}_{1}(Comp([0,1])),$

where we have set

$g_{\nu}:={\frac{\frac{d\nu}{d\mu_{W}}}{\sqrt{E_{\mu_{W}}[({\frac{d\nu}{d\mu_{W}})^{2}]}}}},$

and where $\rho_{g_{\nu}}$ denotes the Tsirelson spectral measure (see
Definition (2.1)) of $\mu_{W}$ with respect to $g_{\nu}\in L^{2}(\mu_{W})$.

Notice that the latter is well defined since from Jensen’s inequality, it
follows that $\|\frac{d\nu}{d\mu_{W}}\|_{L^{2}(\mu_{W})}>0$, for any
$\nu<<\mu_{W}$ (absolutely continuous). In this way, under the assumptions of
Definition 6.1, any such $\nu$ is associated to a random compact set ; in the
limit case where $\nu=\mu_{W}$ this compact set is a.s. empty. Then, we obtain
the following ersatz for part of Theorem 4.1, which notably provides a
probabilistic insight on the previous criterion of existence of a realizable
inverse to a real-time noise filter, thanks to some spectral random set :

######  Corollary 6.1.

Given ${\bf u}\in L^{2}_{a}(\mu_{W},H^{1})$ which satisfies to (3.6) with
$t=1$, further define ${\bf Y}$ by (2.4), and set ${\bf\widetilde{u}}:={\bf
Y}-{\bf B}$. Further assume that we have $\frac{d\nu}{d\mu_{W}}\in
L^{2}(\mu_{W})$, the latter denoting the Radon-Nikodym derivative of the
absolutely continuous probability $\nu:=p_{\bf Y}$ with respect to the Wiener
measure $\mu_{W}$.Then, the following lower bound holds

$\mathbb{P}_{\nu}(\\{\emptyset\\})\geq\frac{1}{E_{\mathds{P}}\left[\exp\left(\int_{0}^{1}\dot{\widetilde{u}}_{s}dY_{s}-\frac{1}{2}\int_{0}^{1}\dot{\widetilde{u}}_{s}^{2}ds\right)\right]},$

$\mathbb{P}_{\nu}$ denoting the normalized Tsirelson spectral measure (6.31)
of the probability $p_{\bf Y}$. Moreover, there exists ${\bf H}^{\bf
u}:C([0,1],\mathbb{R})\to C([0,1],\mathbb{R})$ which satisfies to the
hypothesis of Theorem 4.1 with $t=1$, if and only if,

$\mathbb{P}_{\nu}(\\{\emptyset\\})=\frac{1}{E_{\mathds{P}}\left[\exp\left(\int_{0}^{1}\dot{\widetilde{u}}_{s}dY_{s}-\frac{1}{2}\int_{0}^{1}\dot{\widetilde{u}}_{s}^{2}ds\right)\right]}.$

Proof: Since $Comp([0,1])=J([0,1])$ (see (3.13) above), from Definition 6.1,
we get $\mathbb{P}_{\nu}(\\{\emptyset\\})=\frac{\rho_{f_{\bf
Y}}(\\{\emptyset\\})}{\rho_{f_{\bf Y}}(J([0,1]))},$ where $f_{\bf
Y}:=\frac{dp_{\bf Y}}{d\mu_{W}}$. On the other hand, (2.1) implies

$\rho_{f_{\bf Y}}(\\{\emptyset\\})=\left(E_{\mu_{W}}\left[f_{\bf
Y}\right]\right)^{2}=1,$

since $f_{\bf Y}$ is the Radon-Nikodym derivative of a probability measure
with respect to the probability $\mu_{W}$. Thus, the result follows from
Theorem 4.1.

Remark:  Within this viewpoint, the above results can be extended in several
fashions to a class of Borel probability measures absolutely continuous with
respect to the Wiener measure whose Radon-Nikodym derivatives are also square
integrable, which will be done in some forthcoming works.

.

## References

  * [1] Airault, H., Malliavin, P. : Intégration et analyse de Fourier, Probabilités et analyse gaussienne. $2^{e}$ édition revue et augmentée. MASSON, Paris Milan Barcelone (1994)
  * [2] Allinger, D., Mitter S.K. : New results on the innovations problem for nonlinear filtering. Stochastics 4, no.4, 339$-$348 (1980)
  * [3] Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Stochastic Modeling And Applied Probability 60. Springer (2009)
  * [4] Binia, J., Zakai, M., Ziv, J. : On the epsilon-entropy and the rate-distortion function of certain non-Gaussian processes. IEEE Transactions on Information Theory, vol. IT$-$20, no. 4, pp. 517-524 (1974).
  * [5] Cameron, R.H., Martin, W.T. : Transformation of Wiener integral under translation. Ann. Math. 45 (1944)
  * [6] Cameron, R.H., Martin, W.T.: The transformation of Wiener integrals by nonlinear transformation. Trans. Am. Math. Soc. 66 253-283 (1949)
  * [7] Cont, R., Tankov, P. : Financial Modelling with Jump Processes Chapman and Hall/CRC Boca Ra ton London New York Washington, D.C. (2004)
  * [8] Dellacherie, C. and Meyer, P. A. : Probabilités et Potentiel Ch. 1 à 4. Paris, Hermann. (1975)
  * [9] Dellacherie, C., Maisonneuve, B., Meyer, P. A. : Probabilités et Potentiel V, processus de Markov, Ch. 17 à 24. Paris, Hermann. (1992)
  * [10] Duncan, T.E. : Evaluation of likelihood functions. Inform. Contr., vol. 13, pp.62-74, (1968)
  * [11] Feyel, D., Üstünel, A.S. : Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space. Probab. Theory Relat. Fields 128, 347?385 (2004).
  * [12] Feyel, D., Üstünel, A.S., Zakai, M. : The realization of positive random variables via absolutely continuous transformations of measure on Wiener space. Probab. Surveys 3 170 - 205 (2006).
  * [13] Föllmer, H. : Random fields and diffusion processes. In: Hennequin PL. (eds)  École d’été de Saint Flour XV$-$XVII, 1985$-$1987. Lecture Notes in Mathematics, vol 1362. Springer, Berlin, Heidelberg (1988)
  * [14] Garban, C. : Oded Schramm’s contributions to noise sensitivity. Ann. Probab., Vol. 39, No 5, 1702-1767 (2011)
  * [15] Girsanov, I.V. : On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probab. Appl 5 285-301 (1960)
  * [16] Ikeda, N., Watanabe, S. : Stochastic Differential Equations and Diffusion Processes. Second Edition. North-Holland Mathematical Library, Volume 24. North Holland Publishing Company, Amsterdam (Kodansha Ltd., Tokyo) (1989)
  * [17] Itô, K. : Stochastic integral. Proc. Imp. Acad. Tokyo 20, no. 8, 519-524 (1944)
  * [18] Kailath, T. : Some Extensions of the Innovations Theorems. B.S.T.J., 50, p.1487-1494 (1971)
  * [19] Kailath, T., Zakai, M. : Absolute continuity and Radon-Nikodym derivatives for certain measures relative to Wiener measure. The Annals of Mathematical Statistics, Vol. 42, No 1, 130-140 (1971)
  * [20] Kailath, T. : The structure of Radon-Nikodym derivatives with respect to Wiener and related measures. The Annals of Mathematical Statistics, Vol. 42, No 3, 1054-1067 (1971)
  * [21] Koopmans, L.H. : The spectral analysis of time series. Academic press. (1974)
  * [22] Kuo, H. : Gaussian Measures in Banach Spaces . Lecture Notes in Math. Springer$-$Verlag Berlin Heidelberg New York (1975)
  * [23] Lassalle, R., Cruzeiro, A.B. : An intrinsic calculus of variations for functionals of laws of semi-martingales. Stochastic Processes and their Applications, Volume 129, Issue 10, 3585-3618 (2019)
  * [24] Lassalle, R. : Local invertibility of adapted shifts on Wiener space, under finite energy condition, Stochastics Volume 85, 2013 - Issue 6, 987-996. (2013)
  * [25] Lassalle, R., Üstünel, A.S. : Local Invertibility of Adapted Shifts on Wiener Space, and Related Topics. In : Viens F., Feng J., Hu Y., Nualart E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol. 34. Springer, Boston, MA (2013)
  * [26] Lassalle, R. : Invertibility of adapted perturbations of the identity on abstract Wiener space. In Journal of Functional Analysis, volume 262, Issue 6, 2734$-$2776 (2012)
  * [27] Le Jan, Y., Raimond, O. : Flows, coalescence and noise. Ann. Probab. 32 (2) 1247 - 1315 (2004)
  * [28] Malliavin, P. : Stochastic Analysis. Springer$-$Verlag Berlin Heidelberg, Grundlehren der mathematischen Wissenschaften, volume 313 (1997)
  * [29] Meyer, P.-A. : Sur un problème de filtration. Séminaire de probabilités de Strasbourg, Tome 7, pp. 223-247 (1973)
  * [30] Meyer, P.-A. : Quantum probability for probabilists. Second edition. Lecture notes in mathematics 1538. Springer-Verlag Berlin Heidelberg (1995)
  * [31] Nourdin, I. Peccati, G. Normal approximations with Malliavin Calculus. Cambridge University Press. 192. (2012)
  * [32] Privault, N. : Stochastic analysis in discrete and continuous settings Lecture notes in mathematics 1982. Springer-Verlag Berlin Heidelberg (2009)
  * [33] Protter, P.E. : Stochastic integration and differential equations. Stochastic Modeling and Applied Probability 21. Second edition, version 2.1. Springer Berlin Heidelberg New York. Second edition, 3rd Printing (2005)
  * [34] Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Springer Grundlehren der mathematischen Wissenshaften 293. 3rd edition. Springer Berlin Heidelberg New York (1999)
  * [35] Stroock, D.W. : Probability theory, an analytic view. Cambridge university press, New York. second edition (2011)
  * [36] Tsirelson, B. : Within and beyond the reach of Brownian innovation, Documenta Mathematica Extra Volume ICM, III, 311-326. (1998)
  * [37] Tsirelson, B. : Nonclassical stochastic flows and continuous products, Probability Surveys, 1 (2004), 173-298.
  * [38] Tsirelson, B. : Scaling Limit, Noise, Stability. In: Picard, J. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1840. Springer, Berlin, Heidelberg. (2004)
  * [39] Üstünel, A.S., Zakai, M. : Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139, 207-234 (2007).
  * [40] Üstünel, A.S., Zakai, M. : Transformation of Measure on Wiener Space Springer monographs in Mathematics. Springer-Verlag Berlin Heidelberg (2000)
  * [41] Üstünel, A.S. : Entropy, invertibility and variational calculus of adapted shifts on Wiener space. J. Funct. Anal. 257, no. 11, 3655-3689 (2009)
  * [42] Üstünel, A.S. : Entropic solution of the innovation conjecture of T. Kailath. Kyoto J. Math. 55(3): 555-566. (2015)
  * [43] Warren, J., Watanabe, S. : On Spectra of Noises Associated with Harris Flows. Adv. Stud. Pure Math., 2004: 351-373 (2004)
  * [44] Watanabe, S. : Itô’s theorems on chaos expansions and martingale representations. Journal of the Mathematical Society of Japan. Vol. 67, No. 4 (2015)
  * [45] Wiener, N. : Differential spaces. 131-174 J. Math. Phys. 2 (1923)


# Symmetry groups of pfaffians of symmetric matrices

A.S. Dzhumadil’daev  Kazakh-British Technical University, Tole bi 59, Almaty,
050000,Kazakhstan<EMAIL_ADDRESS>
###### Abstract.

We prove that symmetry group of the pfaffian polynomial of a symmetric matrix
is a dihedral group. We calculate pfaffians of symmetric matrices with
components $(x_{i}-x_{j})^{2}$ and $\cos(x_{i}-x_{j})$ for $i<j.$

###### Key words and phrases:

Pfaffians, determinants, symmetry groups, dihedral invariants

###### 2000 Mathematics Subject Classification:

15A15.

##  1\. Introduction

In first glance the title of the paper is not correct. Pfaffians usually are
connected with determinants of skew-symmetric matrices. If $a_{i,j}=-a_{j,i},$
for any $1\leq i,j\leq 2n,$ then determinant of skew-symmetric matrix
$A=(a_{i,j})$ is complete square and square root of determinant is pfaffian,

$det\,A=(pf_{2n}A)^{2}.$

In fact, pfaffian is defined not for whole matrix $A.$ To construct pfaffians
it is enough to know upper triangular part of $A.$

Connection between determinants of skew-symmetric matrices and pfaffians was
firstly noted in [1]. Details of pfaffian constructions see also [2] and [3].

Let $S_{2n}$ be set of permutations on the set $[2n]=\\{1,2,\ldots,2n\\}$ and
$S_{2n,pf}$ its subset of permutations called Pfaff permutations,

$S_{2n,pf}=\\{\sigma=(i_{1},j_{1},\ldots,i_{n},j_{n})\in
S_{2n}|i_{1}<i_{2}<\cdots<i_{n},i_{s}<j_{s},1\leq s\leq n\\}.$

For any $\sigma\in S_{2n,pf}$ we define Pffaf aggregates $a_{\sigma}$ by

$a_{\sigma}=a_{i_{1},j_{1}}\cdots a_{i_{n},j_{n}}.$

We see that pfaffian aggregates are defined for any triangular array
$\bar{A}=(a_{i,j})_{1\leq i<j\leq 2n}.$ Then pffaffian of order $2n$ is a
polynomial defined as an alternating sum of pfaff aggregates

$pf_{n}=\sum_{\sigma\in S_{2n,pf}}sign\;\sigma\;a_{\sigma}.$

Now suppose that $\\{a_{i,j},1\leq i,j\leq 2n\\}$ are $n^{2}$ generators and
endow space polynomials $K[a_{i,j}|1\leq i,j\leq n]$ by structure of
$S_{2n}$-module by the following action on generators

$\sigma a_{i,j}=a_{\sigma^{-1}(i),\sigma^{-1}(j)}.$

In particular, if $A=(a_{i,j})$ with skew-symmetric set of generators,
$a_{i,j}=-a_{j,i}$ then this action induces structure of $S_{2n}$-module
structure on the space of polynomials with ${n\choose 2}$ generators
$K[a_{i,j}|1\leq i<j\leq n].$ Similarly, we obtain one more structure of
$S_{2n}$-module on this space if generators are symmetric, $a_{i,j}=a_{j,i}.$
In both cases appear natural questions about invariants under these actions of
permutation groups. In particular, we can ask about symmetry and skew-symmetry
groups of given polynomial $f\in K[a_{i,j}],$

$Sym\;f=\\{\sigma\in S_{n}|\sigma\;f=f\\},$ $SSym\;f=\\{\sigma\in
S_{n}|\sigma\;f=sign\,\sigma\;f\\}.$

For example, determinant polynomial $det\,A$ for $A=(a_{ij})_{1\leq i,j\leq
n}$ is a polynomial of degree $n$ and its symmetry group is isomorphic to
$S_{n}.$ Another example: if matrix $A$ is skew-symmetric, then pfaff
polynomiaml$pf_{2n}=pf_{2n}A$ is a polynomial of degree $n$ and

$SSym\;pf_{2n}\cong S_{2n}.$

Let characteristic of main field is $p\neq 2.$ The aim of our paper is to
prove the following result.

######  Theorem 1.1.

If generators $a_{i,j}$ are symmetric, $a_{i,j}=a_{j,i},$ then symmetry group
of pffaffian polynomial $pf_{2n}=pf_{2n}\bar{A}$ is isomorphic to a dihedral
group

$Sym\;pf_{2n}\cong D_{2n}.$

In the proof of this fact we use the following results.

######  Theorem 1.2.

Let $\psi$ be symmetric function,

$\psi(x,y)=\psi(y,x),$

such that

(1) $\psi(x+z,y+z)=\psi(x,y),$

for any $x,y,z.$ Let $pf_{2n}(x_{1},\ldots,x_{2n})$ be pfaffian polynomial
$pf_{2n}\bar{A},$ where $a_{i,j}=\psi(x_{i},x_{j}).$ Then for any $1\leq s\leq
2n,$

$pf_{2n}(x_{1},\ldots,x_{s-1},x_{s},x_{s},x_{s+2},\ldots,x_{2n})=c\;pf_{2n-2}(x_{1},\ldots,x_{s-1},x_{s+2},\ldots,x_{2n}),$

where

$c=\psi(0,0).$

Here we set $x_{2n+1}=x_{1}.$

######  Theorem 1.3.

Let $\bar{A}=(a_{i,j})_{1\leq i<j\leq 2n}$ be triangular array with components
$a_{i,j}=(x_{i}-x_{j})^{2}$ for $1\leq i<j\leq 2n.$ Then

$pf_{2n}\;\bar{A}=-(-2)^{n-1}\;(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{1}).$

We give one more result on symmetric pfaffians.

######  Theorem 1.4.

If $a_{i,j}=cos(x_{i}-x_{j}),$ $1\leq i<j\leq 2n,$ then

$pf_{2n}A=\cos(x_{1}-x_{2}+x_{3}-x_{4}+\cdots+x_{2n-1}-x_{2n}).$

##  2\.  Proof of Theorem 1.2

Let $\bar{A}=(a_{i,j})_{1\leq i<j\leq 2n}$ be triangular array and

$hk_{s}\bar{A}=(a_{1,s},a_{2,s},\ldots,a_{s-1,s},0,a_{s,s+1},\ldots,a_{s,2n})$

be $s$-th hook of triangle $\bar{A}.$ If generators are skew-symmetric, then
we can present it as $s$-th row of skew-symmetric matrix $A,$ constructed by
triangular $\bar{A},$

$hk_{s}A=(-a_{s,1},-a_{s,2},\ldots,-a_{s,s-1},0,a_{s,s+1},\ldots,a_{s,2n})$

Then decomposition formula for pfaffians by $s$-th hook can be written as
follows

$pf_{2n}A=\sum_{1\leq j\leq 2n,j\neq
s}(-1)^{s+j+1+H(s-j)}a_{s,j}pf_{2n-2}A_{\hat{s}.\hat{j}},$

where $A_{\hat{i},\hat{j}}$ denotes the matrix $A$ with both the $i$-th and
$j$-th rows and columns removed. Here $\theta(i-j)$ is Heaviside step
function,

$H(s)=1,\mbox{ if $s>0$ and $0,$ if $s\leq 0.$}$

If generators $a_{i,j}$ are symmetric, $a_{i,j}=a_{j,i},$ then decomposition
formula can be written without Heaviside function,

(2) $pf_{2n}\bar{A}=\sum_{1\leq j\leq 2n,j\neq
s}(-1)^{s+j+1}a_{s,j}pf_{2n-2}\bar{A}_{\hat{s},\hat{j}}.$

Proof of Theorem 1.2. Take $x=y=0$ in (1) . We obtain

(3) $\psi(z,z)=\psi(0,0)=c.$

By (2) we can decompose pfaffian by $s$-th hook

(4) $pf_{2n}\bar{A}=$
$\sum_{i=1}^{s-1}(-1)^{s+i+1}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\hat{s}\ldots,2n\\}+$
$a_{s,s+1}\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,2n\\}+$
$\sum_{i=s+2}^{2n}(-1)^{s+i+1}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{s}\ldots,\hat{i},\ldots,2n\\}.$

Here notation $\hat{i}$ means that corresponding index $i$ is omitted.

Similarly decomposition of pfaffian by $s+1$-th hook gives us the following
relation

(5) $pf_{2n}\bar{A}=$
$\sum_{i=1}^{s-1}(-1)^{s+i}a_{s+1,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\widehat{s+1},\ldots,2n\\}+$
$a_{s+1,s}\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,2n\\}+$
$\sum_{i=s+2}^{2n}(-1)^{s+i}a_{s+1,i}\;pf\;\\{1,2,\ldots,\widehat{s+1},\ldots,\hat{i},\ldots,2n\\}.$

Take here $x_{s+1}=x_{s}.$ Then $a_{s+1,i}=a_{s,i},$ for any $i\neq s$ and by
(3)

$a_{s+1,s}=a_{s,s+1}=\psi(x_{s},x_{s})=\psi(0,0)=c.$

Therefore by (5)

$pf_{2n}(x_{1},\ldots,x_{s-1},x_{s},x_{s},x_{s+2},\ldots,x_{2n})=$
$\sum_{i=1}^{s-1}(-1)^{s+i}a_{s+1,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\widehat{s+1}\ldots,2n\\}+$
$\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\}+$
$\sum_{i=s+2}^{2n}(-1)^{s+i}a_{s,i}\;pf\;\\{1,2,\ldots,\widehat{s+1},\ldots,\hat{i},\ldots,2n\\}=$
$\sum_{i=1}^{s-1}(-1)^{s+i}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\widehat{s+1},\ldots,2n\\}+$
$\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\}+$
$\sum_{i=s+2}^{2n}(-1)^{s+i}a_{s,i}\;pf\;\\{1,2,\ldots,\widehat{s+1},\ldots,\hat{i},\ldots,2n\\}.$

Hence by (4)

$pf_{2n}(x_{1},\ldots,x_{s-1},x_{s},x_{s},x_{s+2},\ldots,x_{2n})=$

$\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\}+$
$\sum_{i=1}^{s-1}(-1)^{s+i}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\hat{s}\ldots,2n\\}+\sum_{i=s+2}^{2n}(-1)^{s+i}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{s}\ldots,\hat{i},\ldots,2n\\}=$

$\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\}+$
$-\sum_{i=1}^{s-1}(-1)^{s+i+1}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{i}\ldots,\hat{s}\ldots,2n\\}-\sum_{i=s+2}^{2n}(-1)^{s+i+1}a_{s,i}\;pf\;\\{1,2,\ldots,\hat{s}\ldots,\hat{i},\ldots,2n\\}=$

$2\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\}-pf_{2n}(x_{1},\ldots,x_{s-1},x_{s},x_{s},x_{s+2},\ldots,x_{2n}).$

So,

$pf_{2n}(x_{1},\ldots,x_{s-1},x_{s},x_{s},x_{s+2},\ldots,x_{2n})=\psi(0,0)\;pf\;\\{1,2,\ldots,\hat{s},\widehat{s+1},\ldots,\ldots,2n\\},$

if characteristic of main field is not $2.$ $\square$

##  3\. Pfaffian of $(x_{i}-x_{j})^{2}$

If $A=(a_{i,j})$ is skew-symmetric, then

$det\,A=(pf_{2n}\bar{A})^{2}.$

If matrix $A$ is not skew-symmetric, say if $A$ is symmetric, then determinant
polynomial $det\,A$ and pfaffian polynomial $pf_{2n}(\bar{A})$ have no such
connection. For example, if $A_{n}=((x_{i}-x_{j})^{2})_{1\leq i,j\leq n},$
then

$det\,A_{n}=\left\\{\begin{array}[]{ll}-(x_{1}-x_{2})^{2},&\mbox{ if
$n=2$,}\\\ 2((x_{2}-x_{2})(x_{2}-x_{3})(x_{3}-x_{1}))^{2},&\mbox{ if $n=3$
and,}\\\ 0,&\mbox{ otherwise}\end{array}\right.$

in that time, by Theorem 1.3, pfaffians are non-trivial for any even $n.$

Proof of Theorem 1.3. Let $\psi(x,y)=(x-y)^{2}$ and
$a_{i,j}=\psi(x_{i},x_{j})=(x_{i}-x_{j})^{2}.$ Note that pfaffian $pf_{2n}A$
is a polynomial on variables $x_{1},\ldots,x_{2n}.$ Let us construct
polynomial $g_{2n}=g_{2n}(x_{1},x_{2},\ldots,x_{2n})$ by

$g_{2n}=(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{1}).$

We have to prove that

$pf_{2n}=-(-2)^{n-1}g_{2n}.$

Since $\psi(0,0)=0,$ by Theorem 1.2 the polynomial
$pf_{2n}(x_{1},\ldots,x_{s},x_{s+1},\ldots,x_{2n})$ is divisable by
$x_{s}-~{}x_{s+1}$ for any $1\leq s\leq 2n.$ Here we set $x_{2n+1}=x_{1}.$
Note that degree of the polynomial $g_{2n}(x_{1},\ldots,x_{2n})$ is $2n$ and
degree of $pf_{2n}((x_{i}-x_{j})^{2})$ is also $2n.$ Therefore,

$pf_{2n}(x_{1},x_{2},\ldots,x_{2n})=c\;g_{2n}(x_{1},x_{2},\ldots,x_{2n}),$

for some constant $c.$ Take $x_{i}=i.$ It is easy to see that

$g_{2n}(1,2,\ldots,2n)=(1-2)(2-3)\cdots(2n-1-2n)(2n-1)=-(2n-1).$

It remains to prove that

(6) $pf_{2n}(1,2,\ldots,{2n})=(-2)^{n-1}(2n-1).$

to obtain $c=-(-2)^{n-1}.$

By induction on $n$ we will prove that

$pf_{2n}(x_{1},x_{2},\ldots,x_{2n})=-(-2)^{n-1}g_{2n}(x_{1},x_{2},\ldots,x_{2n}).$

For $n=1$ our statement is evident:

$pf_{2}\bar{A}=a_{1,2}=-(x_{1}-x_{2})(x_{2}-x_{1}).$

Suppose that our statement is true for $n-1,$

$pf_{2n-2}\bar{A}=-(-2)^{n-2}(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{2n-3}-x_{2n-2})(x_{2n-2}-x_{1}).$

Let us prove it for $n.$

We decompose pfaffian by the first row. By (2)

$pf_{2n}\bar{A}=\sum_{i=2}^{2n}(-1)^{i}a_{1,i}\;pf_{2n-2}A_{\hat{1},\hat{i}}.$

We see that

$pf_{2n}=R_{1}+R_{2}+R_{3},$

where

$R_{1}=a_{1,2}\;pf_{2n}\bar{A}_{\hat{1}\hat{2}},$

$R_{2}=\sum_{i=3}^{2n-1}(-1)^{i}a_{1,i}\;pf_{2n-2}\bar{A}_{\hat{1},\hat{i}},$

$R_{3}=a_{1,2n}\;pf_{2n-2}\bar{A}_{\hat{1},\hat{2n}}.$

By inductive suggestion

$R_{1}=-(-2)^{n-2}(x_{1}-x_{2})^{2}(x_{3}-x_{4})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{3}).$

Hence

$R_{1}|_{x_{i}\rightarrow i}=(1-2)^{2}(3-4)\cdots(2n-1-2n)(2n-3)=$

$(-1)^{2n-3}(2n-3)=(-2)^{n-2}(2n-3),$

$R_{3}|_{x_{i}\rightarrow
i}=-(-2)^{n-2}(1-2n)^{2}(2-3)(3-4)\cdots(2n-2-2n+1)(2n-1-2)=$

$-(-2)^{n-2}(2n-1)^{2}(-1)^{2n-3}(2n-3)=(-2)^{n-2}(2n-3)(2n-1)^{2}.$

Further, if $2<i<2n,$ then

$(-1)^{i}a_{1,i}\;pf_{2n}\bar{A}_{\hat{1},\hat{i}}|_{x_{j}\rightarrow j}=$

$(-1)^{i}(-(-2)^{n-2})(x_{1}-x_{i})^{2}(x_{2}-x_{3})(x_{3}-x_{4})\cdots(x_{i-1}-x_{i+1})(x_{i+1}-x_{i+2})\times\cdots$
$\times(x_{2n-1}-x_{2n})(x_{2n}-x_{2})|_{x_{j}\rightarrow j}=$

$(-1)^{i}(-(-2)^{n-2})(i-1)^{2}(-2)(2n-2)=(-1)^{i}(i-1)^{2}(-2)^{n-2}4(n-1).$

Hence

$R_{2}|_{x_{i}\rightarrow
i}=-(-2)^{n-2}\sum_{i=3}^{2n-1}-(-1)^{i}(i-1)^{2}4(n-1)=(-2)^{n-2}4(n-1)(2n^{2}-3n+2).$

So, we see that (6) is true for $n,$

$f_{2n}(1,2,\ldots,2n)=R_{1}+R_{2}+R_{3}=$

$(-2)^{n-2}[(2n-3)-4(n-1)(2n^{2}-3n+2)+(2n-3)(2n-1)^{2}]=$

$-(-2)^{n-2}2(2n-1)=(-2)^{n-1}(2n-1).$

Theorem 1.3 is proved. $\square$

##  4\. Symmetry group of the polynomial $g_{2n}$

######  Lemma 4.1.

Let

$g_{2n}(x_{1},\ldots,x_{2n})=(x_{1}-x_{2})(x_{2}-x_{3})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{1})$

as in the proof of Theorem 1.3 and

$Sym(g_{2n})=\\{\sigma\in
S_{2n}|g_{2n}(x_{\sigma(1)},\ldots,x_{\sigma(2n)})=g_{2n}(x_{1},\ldots
x_{2n})\\}$

symmetric group of the polynomial $g_{2n}.$ Then

$Sym(g_{2n})\cong D_{2n}.$

Proof. First we check that any dihedral permutation $\sigma\in D_{2n}$ is
symmetry of the polynomial $g_{2n}.$

Let us take realization of dihedral group as a symmetry group of regular
$n$-gon whose vertices are clockwise labelled by $1,2,\ldots,2n.$ Elements of
dihedral group might have:

  * I.

one up-run: $\sigma=1\;2\;\cdots(2n).$

  * II.

one down-run: $\sigma=(2n)\;(2n-1)\;\cdots\;1.$

  * III.

two up-runs

$\sigma(1)=s<\sigma(2)=s+1<\cdots<\sigma(2n-s+1)=2n,\;\;\sigma(2n-s+2)=1<\cdots<\sigma(2n)=s-1,$

for some $1<s\leq 2n,$

  * IV.

or two down-runs

$\sigma(1)=s>\sigma(2)=s-1>\cdots>\sigma(s)=1,\;\;\sigma(s+1)=2n>\cdots>\sigma(2n)=s+1.$

for some $1\leq s<2n.$

In cases I and II our statement is evident.

In case III we have

$g_{2n}(x_{\sigma(1)},\ldots,x_{\sigma(2n)})=$
$(x_{s}-x_{s+1})(x_{s+1}-x_{s+2})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{1})\;(x_{1}-x_{2})\cdots(x_{s-2}-x_{s-1})(x_{s-1}-x_{s})=$
$(x_{1}-x_{2})\cdots(x_{2n-1}-x_{2n})(x_{2n}-x_{1})=g_{2n}(x_{1},\ldots,x_{2n}).$

In case IV

$g_{2n}(x_{\sigma(1)},\ldots,x_{\sigma(2n)})=$
$(x_{s}-x_{s-1})(x_{s-1}-x_{s-2})\cdots(x_{2}-x_{1})(x_{1}-x_{2n})\;(x_{2n}-x_{2n-1})\cdots(x_{s+1}-x_{s})=$

$(-1)^{s}(x_{s-1}-x_{s})(x_{s-2}-x_{s-1})\cdots(x_{1}-x_{2})(x_{2n}-x_{1})\;(-1)^{2n-s}(x_{2n-1}-x_{2n})\cdots(x_{s}-x_{s+1})=$

$(x_{1}-x_{2})\cdots(x_{2n-1}-x_{2n})=g_{2n}(x_{1},\ldots,x_{2n}).$

So,

$D_{2n}\subseteq Sym(g_{2n}).$

Now we will prove that any $\sigma\in Sym(g_{2n})$ is a dihedral permutation.

Let $M_{2n}=\\{1,2,\ldots,2n\\}.$ For $i,j\in M_{2n}$ say that they are
connected, if $|i-j|=1$ or $|i-j|=2n-1.$ So, if $i<j<2n,$ then $i,j$ are
connected iff $j=i+1.$ If $j=2n,$ and $i,j$ are connected, then $i=2n-1$ or
$i=1.$ It is clear that this relation is symmetric: $i,j$ are connected iff
$j,i$ are connected. So, $i,j\in M_{2n}$ are connected, if $|i-j|=1$ or
$(i,j)=(1,2n)$ or $(i,j)=(2n,1).$

Note that he polynomial $g_{2n}(x_{1},\ldots,x_{2n})$ is a product of
polynomials $x_{i}-x_{j},$ $i<j,$ where $i$ and $j$ are connected. Therefore,
any symmetry $\sigma\in Sym(g_{2n})$ has the following property: if $i$ and
$j$ are connected, then $\sigma(i)$ and $\sigma(j)$ are also connected.

Let $\sigma\in Sym(g_{2n})$ and $\sigma(1)=i_{1}.$ Might happen the following
possibilities.

Case A. Suppose that $\sigma(1)=i_{1}<\sigma(2).$ Take $k>1,$ such that
$\sigma(k-1)<\sigma(k)$ and $\sigma(k+1)<\sigma(k).$ Since $\sigma(1)$ and
$\sigma(2)$ are connected and $\sigma(2)>\sigma(1),$ then $\sigma(2)=i_{1}+1.$
By similar arguments,

$\sigma(3)=i_{1}+2,\ldots,\sigma(k)=i_{1}+k-1,$

but $\sigma(k+1)\neq i_{1}+k.$ Such situation is possible only in one case:
$i_{1}=~{}2n-~{}k+~{}1$ and $\sigma(k+~{}1)~{}=~{}1.$ So,

$\sigma(k+1)=1,\sigma(k+2)=2,\ldots,\sigma(2n)=i_{1}-1.$

In other words,

$\sigma=i_{1}\;(i_{1}+1)\;\ldots\;(2n)\;1\;2\;\ldots(i_{1}-1).$

We obtain permutation $\sigma$ that has exactly one up-run if $i_{1}=1,$ or
two up-runs if $i_{1}>1.$ So, we obtain permutations of type I or III.
Therefore, $\sigma\in D_{2n}.$

Case B. Now consider the case $\sigma(1)=i_{1}>\sigma(2).$ Take $k>1,$ such
that $\sigma(k-1)>\sigma(k)$ and $\sigma(k+1)>\sigma(k).$

Since $\sigma(1)$ and $\sigma(2)$ are connected and $\sigma(2)<\sigma(1),$
then $\sigma(2)=i_{1}-1.$ By similar arguments,

$\sigma(3)=i_{1}-2,\ldots,\sigma(k)=i_{1}-k+1.$

but $\sigma(k+1)\neq i_{1}-k.$ Such situation is possible only in one case:
$i_{1}=k,\sigma(k+1)=2n.$ So,

$\sigma(k+1)=2n,\sigma(k+2)=2n-1,\ldots,\sigma(2n)=i_{1}+1.$

In other words,

$\sigma=i_{1}\;(i_{1}-1)\;\ldots\;1\;{2n}\;(2n-1)\;\ldots(i_{1}+1)$

We obtain permutation $\sigma$ that has exactly one down-run if $i_{1}=2n$ or
two down-runs if $i_{1}<2n.$ In other words we obtain permutations of type II
or IV. Thus, $\sigma\in D_{2n}.$ $\square$

##  5\. Proof of Theorem 1.1

First we prove that $D_{2n}\subseteq Sym\;pf_{2n}.$

######  Lemma 5.1.

If $A=(a_{i,j})$ is symmetric, then pfaffian is invariant under action of
dihedral group $D_{2n},$

$\mu(pf_{2n})=pf_{2n}$

for any $\mu\in D_{2n}.$

Proof. The dihedral group $D_{2n}$ has order $4n$ and is generated by cyclic
permutation

$\sigma=\left(\begin{array}[]{cccccc}1&2&3&\cdots&2n-1&2n\\\
2&3&4&\cdots&2n&1\end{array}\right)$

and reflection

$\tau=\left(\begin{array}[]{cccccccccc}1&2&3&\cdots&n&n+1&n+2&\cdots&2n-1&2n\\\
1&2n&2n-1&\cdots&n+2&n+1&n&\cdots&3&2\end{array}\right)$

To prove our lemma it is enough to establish that

$\sigma(pf_{2n})=pf_{2n},$ $\tau(pf_{2n})=pf_{2n},$

if $a_{i,j}=a_{j,i},$ for any $1\leq i<j\leq 2n.$

Recall that $\alpha=(i_{1},i_{2},\ldots,i_{2n-1},i_{2n})$ is Pfaff
permutation, if

$i_{1}<i_{3}<i_{5}<\cdots<i_{2n-1},$
$i_{1}<i_{2},i_{3}<i_{4},\ldots,i_{2n-1}<i_{2n}.$

Let $S_{2n,pf}$ be set of Pfaff permutations. Below we use one-line notation
for permutations. We write $\alpha=(i_{1},i_{2},\ldots,i_{2n-1},i_{2n})$
instead of

$\alpha=\left(\begin{array}[]{ccccc}1&2&\cdots&2n-1&2n\\\
i_{1}&i_{2}&\ldots&i_{2n-1}&i_{2n}\end{array}\right).$

Note that

$\tau(i)+i=\left\\{\begin{array}[]{ll}2n+2,&\mbox{ if $1<i\leq 2n.$}\\\
2&\mbox{ if $i=1$}\end{array}\right.$

Set

$\bar{i}=2n+2-i,$

if $i>1.$

Now we study action of the generator $\sigma$ on pfaffian polynomials, when
generators are symmetric, $a_{i,j}=a_{j,i},$ for any $1\leq i,j\leq 2n.$

Let $\alpha=(1,i_{2},i_{3},\ldots,i_{2n})\in S_{2n,pf},$ and
$l=\alpha^{-1}(2n).$ Then $l$ is even, $l=2k,$ and

$\sigma(a_{\alpha})=\sigma(a_{i_{1},i_{2}}\cdots a_{i_{2n-1}i_{2n}})=$

$a_{i_{1}+1,i_{2}+1}\cdots
a_{i_{2k-3}+1,i_{2k-2}+1}a_{i_{2k-1}+1,1}a_{i_{2k+1}+1,i_{2k+2}+1}\cdots
a_{i_{2n-1}+1,i_{2n}+1}=$

$a_{\tilde{\alpha}},$

where

$\tilde{\alpha}=(1,i_{2k-1}+1,i_{1}+1,i_{2}+1,\ldots,{i_{2k-3}+1,i_{2k-2}+1},{i_{2k+1}+1,i_{2k+2}+1},\ldots,{i_{2n-1}+1,i_{2n}+1}).$

Here we change $a_{i_{2k-1}+1,1}$ to $a_{1,i_{2k-1}+1}.$ We see that the map

$S_{2n,pf}\rightarrow S_{2n,pf},\quad\alpha\mapsto\tilde{\alpha}$

is bijection and

$sign\;\tilde{\alpha}=sign\;\alpha.$

Hence

$\sigma(pf_{2n})=\sum_{\alpha\in
S_{2n,pf}}sign\,\alpha\,\sigma(a_{\alpha})=\sum_{\alpha\in
S_{2n,pf}}sign\,\tilde{\alpha}\,a_{\tilde{\alpha}}=pf_{2n}.$

So, we have estabilshed that pfaffian is invariant under action $\sigma\in
D_{2n}.$

Let us study action of the generator $\tau$ on pfaffian polynomials.

We have

$\tau:a_{\alpha}\mapsto
a_{1,\;\overline{i_{2}}}a_{\overline{i_{3}},\;\overline{i_{4}}}\cdots
a_{\overline{i_{2n-1}},\;\overline{i_{2n}}}.$

Since

$\overline{i_{2k-1}}>\overline{i_{2k}},\quad 1<k\leq n,$

we have to change $a_{\overline{i_{2k-1}},\;\overline{i_{2k}}}$ to
$a_{\overline{i_{2k}},\;\overline{i_{2k-1}}}.$ Further,

$\overline{i_{3}}>\overline{i_{5}}>\cdots>\overline{i_{2n-1}}>1.$

Therefore,

$\tau:a_{\alpha}\mapsto a_{\bar{\alpha}},$

where

$a_{\bar{\alpha}}=a_{1,\;\overline{i_{2}}}a_{\overline{i_{2n}},\;\overline{i_{2n-1}}}a_{\overline{i_{2n-2}},\;\overline{i_{2n-3}}}\cdots
a_{\overline{i_{4}},\;\overline{i_{3}}}.$

We see that

$sign\;\alpha=sign\;\bar{\alpha}.$

Note that the map

$S_{2n,pf}\rightarrow S_{2n,pf},\quad\alpha\mapsto\bar{\alpha},$

is bijection. Therefore,

$\tau(pf_{2n})=\sum_{\alpha\in
S_{2n,pf}}sign\,\alpha\;\tau(a_{\alpha})=\sum_{\alpha\in
S_{2n,pf}}sign\,\bar{\alpha}\;a_{\bar{\alpha}}=pf_{2n}.$

So, we have proved that pfafian $pf_{2n}$ is invariant under action of
dihedral group $D_{2n}$ of order $4n,$ if the matrix $(a_{i,j})_{1\leq i,j\leq
2n}$ is symmetric. $\square$

Example. Let

$\tau=\left(\begin{array}[]{cccc}1&2&3&4\\\
1&4&3&2\end{array}\right),\mu=\left(\begin{array}[]{cccc}1&2&3&4\\\
4&3&2&1\end{array}\right).$

Then

$\tau(pf_{4})=\tau(a_{1,2}a_{3,4}-a_{1,3}a_{2,4}+a_{1,4}a_{2,3})=$
$a_{1,4}a_{3,2}-a_{1,3}a_{4,2}+a_{1,2}a_{4,3}=$
$a_{1,4}a_{2,3}-a_{1,3}a_{2,4}+a_{1,2}a_{3,4}=pf_{4},$

$\mu(pf_{4})=\mu(a_{1,2}a_{3,4}-a_{1,3}a_{2,4}+a_{1,4}a_{2,3})=$
$a_{4,3}a_{2,1}-a_{4,2}a_{3,1}+a_{4,1}a_{3,2}=$
$a_{3,4}a_{1,2}-a_{2,4}a_{1,3}+a_{1,4}a_{2,3}=pf_{4}.$

Proof of Theorem 1.1. Let $\sigma\in Sym\,pf_{2n}$ i.e.,

$\sigma\,(pf_{2n})=pf_{2n}$

for any $a_{i,j},$ such that $a_{i,j}=a_{j,i}.$ In particular, $\sigma$ is a
symmetry of the pfaffian polynomial $pf_{2n}((x_{i}-x_{j})^{2})_{1\leq i<j\leq
2n}.$ By Theorem 1.2 the polynomial $g_{2n}(x_{1},\ldots,x_{2n})$ is pfaffian
polynomial $pf_{2n}((x_{i}-x_{j})^{2})_{1\leq i<j\leq 2n}$ up to non-zero
constant. Therefore by Lemma 4.1 $\sigma\in D_{2n}.$

In remains to use Lemma 5.1 to finish the proof. $\square$

##  6\.  Proof of Theorem 1.4

Let $a_{i,j}=\cos(x_{i}-x_{j}).$ We have to prove that

$pf_{2n}=\cos(x_{1}-x_{2}+x_{3}-x_{4}+\cdots+x_{2n-1}-x_{2n}).$

The proof is based on the following elementary trigonometric facts.

######  Lemma 6.1.

$-\cos\alpha\cos(\theta-\alpha)+\cos\beta\cos(\theta-\beta)=\sin(\alpha-\beta)\sin(\alpha+\beta-\theta).$

Proof. We have

$\sin(\alpha-\beta)\sin(\alpha+\beta-\theta)=$
$\sin(\alpha-\theta+\theta-\beta)\sin(\alpha+\beta-\theta)=$

$(\sin(\alpha-\theta)\cos(\theta-\beta)+\cos(\alpha-\theta)\sin(\theta-\beta))\sin(\alpha+\beta-\theta)=$

$\sin(\alpha-\theta)\cos(\theta-\beta)\sin(\alpha+\beta-\theta)+\cos(\alpha-\theta)\sin(\theta-\beta)\sin(\alpha+\beta-\theta)=$

$\sin(\alpha-\theta)\cos(\theta-\beta)\sin(\alpha+\beta-\theta)+\cos(\alpha-\theta)\cos(\theta-\beta)\cos(\alpha+\beta-\theta)+$
$\cos(\alpha-\theta)\sin(\theta-\beta)\sin(\alpha+\beta-\theta)-\cos(\alpha-\theta)\cos(\theta-\beta)\cos(\alpha+\beta-\theta)=$

$\cos(\theta-\beta)\left\\{\sin(\alpha-\theta)\sin(\alpha+\beta-\theta)+\cos(\alpha-\theta)\cos(\alpha+\beta-\theta)\right\\}+$
$\cos(\alpha-\theta)\\{\sin(\theta-\beta)\sin(\alpha+\beta-\theta)-\cos(\theta-\beta)\cos(\alpha+\beta-\theta)\\}=$

$\cos(\theta-\beta)\cos(\alpha-\theta-(\alpha+\beta-\theta))$
$-\cos(\alpha-\theta)\cos(\theta-\beta+(\alpha+\beta-\theta))=$

$\cos(\theta-\beta)\cos\beta-\cos(\theta-\alpha)\cos(\alpha).$

$\square$

######  Lemma 6.2.

$\sum_{i=1}^{n}(-1)^{i}\sin\alpha_{i}\,\sin(\sum_{j=1}^{i-1}(-1)^{j}\alpha_{j}-\sum_{j=i+1}^{n}(-1)^{j}\alpha_{j})=0,$

$\sum_{i=1}^{n}(-1)^{i}\cos\alpha_{i}\,\sin(\sum_{j=1}^{i-1}(-1)^{j}\alpha_{j}-\sum_{j=i+1}^{n}(-1)^{j}\alpha_{j})=\left\\{\begin{array}[]{cc}0&\mbox{
if $n$ is odd}\\\ -\sin(\sum_{j=1}^{n}(-1)^{j}\alpha_{j})&\mbox{ if $n$ is
even}\end{array}\right.$

Proof. Follows from known relation

$\sin\,\sum_{j=1}^{n}\theta_{i}=\sum_{\mbox{\small odd $k\geq
1$}}(-1)^{\frac{(k-1)}{2}}\sum_{A\subseteq[n],|A|=k}\;(\prod_{i\in
A}\sin\theta_{i}\prod_{i\notin A}\cos\theta_{i}).$

$\square$

We use induction on $n.$ For $n=2$ our statement is evident. Suppose that for
$n-1$ our statement is true,

$pf_{2n-2}(2,\ldots,\hat{i},\ldots,2n)=\cos(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-1}-x_{2n}).$

Let us decompose pfaffian by first row

$pf_{2n}=\sum_{i=2}^{2n}(-1)^{i}a_{1,i}\;pf_{2n-2}(2,\ldots,\hat{i},\ldots,2n).$

Therefore,

$pf_{2n}=\sum_{i=2}^{2n}(-1)^{i}\cos(x_{1}-x_{i})\;\cos(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-1}-x_{2n})=$
$R_{1}+R_{2},$

where

$R_{1}=$
$-\cos(x_{1}-x_{2n-1})\cos(x_{2}-x_{3}+\cdots+x_{2n-2}-x_{2n})+\cos(x_{1}-x_{2n})\cos(x_{2}-x_{3}+\cdots+x_{2n-2}-x_{2n-1}),$

$R_{2}=$
$\sum_{i=2}^{2n-2}(-1)^{i}\cos(x_{1}-x_{i})\;\cos(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-1}-x_{2n}).$

We have

$R_{2}=\sum_{i=2}^{2n-2}(-1)^{i}\cos(x_{1}-x_{i})\;\cos(x_{2n-1}-x_{2n})R_{2,i}^{\prime}-(-1)^{i}\cos(x_{1}-x_{i})\sin(x_{2n-1}-x_{2n})R_{2,i}^{\prime\prime},$

where

$R_{2,i}^{\prime}=\cos(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-3}-x_{2n-2}),$
$R_{2,i}^{\prime\prime}=\sin(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-3}-x_{2n-2}).$

By Lemma 6.2

$\sum_{i=2}^{2n-2}-(-1)^{i}\cos(x_{1}-x_{i})\sin(x_{2n-1}-x_{2n})R_{2,i}^{\prime\prime}=$

$-\sin(x_{2n-1}-x_{2n})\sum_{i=2}^{2n-2}(-1)^{i}\cos(x_{1}-x_{i})\sin(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-3}-x_{2n-2})=0,$

and

$R_{2}=\cos(x_{2n-1}-x_{2n})\;\sum_{i=2}^{2n-2}(-1)^{i}\cos(x_{1}-x_{i})R_{2,i}^{\prime}=$

$\cos(x_{2n-1}-x_{2n})\times$
$\sum_{i=2}^{2n-2}(-1)^{i}\cos(x_{1}-x_{i})\cos(x_{2}-x_{3}+\cdots+(-1)^{i-1}x_{i-1}+(-1)^{i}x_{i+1}+\cdots+x_{2n-3}-x_{2n-2}).$

Therefore by inductive suggestion,

$R_{2}=\cos(x_{2n-1}-x_{2n})\;pf_{2n-2}(1,2,\ldots,2n-2)=\cos(x_{2n-1}-x_{2n})\cos(\sum_{i=1}^{2n-2}(-1)^{j}x_{j}).$

By Lemma 6.1 applied for

$\alpha=x_{1}-x_{2n-1},\quad\beta=x_{1}-x_{2n},\quad\theta=x_{1}+x_{2}-x_{3}+x_{4}-\cdots-
x_{2n-3}+x_{2n-2}-x_{2n-1}-x_{2n},$

we obtain

$R_{1}=\sin(-x_{2n-1}+x_{2n})\sin(x_{1}-x_{2}+x_{3}-x_{4}+\cdots-x_{2n-2}).$

Thus,

$pf_{2n}=R_{1}+R_{2}=$
$-\sin(x_{2n-1}-x_{2n})\sin(x_{1}-x_{2}+x_{3}-x_{4}+\cdots-
x_{2n-2})+\cos(x_{2n-1}-x_{2n})\cos(\sum_{i=1}^{2n-2}(-1)^{j}x_{j})=$
$\cos(\sum_{i=1}^{2n}(-1)^{i}x_{i}).$

$\square$

Acknowledgment.

The work was supported by grant AP08855944.GF Ministry of Education and
Science of Kazakhstan Republic.

## References

  * [1] A. Cayley, Sur les déterminants gauches. J. reine und angew. Math. 1849, 38: 93-96.
  * [2] N. Bourbaki, Elements of mathematics, v. 2. Linear and multilinear algebra, Addison-Wesley (1973) pp. Chapt. 2
  * [3] R.Vein, P. Dale, Determinants and their Applications in Mathematical Physics, Springer-Verlag, New York, 1999.