#  Long-term properties of finite-correlation time isotropic stochastic
systems

A.S. Il’yn1,2, A.V. Kopyev1, V.A. Sirota1, and K.P. Zybin1,2 Electronic
addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1
P.N.Lebedev Physical Institute of RAS, 119991, Leninskij pr.53, Moscow, Russia
2 National Research University Higher School of Economics, 101000,
Myasnitskaya 20, Moscow, Russia

###### Abstract

We consider finite-dimensional systems of linear stochastic differential
equations
${\partial_{t}}{x_{k}}\left(t\right)={A_{kp}}\left(t\right){x_{p}}\left(t\right)$,
${\bf A}(t)$ being a stationary continuous statistically isotropic stochastic
process with values in real $d\times d$ matrices. We suppose also that the
laws of ${\bf A}(t)$ satisfy the large deviation principle. For these systems,
we find exact expressions for the Lyapunov and generalized Lyapunov exponents
and show that they are determined in a precise way only by the rate function
of the diagonal elements of ${\bf A}$.

##  I Introduction

Finite-dimensional first order systems of linear differential equations with
stochastic coefficients appear in stochastic dynamics, theory of turbulence
and turbulent transport, kinematic dynamo and many other physical, chemical
and biological problems (see,e.g., FGV ; Sab ). The long-term solutions of
these systems have been investigated for the case of Gaussian and
$\delta$-correlated in time stochastic noise Kraichnan ; Kazantsev . In this
paper, we study the case of isotropic non-Gaussian statistics and finite
correlation time of the noise.

Consider a system of linear equations

${\partial_{t}}{x_{k}}\left(t\right)={A_{kp}}\left(t\right){x_{p}}\left(t\right),\
\ k=1\dots d$ (1)

where ${\bf A}(t)$ is a stationary continuous real $d\times d$ matrix
stochastic process with known law. It is assumed to be statistically
isotropic, i.e. ${\bf OA}(t){\bf O}^{T}$ and ${\bf A}(t)$ are identically
distributed at any $t$ and have the same time correlations for any ${\bf O}\in
O(d)$.

The formal solution of (1) can be written as

${x_{k}}\left(t\right)={Q_{kp}}\left(t\right){x_{p}}\left(0\right),$

where the evolution matrix $\bf Q$ satisfies the equation:

${\partial_{t}}{\bf Q}\left(t\right)={\bf A}\left(t\right){\bf
Q}\left(t\right),\ \ {\bf Q}\left(0\right)={\bf 1}$ (2)

For each continuous realization of ${\bf A}(t)$ there exists a well-defined
solution of (2) given by the Volterra product integral Gantmacher-Volterra
(in quantum-mechanical terminology this is also called T-exponential):

${\bf Q}\left(t\right)=\mathop{\prod}\limits_{\tau=0}^{t}\left({{\bf 1}+{\bf
A}\left(\tau\right)d\tau}\right)$ (3)

So, the solutions of (2) can be interpreted as continual products of random
matrices.

In many physical problems (e.g., stochastic dynamics and turbulent transport)
one is basically interested in the infinite-time asymptotic behavior of the
norms of the basis multivectors Ottino ; Pope :

${E_{1}}=||{\bf Q}{\vec{e}_{1}}||\ ,\ \ {E_{2}}=||{\bf
Q}{\vec{e}}_{1}\wedge{\bf Q}{\vec{e}}_{2}||,\ \ {E_{d}}=||{\bf
Q}{\vec{e}}_{1}\wedge...\wedge{\bf Q}{\vec{e}}_{d}||$

For instance, the absolute value of a passive vector advected by a random
smooth flow is $E_{1}$; the density of an advected passive scalar in presence
of weak molecular diffusion is $\prod_{k=1}^{d}\mbox{min}\\{E_{k}^{-1},1\\}$
BF ; PRE2017 ; the energy of magnetic field generated by a turbulent MHD flow
with strong conductivity is
$\mbox{min}\\{E_{1}^{2}/E_{2}^{2},E_{2}^{2}/E_{3}^{2}\\}$ zeld ; Chertkov ;
PhFlu21 .

From the multiplicative Oseledets theorem Oseledets  it follows that almost
surely, there exist the limits

$\mathop{\lim}\limits_{T\to\infty}\left(\frac{1}{T}\ln
E_{k}\right)=\lambda_{1}+\dots+\lambda_{k},\ \ k=1\dots d$ (4)

The constants $\lambda_{k}$ are called Lyapunov exponents (LE). Complete
information about asymptotic behavior of the set $\\{E_{k}\\}$ is contained in
the generalized Lyapunov exponents (GLE) CrisantiPaladinVulp  defined by

$w\left({{\eta_{1}},...,{\eta_{d}}}\right)=\mathop{\lim}\limits_{T\to\infty}\frac{1}{T}\ln{\left\langle
E_{1}^{\eta_{1}-\eta_{2}}E_{2}^{\eta_{2}-\eta_{3}}...E_{d-1}^{\eta_{d-1}-\eta_{d}}E_{d}^{\eta_{d}}\right\rangle}$
(5)

where $\langle...\rangle$ is the average over all realizations of ${\bf
A}(t)$, $\eta_{k}\in\mathbb{R}$. For example, the LE can be expressed in terms
of GLE:

${\lambda_{k}}=\frac{\partial}{{\partial{\eta_{k}}}}w\left(0\right)$ (6)

The calculation of the values of (4) and (5) is the subject of this paper.

The most part of previous investigations considered the case of Gaussian ${\bf
A}(t)$ that is delta-correlated in time. For this case, the exact expressions
for LE were found in Le Jan ; Baxendale ; Newman . The delta-correlated
process typically has nowhere continuous paths, so Eq. (2) requires a
stochastic convention; the known results for that case refer to the
Stratonovich convention Gamba ; Liao .

However, in many physical applications random processes acting as
multiplicative noise are, generally, non-Gaussian, and have finite (non-zero)
correlation time. For continuous processes the solutions (3) are well defined,
so there is no need in stochastic conventions. It is well known Van-Kampen ;
JOSS1  that for such systems, the central limit theorem ’is not valid’ in the
sense that all connected correlators (cumulants) of ${\bf A}(t)$ contribute
equivalently to the LE and GLE. So, the result may differ essentially from the
Gaussian case.

To deal with non-zero correlation time, the renovation model has been
developed by different authors renov-model1 ; IS-renov : non-stationary
piecewise constant process is substituted for the stationary continious
process ${\bf A}(t)$. Alternatively, we stay in the frame of stationarity and
use the large deviation principle; in quantum mechanics and quantum field
theory the corresponding technics is known as the low-frequency limit. For the
processes $A_{ij}(t)$ that satisfy the large deviations principle, this allows
to get exact expressions for LE and GLE: it appears that they are completely
determined by the rate function of the diagonal elements $A_{kk}$.

###  I.1 Iwasawa decomposition

To get explicit expressions for the LE and GLE, one performs the Iwasawa
decomposition of the evolution matrix:

${\bf Q}={\bf RDZ}$

where ${\bf R}$ is orthogonal matrix, ${\bf D}$ is positive diagonal, and
${\bf Z}$ is upper-triangular unipotent matrix:

${\bf RR}^{T}=\hat{\bf I}\ ,\ \ D_{ij}=\delta_{ij}D_{i},\ D_{i}>0,\ \
Z_{i<j}=0\ ,Z_{jj}=1$

Then

$E_{k}=D_{1}\dots D_{k}$

and the LE can be expressed by

${\lambda_{k}}=\mathop{\lim}\limits_{T\to\infty}\frac{1}{T}\ln{D_{k}}\ ,\ \
\lambda_{1}\geq\dots\geq\lambda_{k}$ (7)

The expression for the GLE takes the form

$w\left({{\eta_{1}},...,{\eta_{d}}}\right)=\mathop{\lim}\limits_{T\to\infty}\frac{1}{T}\ln{\left\langle
D_{1}^{\eta_{1}}...D_{d}^{\eta_{d}}\right\rangle}$ (8)

The statistical properties of $\bf R$, $\bf D$ and $\bf Z$ behave differently
during the evolution (2); we are interested only in the statistics of $\bf D$.

In the next Section we consider the main strategy and the application of the
large deviation theory on a simple one-dimensional example. Then we proceed to
the multi-dimensional case, and make use of the isotropy to calculate the GLE.
In conclusion, we reformulate the results in terms of the ’effective delta-
process’, which is a useful tool for physical applications.

##  II One-dimensional case

###  II.1 Rate function and GLE

Consider one-dimensional differential equation with multiplicative noise:

${\partial_{t}}x\left(t\right)=\xi\left(t\right)x\left(t\right)\ ,\ \ x(0)=1$
(9)

where $\xi(t)$ is a continuous stationary random process with given law. We
are interested in the moments $\langle{x^{\eta}}\rangle$, $\eta\in\mathbb{R}$.

For any realization of $\xi(t)$, the solution of (9) is

$x\left(T\right)={e^{\mathop{\smallint}\limits_{0}^{T}\xi\left(t\right)dt}}$

We assume that $\xi(t)$ satisfies the large deviation principle Varadhan ,
i.e., that the probability density of its time average

$\frac{1}{T}\int\limits_{0}^{T}\xi(t)dt=\bar{\xi}$

satisfies at large $T\to\infty$ the relation:

$\rho_{\bar{\xi}}(a)\sim e^{-TJ(a)}$

where the sign $\sim$ means that there exists the limit

$\lim\limits_{T\to\infty}\frac{1}{T}\ln{\rho}_{\bar{\xi}}(a)=-J(a)$

Here $J(a)$ is the rate function (Cramer function). Then

$\langle{x^{\eta}(T)}\rangle=\int e^{T\eta a}\rho_{\bar{\xi}}(a)da\sim
e^{T\,w(\eta)}$ (10)

where $w(\eta)$ is the Legendre transform of the rate function,

$w(\eta)=\sup\limits_{a}\left(\eta a-J(a)\right)$ (11)

This proves the existence of the limit

$\lim\limits_{T\to\infty}\frac{1}{T}\ln\langle{x^{\eta}}(T)\rangle=w(\eta)\ ,$
(12)

$w(\eta)$ is the GLE of the process $\xi(t)$. We note that, according to (10),
the function $w(\eta)T$ at large $T$ coincides with the cumulant generating
function of the integral $\int\limits_{0}^{T}\xi(t)dt$.

Let the cumulants (or connected correlators)

$\left\langle\xi\left({{t_{1}}}\right)...\xi\left({{t_{n}}}\right)\right\rangle_{c}={W^{\left(n\right)}}\left({{t_{1}}-{t_{2}},\ldots,{t_{1}}-{t_{n}}}\right)$
(13)

be regular fast-decaying functions (i.e., let $\int W^{(n)}dt_{2}...dt_{n}$
converge). Then the cumulant-generating functional

$W\left[{\eta\left(t\right)}\right]=\ln\left\langle{e^{\mathop{\int}dt\eta(t)\xi(t)}}\right\rangle$
(14)

can be expanded into the infinite series in $\eta(t)$:

$W\left[{\eta\left(t\right)}\right]=\mathop{\sum}\limits_{n}\frac{1}{{n!}}\mathop{\int}{W^{\left(n\right)}}\left({{t_{1}}-{t_{2}},\ldots,{t_{1}}-{t_{n}}}\right)\eta\left({{t_{1}}}\right)...\eta\left({{t_{n}}}\right)d{t_{1}}...d{t_{n}}$
(15)

In accordance with (14),

$\langle{x^{\eta}}\left(T\right)\rangle=\left\langle{e^{\eta\mathop{\int}\limits_{0}^{T}\xi\left(t\right)dt}}\right\rangle=e^{W\left[{\eta{{\rm{I}}_{\left[{0,1}\right]}}\left(t/T\right)}\right]}$
(16)

where ${\rm{I}}_{\left[{0,1}\right]}$ is the indicator function of the segment
$\left[0,1\right]$. Substituting (15) we find

$\lim\limits_{T\to\infty}\frac{1}{T}\ln\langle{x^{\eta}(T)}\rangle=\mathop{\sum}\limits_{n}\frac{w^{(n)}}{{n!}}\eta^{n}$
(17)

where

$w^{(n)}=\mathop{\int}{W^{\left(n\right)}}\left({{t_{1}}-{t_{2}},\ldots,{t_{1}}-{t_{n}}}\right)d{t_{2}}...d{t_{n}}$
(18)

Comparing (10) with (17) we get

$w\left(\eta\right)=\mathop{\sum}\limits_{n}\frac{w^{(n)}}{{n!}}\eta^{n}$ (19)

From (13), (19) we see that the long-time asymptotics of the moments
$\langle{x^{\eta}}\rangle$, as well as the GLE, do not depend on the details
of the cumulants and are determined only by their integrals (18).

###  II.2 Low-frequency limit.

We now consider an alternative way to find the GLE based on the Lagrangian
formalism. The aim is to derive the relation between the Lagrangian density
and the rate function.

So, now we start from the probability density functional
$\wp\left[\xi(t)\right]$, which can be presented in the form

${\wp}\left[\xi\right]\sim e^{-{\cal S}\left[\xi\right]}\ ,\qquad{\cal
S}\left[{\xi\left(t\right)}\right]=\mathop{\int}dt{\cal
L}(\xi,\partial_{t}\xi,\partial_{t}^{2}\xi,\dots)$

Here ${\cal L}$ is the Lagrangian density. For example, the case ${\cal
L}=\frac{1}{2}{\left(\epsilon{{\partial_{\tau}}\xi}\right)^{2}}+\frac{1}{2}{\xi^{2}}$
corresponds to Ornstein-Uhlenbeck process with the only non-zero cumulant
${W^{\left(2\right)}}\left({{t_{1}}-{t_{2}}}\right)=\frac{1}{2\epsilon}{e^{-\left|{{t_{1}}-{t_{2}}}\right|/\epsilon}}$.
Generally, ${\cal L}$ is non-local, and contains the derivatives of all
orders. The generating functional can then be written in the form of the
Feynman-Kac integral Zinn-Justin :

${e^{W\left[{\eta\left(t\right)}\right]}}\sim\mathop{\int}\left[d\xi\right]e^{-\int
dt\left({{\cal L}\left(t\right)-\eta\left(t\right)\xi\left(t\right)}\right)}$
(20)

In the previous subsection, we have seen that the logarithm of the moments
$\langle x^{m}\rangle$ is determined by the functions $\eta(t)$ that are
constant in the time range $\left[0,T\right]$ (16). For this reason, we are
interested in the values of $W\left[{\eta\left(t\right)}\right]$ on ’slow-
changing’ functions with characteristic time scale $T$:
$\eta_{T}(t)=\eta(t/T)$ (the low-frequency limit). In (20), we rescale the
time $t=\tau T$ and change accordingly the integration variable:
$\xi\left(t\right)={\xi_{T}}\left(\tau\right)$. Then from (20) we obtain

${e^{W\left[\eta_{T}\left(t\right)\right]}}\sim\mathop{\int}\left[d{\xi_{T}}\right]e^{-T\int
d\tau{\left({{{\cal
L}_{T}}\left(\tau\right)-\eta\left(\tau\right){\xi_{T}}\left(\tau\right)}\right)}}$
(21)

where

${{\cal L}_{T}}\left(\tau\right)={\cal
L}\left(\xi_{T},\;{\frac{1}{T}{\partial_{\tau}}{\xi_{T}}},\;\frac{1}{T^{2}}{\partial_{\tau}^{2}}{\xi_{T}},\dots\right)$

As $T\to\infty$, one can substitute the rate function

$J(\xi_{T})={{\cal L}(\xi_{T},0,0,\dots)}+C$ (22)

for ${{\cal L}_{T}}$. Here $C$ is a normalization constant. Then we estimate
the integral by means of the saddle point method:

$\mathop{\int}\left[d{\xi_{T}}\right]e^{-T\int
d\tau{\left(J(\xi_{T}(\tau))-\eta(\tau){\xi_{T}}(\tau)\right)}}\sim e^{-T\int
d\tau\left(J(\xi_{s}(\tau))-\eta\left(\tau\right)\xi_{s}(\tau)\right)}$

where $\xi_{s}(\tau)$ is defined by the minimum condition

$\frac{{\partial
J\left({{\xi_{s}}}\right)}}{{\partial\xi}}=\eta\left(\tau\right)$ (23)

Eventully, we get

${e^{W\left[\eta_{T}\left(t\right)\right]}}={e^{W\left[\eta\left(t/T\right)\right]}}\sim
e^{T\int w(\eta(\tau))d\tau}$ (24)

where

$w(\eta)=\sup\limits_{\xi}\left(\eta\xi-J(\xi)\right)$ (25)

The normalization condition $W[0]=0$ results in the claim $w(0)=0$. Now,
substituting (24) in (16) we find

$\langle{x^{\eta}}\left(T\right)\rangle\sim
e^{T\int\limits_{0}^{1}w(\eta)d\tau}=e^{Tw(\eta)}$

Comparing this with (10), (13) we see that the function $w(\eta)$ defined in
this subsection coincides with the GLE found in (11) and (19). From (11) and
(25) it also follows that the function $J(\xi)$ defined in (22) coincides with
the rate function.

So, both ways to determine the rate function are equivalent. In multi-
dimensional case the second way appears to be more convenient.

##  III Multi-dimensional equation

###  III.1 Equation for the Iwasawa components

Now we return to the matrix equation (2) where $A(t)$ is a stationary
continuous stochastic process with regular fast-decaying connected
correlations:

$\left\langle
A_{{i_{1}}{j_{1}}}\left({{t_{1}}}\right)...A_{{i_{n}}{j_{n}}}\left({{t_{n}}}\right)\right\rangle_{c}=W_{{i_{1}}{j_{1}}...{i_{n}}{j_{n}}}^{\left(n\right)}\left({{t_{1}}-{t_{2}},\ldots,{t_{1}}-{t_{n}}}\right)$
(26)

and non-local Lagrangian density: ${{\cal L}_{A}}\left({\bf
A},{{\partial_{t}}{\bf A}},\ {{\partial^{2}_{t}}{\bf A}},\ \dots\right)$. The
well-defined solution (3) exists for any continuous realization of ${\bf
A}(t)$, but noncommutativity makes it difficult to use: there is a
$T$-exponential instead of a usual exponential, and it seems impossible to
apply the large deviations approach for $\int{\bf A}dt$. However, we will see
in what follows that this is possible in the case of isotropic law of ${\bf
A}$ at least for the important ’diagonal’ (in the sense of Iwasawa
decomposition) part of the evolution matrix.

To separate the Iwasawa components, we rewrite the equation (2) in the form

${\bf A}=\partial_{t}{\bf Q}\;{\bf Q}^{-1}$

and substitute the Iwasawa decomposition for ${\bf Q}$. We obtain

${\bf A}={\bf R}\,{\bf X}\,{{\bf R}^{-1}}\ ,\qquad{\bf
X}={\boldsymbol{\xi}+\boldsymbol{\zeta}+\boldsymbol{\theta}}$ (27)

where

$\boldsymbol{\xi}=\left({{\partial_{t}}{\bf D}}\right){{\bf D}^{-1}}\
,\qquad\boldsymbol{\zeta}={\bf D}\left({{\partial_{t}}{\bf Z}}\right){{\bf
Z}^{-1}}{{\bf D}^{-1}}\ ,\qquad\boldsymbol{\theta}={{\bf
R}^{-1}}\left({{\partial_{t}}{\bf R}}\right)$ (28)

The matrices $\boldsymbol{\xi}$, $\boldsymbol{\zeta}$ and
$\boldsymbol{\theta}$ are diagonal, nilpotent upper triangular and
antisymmetric, respectively. The equation (28) can be rewritten as

$\displaystyle{\partial_{t}}{\bf D}=\boldsymbol{\xi}{\bf D}$ (29)
$\displaystyle{\partial_{t}}{\bf Z}={{\bf D}^{-1}}\boldsymbol{\zeta}{\bf
D}\,{\bf Z}$ (30) $\displaystyle{\partial_{t}}{\bf R}={\bf
R}\boldsymbol{\theta}$ (31)

Thus, treating $\boldsymbol{\xi}$, $\boldsymbol{\zeta}$ and
$\boldsymbol{\theta}$ as independent variables, one could separate the
equations for ${\bf D}$ and for ${\bf R}$. Moreover, the elements $D_{i}$
satisfy one-dimensional equations (29) same as the equation (9). So, (8) takes
the form

$w(\eta_{1},...,\eta_{d})\equiv
w_{\xi}(\eta_{1},...,\eta_{d})=\lim\limits_{T\to\infty}\frac{1}{T}\ln\left\langle
e^{\int\limits_{0}^{T}(\xi_{1}\eta_{1}+...+\xi_{d}\eta_{d})dt}\right\rangle$
(32)

To calculate this, it would be enough to know the rate function of
$\xi=\\{\xi_{1},...,\xi_{d}\\}$ and make use of (25). So, in the next
subsection we discuss the relation between the Lagrangian densities of ${\bf
X}$ and ${\bf A}$.

###  III.2 Change of variables

One can consider (27) as a functional transformation from $A$ to
$X$-variables,

${\bf A}={\bf R}[{\bf X}]{\bf X}{\bf R}^{-1}[{\bf X}]$ (33)

where the dependence ${\bf R}({\bf X})$ is determined by (31). To find the
probability density of ${\bf X}(t)$, one has to calculate the Jacobian:

${\cal J}\left[X\right]={\rm
Det}\left({\frac{{\delta{A_{ij}}\left(t\right)}}{{\delta{X_{kp}}\left({t^{\prime}}\right)}}}\right)$
(34)

It was calculated, e.g., in JOSS1 :

${\cal J}[X]=e^{\displaystyle\int{{\,\rm tr\,}\left({\boldsymbol{\eta}_{0}{\bf
X}\left(t\right)}\right)dt}}$ (35)

where

${\left({{\eta_{0}}}\right)_{kp}}=\left({\frac{{d+1}}{2}-k}\right){\delta_{kp}}$
(36)

In the Appendix A to this paper we derive this result taking the continuous
limit of a stochastic difference equation.

With account of (34) and (33), from the condition ${\cal P}({\bf
X})\left[d{\bf X}\right]={\cal P}({\bf A})\left[d{\bf A}\right]$ we get the
expression for the Lagrangian density of ${\bf X}$:

${\cal L}_{X}=-{\,\rm tr\,}({\boldsymbol{\eta}_{0}}{\bf X})+{\cal L}_{A}({\bf
A},\partial_{t}{\bf A},\partial^{2}_{t}{\bf A},\dots)$

where ${\bf A}$ is a function of ${\bf X}$ in accordance with (33). From (31)
it follows that for any ${\bf F}(t)$,

$\partial_{t}\left({{\bf R}\left(t\right){\bf F}(t){{\bf
R}^{-1}}\left(t\right)}\right)={\bf R}\left(t\right)\left({{D_{t}}{\bf
F}\left(t\right)}\right){{\bf R}^{-1}}\left(t\right)\ ,$ (37)

where

$D_{t}{\bf F}=\partial_{t}{\bf F}+[\boldsymbol{\theta},{\bf F}]$ (38)

where $[a,b]=ab-ba$. Substituting this for the arguments of ${\cal L}_{A}$, we
obtain

${\cal L}_{X}=-{\,\rm tr\,}({\bf\eta_{0}}{\bf X})+{\cal L}_{A}({\bf
RXR}^{-1},{\bf R}(D_{t}{\bf X}){\bf R}^{-1},{\bf R}(D^{2}_{t}{\bf X}){\bf
R}^{-1},\dots)$ (39)

This expression contains ${\bf R}$, which is the Volterra product integral of
the components of ${\bf X}$, ${\bf R}\left[{{\bf
X},t}\right]=\mathop{\prod}\limits_{\tau=0}^{t}\left({{\bf
1}+\boldsymbol{\theta}\left(\tau\right)d\tau}\right)$. However, we will see
below that in the case of isotropic processes ${\bf R}$ vanishes.

##  IV Isotropic systems

We now make use of the claim that ${\bf A}$ is isotropic, in particular, ${\bf
A}(t)$ has the same probability density as ${\bf OA}(t){\bf O}^{T}$ for any
${\bf O}\in O(d)$. For such processes, the Lagrangian density ${\cal L}_{A}$
can be presented as a sum of different combinations of traces with arguments
containing products of $A$, $A^{T}$, and their derivatives. As one substitutes
${\bf RXR}^{-1}$ for ${\bf A}$, the ’plates’ ${\bf R}$,${\bf R}^{-1}$ in the
expressions vanish; but, in accordance with (37) and (39), all time
derivatives in the expression for the Lagrangian density change to ’long
derivatives’ (38),

${\bf A}\mapsto{\bf X},\quad\partial_{t}^{p}{\bf A}\mapsto D_{t}^{p}{\bf X}$

As we proceed from ${\cal L}_{X}$ to the rate function as in (22), we have to
set all the time derivatives of ${\bf X}$ zero. But the commutators stay in
their places, so the rate function takes the form

$J_{X}({\bf X})={\cal L}_{X}({\bf X},0,0,\dots)+C=-{\,\rm
tr\,}({\boldsymbol{\eta}_{0}}{\bf X})+{\cal L}_{A}({\bf
X},[\boldsymbol{\theta},{\bf
X}],[\boldsymbol{\theta},[\boldsymbol{\theta},{\bf X}]],\dots)+C$ (40)

One can separate the rate function of the matrix ${\bf A}$:

$J_{A}({\bf A})={\cal L}_{A}({\bf A},0,\dots)$

and present $J_{X}$ in the form:

$J_{X}({\bf X})=-{\,\rm tr\,}({\boldsymbol{\eta}_{0}}{\bf X})+J_{A}({\bf
X})+\delta J({\bf X})+C\ ,$ (41) $\delta J({\bf X})={\cal L}_{A}({\bf
X},[\boldsymbol{\theta},{\bf
X}],[\boldsymbol{\theta},[\boldsymbol{\theta},{\bf X}]],\dots)-{\cal
L}_{A}({{\bf X}},0,\dots)$

We define $w_{X}(\boldsymbol{\mu})$ for the matrix process ${\bf X}$ in the
same way as in (11),(25),

$w_{X}(\boldsymbol{\mu})=\sup\limits_{{\bf X}}\left({\,\rm
tr\,}\left(\boldsymbol{\mu}{\bf X}\right)-J_{X}({\bf
X})\right)=\lim\limits_{T\to\infty}\frac{1}{T}\ln\left\langle
e^{\int\limits_{0}^{T}{\,\rm tr\,}\left(\boldsymbol{\mu}{\bf
X}\right)dt}\right\rangle$ (42)

To this purpose, we have to find the extremum point ${\bf X}_{s}$ analogous to
$\xi_{s}$ in (23):

$\frac{\partial J_{X}({\bf X}_{s})}{\partial X_{qr}}=\mu_{qr}$ (43)

This equations system is very complicated. However, it can be simplified
significantly as we are interested only in the GLE of the diagonal elements
$X_{kk}=\xi_{k}$. From (32) it follows that

$w_{\xi}(\eta_{1},\dots,\eta_{d})=w_{X}(\boldsymbol{\eta})$ (44)

where $\boldsymbol{\eta}=diag(\eta_{1},\eta_{2},...,\eta_{d})$. So, to find
$w_{\xi}$ we can restrict ourself to the diagonal matrices
$\boldsymbol{\mu}=\boldsymbol{\eta}$ in (43), so the extremum condition takes
the form:

$\frac{\partial J_{X}({\bf X}_{s})}{\partial X_{qr}}=0,\ q\neq
r;\qquad\frac{\partial J_{X}({\bf X}_{s})}{\partial X_{qq}}=\eta_{q},\ q=1..d$
(45)

This system has the diagonal solution:

${\bf X}_{s}=\boldsymbol{\xi}_{s}$ (46)

where ${\boldsymbol{\xi}_{s}}$ satisfies the relation

$\frac{\partial J_{A}(\boldsymbol{\xi}_{s})}{\partial
X_{qq}}-(\eta_{0})_{qq}=\eta_{q}$ (47)

Indeed, we recall that ${\bf
X}=\boldsymbol{\xi}+\boldsymbol{\theta}+\boldsymbol{\zeta}$, and $J_{X}({\bf
X})$ is a combination of traces ${\,\rm
tr\,}f(\boldsymbol{\xi},\boldsymbol{\theta},\boldsymbol{\zeta})$ where $f$ is
some product of the matrices. Since $\boldsymbol{\xi}$ is diagonal and
$\boldsymbol{\theta},\boldsymbol{\zeta}$ have zero diagonal elements, each
summand in $J_{X}$ contains either zero or more than one of the matrices
$\boldsymbol{\theta},\boldsymbol{\zeta}$. Taking the derivative with respect
to $X_{qr},\ q\neq r$ leaves the summands with at least one multiplier
$\boldsymbol{\theta}$ or $\boldsymbol{\zeta}$, and subsequent setting ${\bf
X}=\boldsymbol{\xi}_{s}$ makes them zero; so, the first equation in (45) holds
automatically. On the other hand, each summand in $\delta J({\bf X})$ contains
at least one $\boldsymbol{\theta}$ as a multiplier, so $\partial\delta
J/\partial X_{qq}=0$.

We note that $\boldsymbol{\theta}_{s}=0$ and thus, $\delta J({\bf X}_{s})=0$.
From (44), (42), (41) it then follows

$w_{\xi}(\eta_{1},\eta_{2},...,\eta_{d})=\
tr((\boldsymbol{\eta}+{\boldsymbol{\eta}_{0}}){\boldsymbol{\xi}_{s}})-J_{A}(\boldsymbol{\xi}_{s})-C$
(48)

The statistical isotropy of ${\bf A}$ implies $\langle{\bf A}_{i\neq
j}\rangle=0$. Thus, $\partial J_{A}/\partial A_{qr}(\boldsymbol{\xi}_{s})=0$
for $q\neq r$, and $J_{A}(\boldsymbol{\xi})$ coincides with the rate function
of the diagonal elements $\alpha_{q}=A_{qq}$:

$J_{\alpha}(\alpha_{1},...,\alpha_{d})=J_{A}(\boldsymbol{\alpha}),$
$\boldsymbol{\alpha}=diag(\alpha_{1},...,\alpha_{d})$

So, (48) proves that the GLE and, as a consequence, also the Lyapunov
exponents are completely determined by the rate function of the diagonal
elements of $\mathbf{A}$.

By analogy with (42), we define the local cumulant-generating function of
these diagonal elements:

$w_{\alpha}(\boldsymbol{\eta})=\sup\limits_{\boldsymbol{\alpha}}\left({\,\rm
tr\,}\left(\boldsymbol{\eta}\boldsymbol{\alpha}\right)-J_{\alpha}(\alpha_{1},...,\alpha_{d})\right)=\lim\limits_{T\to\infty}\frac{1}{T}\ln\left\langle
e^{\int\limits_{0}^{T}{\,\rm
tr\,}\left(\boldsymbol{\eta}\boldsymbol{\alpha}\right)dt}\right\rangle$

It is related to the cumulants of $\alpha_{i}$ by

${w_{\alpha}}\left({{\eta_{1}},...,{\eta_{d}}}\right)=\mathop{\sum}\limits_{n=1}^{\infty}\mathop{\sum}\limits_{{i_{1}}...{i_{n}}}\frac{w_{{i_{1}}....{i_{n}}}^{\left(n\right)}}{{n!}}{\eta_{{i_{1}}}}...{\eta_{{i_{n}}}}$
(49)

where

$w_{{i_{1}}...{i_{n}}}^{\left(n\right)}=\mathop{\int}\nolimits
d{t_{2}}...d{t_{n}}\left\langle\alpha_{{i_{1}}}\left({{t_{1}}}\right)...\alpha_{{i_{n}}}\left({{t_{n}}}\right)\right\rangle_{c}$

From (48) we see that $w_{\xi}(\eta_{1},...,\eta_{d})$ can be reduced to
$w_{\alpha}(\boldsymbol{\eta}+\boldsymbol{\eta}_{0})$; with account of the
normalization condition $w_{\xi}(0)=\ln\langle 1\rangle=0$, we obtain GLE and
the Lyapunov exponents:

$w_{\xi}(\eta_{1},\eta_{2},...,\eta_{d})=w_{\alpha}(\boldsymbol{\eta}+\boldsymbol{\eta}_{0})-w_{\alpha}(\boldsymbol{\eta}_{0})$
(50)
$\lambda_{k}=\frac{\partial}{{\partial{\eta_{k}}}}{w_{\alpha}}\left(\boldsymbol{\eta}_{0}\right)$
(51)
${\left({{\eta_{0}}}\right)_{kp}}=\left({\frac{{d+1}}{2}-k}\right){\delta_{kp}}$

###  IV.1 Gaussian process

Consider now the important particular case: let ${\bf A}(t)$ be Gaussian
continuous process with zero mean and given second-order correlator:

$\langle{A_{ij}}\left(t\right)\rangle=0\
,\qquad\left\langle{A_{ij}}\left({{t_{1}}}\right){A_{kp}}\left({{t_{2}}}\right)\right\rangle_{c}=K_{(ij)(kp)}\Phi(t_{1}-t_{2})$

where

$K_{(ij)(kp)}=-a{\delta_{ij}}{\delta_{kp}}+b{\delta_{ik}}{\delta_{jp}}+c{\delta_{ip}}{\delta_{jk}}$
(52)

where $a,b,c$ are some constants and $\Phi(t)$ is some regular even fast
decaying function, $\int\Phi(t)dt=1$. The form (52) of $K_{(ij)(kp)}$ is
determined by the isotropy.

From (49) it follows the expression for the cumulant generation function of
the diagonal elements of ${\bf A}$:

$w_{\boldsymbol{\alpha}}(\boldsymbol{\eta})=-\frac{a}{2}({\,\rm
tr\,}\boldsymbol{\eta})^{2}+\frac{b+c}{2}{\,\rm tr\,}\boldsymbol{\eta}^{2}$

From (50) we get

$w_{\xi}(\eta_{1},...,\eta_{d})=\left({b+c}\right)\mathop{\sum}\limits_{k=1}^{d}\left({\frac{{d+1}}{2}-k}\right){\eta_{k}}+\frac{1}{2}\mathop{\sum}\limits_{k,p=1}^{d}\left({\left({b+c}\right){\delta_{kp}}-a}\right){\eta_{k}}{\eta_{p}}\
,$ (53) $\lambda_{k}=\left({b+c}\right)\left({d+1}-2k\right)$ (54)

In many applications one considers traceless matrices ${\,\rm tr\,}{\bf A}=0$.
With this additional requirement, the coefficients $a,b,c$ are associated by
the relation $b+c-ad=0$, which can be taken into account in (53).

##  V Conclusion. Effective $\delta$-process

So, in the paper we consider linear stochastic equations systems (1) with
statistically isotropic matrix random process ${\bf A}(t)$ that have regular
fast-decaying connected correlations (26). We find the explicit expressions
(50) for the generalized Lyapunov exponents in terms of rate functions of the
diagonal elements of ${\bf A}$.

Now we reformulate the results in the form that is useful for physical
applications.

We find that the correlations of the diagonal elements of ${\bf A}$ contribute
to GLE only via their integrals:

$w_{{i_{1}}...{i_{n}}}^{\left(n\right)}=\mathop{\smallint}\nolimits
d{t_{2}}...d{t_{n}}W_{{i_{1}}{i_{1}}...{i_{n}}{i_{n}}}^{\left(n\right)}\left({{t_{1}}-{t_{2}},...,{t_{1}}-{t_{n}}}\right)\qquad(no\
summation)$

In (50), the GLE are expressed in terms of the cumulant-generating function of
the diagonal elements of the matrix ${\bf A}$,

${w_{\alpha}}\left({{\eta_{1}},...,{\eta_{d}}}\right)=\mathop{\sum}\limits_{n=1}^{\infty}\mathop{\sum}\limits_{{i_{1}}...{i_{n}}}\frac{w_{{i_{1}}....{i_{n}}}^{\left(n\right)}}{{n!}}{\eta_{{i_{1}}}}...{\eta_{{i_{n}}}}$

So, there exists the sequence of formal random processes ${\bf A}_{\epsilon}$
with connected correlations

$\frac{1}{{{\epsilon^{n-1}}}}W_{{i_{1}}{j_{1}}...{i_{n}}{j_{n}}}^{\left(n\right)}\left({\frac{{{t_{1}}-{t_{2}}}}{\epsilon},...,\frac{{{t_{1}}-{t_{n}}}}{\epsilon}}\right)$

which produce identical GLE. Going to the formal limit $\epsilon\to 0$ one can
define the ’effective $\delta$\- process’ ${\bf A}_{0}$ with singular
correlation functions

${\rm{\Delta}}_{ij...kp}^{\left(n\right)}=w_{ij...kp}^{\left(n\right)}\delta\left({{t_{1}}-{t_{2}}}\right)...\delta\left({{t_{1}}-{t_{n}}}\right)$
(55)

This formal process provides the same GLE and allows to split correlations and
get closed equations for different averages (see Appendix B). Despite its
formal nature, it is a handy instrument for calculations for the problems that
appear in theory of turbulence, turbulent transport, kinematic dynamo in
turbulent flows etc APS22 ; PoF22 ; PoF20 .

In the particular case of the Gaussian isotropic processes, the result (54)
coincides with the well-known expressions Liao ; Gamba  obtained for the
differential equation $\dot{{\bf Q}}=d{\bf W}(t)/dt{\bf Q}$ in the frame of
Stratonovich stochastic convention, $d{\bf Q}(t)=d{\bf W}(t)\circ{\bf Q}(t)$.
Thus, for these processes the effective $\delta$-process has not only formal
sense but can also be expressed in terms of the Wiener process’ derivative.
This corresponds to the Wong-Zakai theorem Wong-Zakai .

We also make some notes on the relation between our approach and the
renovation model IS-renov . In our approach, the noise is stationary for any
correlation time while in the renovation model it becomes stationary only as
$\tau\to 0$. Actually, the results of both approaches coincide as $\tau\to 0$.
Possibly, the non-stationarity can be taken into account in our approach by
means of corrections to $w_{i_{1}\cdots i_{n}}^{(n)}$. However, this is a
subject for separate issue.

It is also important to note that even in the isotropic case, one can
substitute the effective $\delta$-process for the real matrix process only
when calculating the long-term asymptotics of $E_{k}$ and their combinations.
For the quantities that depend on the non-diagonal elements $\theta_{ij}$ and
$\zeta_{ij}$ (e.g., the coordinates $x_{k}$), the asymptotic behavior is
determined not only by the rate function of the matrix elements $A_{ij}$ but
also by the shape of their correlation functions. This illustrates the fact
that the possibility to introduce the effective $\delta$-process is a non-
trivial feature of multi-dimensional isotropic stochastic systems with
multiplicative noise.

###### Acknowledgements.

The authors are grateful to Professor A. V. Gurevich for his permanent
attention to their work.

##  Appendix A: Calculation of the Jacobian (34)

The Jacobian was calculated in JOSS1  by means of operations with continuous
stochastic processes and functional integrals. The notion of Jacobian is
difficult to define for a continuous process. It is natural to define it for a
discrete process and then take the continuous limit. The result must not
depend on the way of discretization. Here we calculate the Jacobian for one
particular discretization of the random process ${\bf Q}$ and the
corresponding stochastic difference equation. Then we show that for another
choice of the discretization with the same continuous limit, the result is the
same. In analogous way, one can check that any difference equation with the
same continuous limit would lead to the same result.

### Discretization $\grave{a}$ la Stratonovich

So, again we start with the equation (2),

${\partial_{t}}{\bf Q}\left(t\right)={\bf A}\left(t\right){\bf
Q}\left(t\right),\ \ {\bf Q}\left(0\right)={\bf 1}$

Although $A(t)$ is a finite - correlation time process and thus the
differential equation is well defined, we still consider its discrete analog
in order to substitute ordinary derivatives for variational derivatives. Thus,
we split T into N discrete intervals, $\Delta t$ being much smaller than the
correlation time of ${\bf A}$, and consider the discrete equation:

$\Delta{\bf Q}_{t}\equiv{\bf Q}_{t+1}-{\bf Q}_{t}={\bf A}_{t}\frac{{\bf
Q}_{t}+{\bf Q}_{t+1}}{2}\Delta t$ (56)

We use the Stratonovich-type discretization because to calculate the Jacobian,
we need the second order accuracy in $\Delta t$, and the Stratonovich choice
allows to get this accuracy for $\Delta{\bf Q}$ without writing the second
order derivative.

Multiplying (56) by ${\bf Q}_{t}^{-1}$ and by ${\bf Q}_{t+1}^{-1}$ and taking
the sum, we get:

${\bf A}_{t}\Delta t=\frac{1}{2}\left({\bf Q}_{t+1}{\bf Q}_{t}^{-1}-{\bf
Q}_{t}{\bf Q}_{t+1}^{-1}\right)+O(\Delta t^{3})$

Now we make use of the Iwasawa decomosition and substitute ${\bf Q}_{t}={\bf
R}_{t}{\bf D}_{t}{\bf Z}_{t}$, ${\bf Q}_{t+1}={\bf R}_{t+1}{\bf D}_{t+1}{\bf
Z}_{t+1}$. With unified notation $\Delta{\bf F}_{t}={\bf F}_{t+1}-{\bf F}_{t}$
and taking into account $\Delta\left({\bf F}^{-1}\right)_{t}=-{\bf
F}_{t}^{-1}\Delta{\bf F}_{t}{\bf F}_{t}^{-1}+{\bf F}_{t}^{-1}\Delta{\bf
F}_{t}{\bf F}_{t}^{-1}\Delta{\bf F}_{t}{\bf F}_{t}^{-1}+O(\Delta t^{3})$ we
obtain

$\displaystyle{\bf A}\Delta t=\frac{1}{2}\left(\Delta{\bf R}_{t}{\bf
D}_{t+1}{\bf Z}_{t+1}+{\bf R}_{t}\Delta{\bf D}_{t}{\bf Z}_{t+1}+{\bf
R}_{t}{\bf D}_{t}\Delta{\bf Z}_{t}\right){\bf Z}_{t}^{-1}{\bf D}_{t}^{-1}{\bf
R}_{t}^{-1}-$ $\displaystyle-\frac{1}{2}{\bf R}_{t}{\bf D}_{t}{\bf
Z}_{1}\left(\Delta({\bf Z}_{t}^{-1}){\bf D}_{t+1}^{-1}{\bf R}_{t+1}^{-1}+{\bf
Z}_{t}^{-1}\Delta({\bf D}_{t}^{-1}){\bf R}_{t+1}^{-1}+{\bf Z}_{t}^{-1}{\bf
D}_{t}^{-1}\Delta({\bf R}_{t}^{-1})\right)=$
$\displaystyle\begin{array}[]{l}={\bf R}_{t}\left[\frac{1}{2}\left({\bf
R}_{t}^{-1}{\bf R}_{t+1}-{\bf R}_{t+1}^{-1}{\bf
R}_{t}\right)+\frac{1}{2}\left({\bf R}_{t}^{-1}{\bf R}_{t+1}\Delta{\bf
D}_{t}{\bf D}_{t}^{-1}+\Delta{\bf D}_{t}{\bf D}_{t}^{-1}{\bf R}_{t+1}^{-1}{\bf
R}_{t}-(\Delta{\bf D}_{t}{\bf D}_{t}^{-1})^{2}\right)\right.+\\\
\left.+\frac{1}{2}\left({\bf R}_{t}^{-1}{\bf R}_{t+1}{\bf D}_{t+1}\Delta{\bf
Z}_{t}{\bf Z}_{t}^{-1}{\bf D}_{t}^{-1}+{\bf D}_{t}\Delta{\bf Z}_{t}{\bf
Z}_{t}^{-1}{\bf D}_{t+1}^{-1}{\bf R}_{t+1}^{-1}{\bf R}_{t}-({\bf
D}_{t}\Delta{\bf Z}_{t}{\bf Z}_{t}^{-1}{\bf D}_{t}^{-1})^{2}\right)\right]{\bf
R}_{t}^{-1}+O(\Delta t^{3})\end{array}$ (59)

We denote

$\boldsymbol{\theta}_{t}\Delta t\equiv\frac{1}{2}\left({\bf R}_{t}^{-1}{\bf
R}_{t+1}-{\bf R}_{t+1}^{-1}{\bf R}_{t}\right)=\frac{1}{2}\left({\bf
R}_{t}^{-1}+{\bf R}_{t+1}^{-1}\right)\Delta{\bf R}_{t}$ (60)

This is an antisymmetric matrix; from (60) it follows that, with accuracy
$O(\Delta t^{3})$,

$\Delta{\bf R}_{t}=\frac{{\bf R}_{t}+{\bf
R}_{t+1}}{2}\boldsymbol{\theta}_{t}\Delta t$ (61)

which is the discrete analog to (31) in accord with the Stratonovich approach.
Analogously, we claim that

$\Delta{\bf D}_{t}=\boldsymbol{\xi}_{t}\frac{{\bf D}_{t}+{\bf
D}_{t+1}}{2}\Delta t$

and arrive at

$\boldsymbol{\xi}_{t}\Delta t=\Delta{\bf D}_{t}{\bf
D}_{t}^{-1}-\frac{1}{2}\left(\Delta{\bf D}_{t}{\bf
D}_{t}^{-1}\right)^{2}+O(\Delta t^{3})$

Demanding

$\Delta{\bf Z}_{t}=\left(\frac{{\bf D}_{t}+{\bf
D}_{t+1}}{2}\right)^{-1}\boldsymbol{\zeta}_{t}\left(\frac{{\bf D}_{t}+{\bf
D}_{t+1}}{2}\right)\left(\frac{{\bf Z}_{t}+{\bf Z}_{t+1}}{2}\right)\Delta t$

we also get (up to $O(\Delta t^{3})$ accuracy):

$\begin{array}[]{ll}\boldsymbol{\zeta}_{t}\Delta t&=({\bf D}+\frac{\Delta{\bf
D}}{2})\Delta{\bf Z}\left(\frac{{\bf Z}_{t}+{\bf Z}_{t+1}}{2}\right)^{-1}({\bf
D}+\frac{\Delta{\bf D}}{2})^{-1}\\\ &=\left({\bf 1}+\frac{\Delta{\bf
D}_{t}{\bf D}_{t}^{-1}}{2}\right){\bf D}_{t}\Delta{\bf Z}_{t}{\bf
Z}_{t}^{-1}{\bf D}_{t}^{-1}\left({\bf 1}-\frac{{\bf D}_{t}\Delta{\bf
Z}_{t}{\bf Z}_{t}^{-1}{\bf D}_{t}^{-1}}{2}\right)\left({\bf
1}-\frac{\Delta{\bf D}_{t}{\bf D}_{t}^{-1}}{2}\right)+O(\Delta
t^{3})\end{array}$

The first summand in the round brackets in (59) is $\boldsymbol{\theta}_{t}$;
now, we note that the summand in the second brackets is

${\bf R}_{t}^{-1}{\bf R}_{t+1}\boldsymbol{\xi}_{t}\Delta
t+\boldsymbol{\xi}_{t}\Delta t{\bf R}_{t}^{-1}{\bf R}_{t+1}+O(\Delta
t^{3})=({\bf 1}+\boldsymbol{\theta}_{t}\Delta t)\boldsymbol{\xi}_{t}\Delta
t+\boldsymbol{\xi}_{t}\Delta t({\bf 1}+\boldsymbol{\theta}_{t}\Delta
t)+O(\Delta t^{3})$

Finally, the third bracket can be written as ${\bf R}_{t}^{-1}{\bf
R}_{t+1}\boldsymbol{\zeta}_{t}\Delta t+\boldsymbol{\zeta}_{t}\Delta t{\bf
R}_{t}^{-1}{\bf R}_{t+1}+O(\Delta t^{3})$.

Summarizing, we rewrite (59) as

${\bf A}_{t}={\bf R}_{t}\left({\bf
X}_{t}+\frac{1}{2}[\boldsymbol{\theta}_{t},{\bf X}_{t}]\Delta t\right){\bf
R}_{t}^{-1}+O(\Delta t^{2})$ (62)

where

${\bf
X}_{t}=\boldsymbol{\theta}_{t}+\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t}$

Eq. (62) is the discrete analog to Eq. (27).

### Calculation of the determinant

So, now we have to calculate the Jacobian ${\cal J}=\left|\partial
A_{ij\;t}/\partial X_{km\;t^{\prime}}\right|$.

First, we note that from (62) and (61) it follows

$\partial A_{ij\,t}/\partial X_{km\,t^{\prime}}=0\quad{\mbox{f}or\ any}\quad
t<t^{\prime}$

(This is the manifestation of causality)Thus, the matrix $(\partial{\bf
A}/\partial{\bf X})_{ij,t;km,t^{\prime}}$ is a block triangular matrix, and
its determinant is equal to the product of the determinants of the diagonal
blocks, $t=t^{\prime}$:

${\cal J}=\prod\limits_{t}\left|\frac{\partial A_{ij\,t}}{\partial
X_{km\,t}}\right|$ (63)

Second, we note that, in accordance with the causality principle, from (61) it
follows that the value of the rotation matrix ${\bf R}_{t}$ depends only on
the ’previous-time’ values of $\boldsymbol{\theta}_{t^{\prime}<t}$ and does
not depend on the ’simultaneous’ value $\boldsymbol{\theta}_{t}$,

$\partial{\bf R}_{t}/\partial\boldsymbol{\theta}_{t^{\prime}\geq t}=0$

Thus, in (62) the derivative must be taken only over the multiplier in the
square brackets. We now introduce the multiindices $\alpha=\\{ij\\}$ and the
$d^{2}\times d^{2}$ matrix

${\cal R}_{ij,km}=R_{ik}R_{jm}$

Then (62) can be written in the form:

$A_{\alpha}={\cal R}_{\alpha\beta}\left(X_{\beta}+M_{\beta}\Delta t\right)$

where

${\bf M}=\frac{1}{2}[\boldsymbol{\theta},{\bf X}]$

So,

$\left|\frac{\partial A_{\alpha}}{\partial X_{\gamma}}\right|=\left|{\cal
R}_{\alpha\beta}\right|\cdot\left|\delta_{\beta\gamma}+\frac{\partial
M_{\beta}}{\partial X_{\gamma}}\Delta t\right|$

Since the matrices ${\bf R}$ are orthogonal, $\cal R$ is also orthogonal,
i.e., ${\cal R}{\cal R}^{T}={\bf 1}$. Thus, $\det{\cal R}=1$. The second
determinant, to an accuracy of $\sim O(\Delta t)$, can be reduced to the trace
of $\partial{\bf M}/\partial{\bf X}$:

$\left|\frac{\partial A_{\alpha}}{\partial
X_{\gamma}}\right|=1+\left(\frac{\partial M_{\beta}}{\partial
X_{\gamma}}\right)\delta_{\beta\gamma}\Delta t+O(\Delta t^{2})$ (64)

Finally, we make use of the fact that only the lower triangular part of ${\bf
X}$ determines the values of $\boldsymbol{\theta}$, while the diagonal and
upper triangular components are ’responsible’ for $\boldsymbol{\xi}$ and
$\boldsymbol{\zeta}$, correspondingly. So,

$\theta_{ij}=\left\\{\begin{array}[]{l}X_{ij}\ \ if\ i>j\ ,\\\ 0\ \ if\ i=j\
,\\\ -X_{ji}\ \ if\ i<j\end{array}\right.$

In particular, $\partial\theta_{ij}/\partial X_{km}=0$ if $k\leq m$. Thus,
taking the derivative of $\bf M$, we obtain

${\,\rm tr\,}(\partial{\bf M}/\partial{\bf
X})=\sum\limits_{i,j}\left(\frac{\partial M_{ij}}{\partial
X_{ij}}\right)=\frac{1}{2}\sum\limits_{i,j}(\theta_{ii}-\theta_{jj})+\sum\limits_{i>j}(X_{jj}-X_{ii})=\sum\limits_{j=1}^{d}(d-2j+1)X_{jj}$

Combining this with (64) and (63), we eventually get

${\cal J}=\prod\limits_{t}(1+\sum\limits_{j=1}^{d}\eta_{j}X_{jj}\Delta
t)+O(\Delta t^{2})=e^{\sum\limits_{t}\sum\limits_{j=1}^{d}\eta_{j}X_{jj}\Delta
t}\ ,\quad\eta_{j}=\frac{(d-2j+1)}{2}$

Taking the continuous limit $\Delta t\to 0$ we arrive at the integral

${\cal J}=e^{\displaystyle\int\sum\limits_{j=1}^{d}\eta_{j}X_{jj}dt}$

which coincides with Eq-s (35), (36).

### Discretization $\grave{a}$ la Ito

The same result can be obtained in other discretization settings; However, in
general case one has to keep the terms up to the second order in $\Delta t$ in
the initial difference equation for ${\bf Q}$. Here we consider the
It$\hat{\mbox{o}}$-type discretization.

We rewrite the initial continuous differential equation (2), $\partial_{t}{\bf
Q}={\bf A}{\bf Q}$, in the integral form:

${\bf Q}(t)={\bf Q}_{0}+\int\limits_{t_{0}}^{t}{\bf A}{\bf Q}dt$

Applying this equation to the time range from $t$ to $t+\Delta t$ and solving
it by means of iterations, after two iterations we get:

${\bf Q}(t+\Delta t)={\bf Q}(t)+\bar{{\bf A}}{\bf Q}(t)\Delta
t+\frac{1}{2}\bar{{\bf A}}^{2}{\bf Q}(t)\Delta t^{2}$

where $\bar{{\bf A}}$ is the time average of ${\bf A}(t)$ over the range
$\Delta t$. Basing on this equation, we write the difference equation:

${\bf Q}_{t+1}={\bf Q}_{t}+\left({{\bf A}}_{t}\Delta t+\frac{1}{2}{{\bf
A}_{t}}^{2}\Delta t^{2}\right){\bf Q}_{t}$ (65)

Multiplying (65) by ${\bf Q}_{t}^{-1}$ and making use of the Iwasawa
decomposition, we present the left-hand side in the form:

${\bf Q}_{t+1}{\bf Q}_{t}^{-1}={\bf R}_{t}\left({\bf 1}+{\bf
R}_{t}^{-1}\Delta{\bf R}_{t}\right)\left({\bf 1}+\Delta({\bf D}{\bf
Z})_{t}({\bf D}{\bf Z})_{t}^{-1}\right){\bf R}_{t}^{-1}$ (66)

where $\Delta F_{t}\equiv F_{t+1}-F_{t}$.

Now we formally define $\boldsymbol{\xi}_{t}$ by

$\boldsymbol{\xi}_{t}\Delta t=\Delta{\bf D}_{t}{\bf
D}_{t}^{-1}-\frac{1}{2}(\Delta{\bf D}_{t}{\bf D}_{t}^{-1})^{2}$

which coincides up to the second order in $\Delta t$ with

$\Delta{\bf D}_{t}={\bf D}_{t+1}-{\bf D}_{t}=(\boldsymbol{\xi}_{t}\Delta
t+\frac{1}{2}\boldsymbol{\xi}_{t}^{2}\Delta t^{2}){\bf D}_{t}$

In accordance with the chosen prescription, this difference equation
corresponds to the differential equation (29). Accordingly, (31) corresponds
to

$\Delta{\bf R}_{t}={\bf R}_{t+1}-{\bf R}_{t}={\bf
R}_{t}(\boldsymbol{\theta}_{t}\Delta
t+\frac{1}{2}\boldsymbol{\theta}_{t}^{2}\Delta t^{2})$

What about the equation (30) for the upper-triangular part, with account of
(29) we rewrite it in the form $\partial_{t}({\bf D}{\bf
Z})=(\boldsymbol{\xi}+\boldsymbol{\zeta}){\bf D}{\bf Z}$. Then, the
corresponding difference equation takes the form:

$\Delta({\bf D}{\bf Z})_{t}={\bf D}_{t+1}{\bf Z}_{t+1}-{\bf D}_{t}{\bf
Z}_{t}=\left((\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t})\Delta
t+\frac{1}{2}(\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t})^{2}\Delta
t^{2}\right){\bf D}_{t}{\bf Z}_{t}$

Substituting these expressions in (66) and keeping the terms of the order of
$\Delta t^{2}$ we obtain:

${\bf Q}_{t+1}{\bf Q}_{t}^{-1}=1+{\bf
R}_{t}\left((\boldsymbol{\theta}_{t}+\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t})\Delta
t+(\boldsymbol{\theta}_{t}^{2}+(\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t})^{2})\Delta
t^{2}/2+\boldsymbol{\theta}_{t}(\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t})\Delta
t^{2}\right){\bf R}_{t}^{-1}$

Combining this with (65), we get

${\bf A}_{t}={\bf R}_{t}\left({\bf
X}_{t}+\frac{1}{2}[\boldsymbol{\theta}_{t},{\bf X}_{t}]\Delta t\right){\bf
R}_{t}^{-1}\ ,\quad{\bf
X}_{t}=\boldsymbol{\xi}_{t}+\boldsymbol{\zeta}_{t}+\boldsymbol{\theta}_{t}$

This result coincides with (62) obtained in the Stratonovich convention. The
rest of the derivation is the same as in the Stratonovich case.

## Appendix B. Correlation splitting for $\delta$-process

In this appendix, we derive the analog of the Furutsu-Novikov formula for the
$\delta$-processes.

The Furutsu-Novikov relation for regular processes has the form Furutsu ;
Novikov :

$\langle
A_{ij}(t)g[\mathbf{A}]\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}\int_{\mathbb{R}^{n}}\bigl{\langle}A_{ij}(t)A_{i_{1}j_{1}}(t_{1})\dots
A_{i_{n}j_{n}}(t_{n})\bigr{\rangle}_{c}\,\biggl{\langle}\frac{\delta^{n}\,g[\mathbf{A}]}{\delta
A_{i_{1}j_{1}}(t_{1})\,\dots\,\delta
A_{i_{n}j_{n}}(t_{n})}\biggr{\rangle}\mathrm{d}t_{1}\dots\mathrm{d}t_{n}$ (67)

for any regular functional $g[{\bf A}]$. For the $\delta$-process with
correlation functions (55),

${\rm{\Delta}}_{ij...kp}^{\left(n\right)}=w_{ij...kp}^{\left(n\right)}\delta\left({{t_{1}}-{t_{2}}}\right)...\delta\left({{t_{1}}-{t_{n}}}\right)\
,$ (68)

it takes the form:

$\langle
A_{ij}(t)g[\mathbf{A}]\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}w^{(n+1)}_{iji_{1}j_{1}...i_{n}j_{n}}\biggl{\langle}\frac{\delta^{n}\,g[\mathbf{A}]}{\delta
A_{i_{1}j_{1}}(t_{1})\,\dots\,\delta A_{i_{n}j_{n}}(t_{n})}\biggr{\rangle}$
(69)

Here all the variational derivatives are taken at the same moment $t$.

However, in physical applications one often has to deal with ’causal’
functionals, i.e., the functionals that depend explicitly on time and satisty
the causality principle:

$\frac{\delta\,g[t,\mathbf{A}]}{\delta A_{ij}(t^{\prime})}=0\ \ \mbox{if}\ \
t^{\prime}>t$

For these functionals,

$\frac{\delta^{n}\,g[t,\mathbf{A}]}{\delta
A_{i_{1}j_{1}}(t_{1})\,\dots\,\delta A_{i_{n}j_{n}}(t_{n})}={\rm
I}_{[0,\infty)}(t-t_{1})...{\rm
I}_{[0,\infty)}(t-t_{n})G^{(n)}_{iji_{1}j_{1}...i_{n}j_{n}}[t,t_{1},...,t_{n};{\bf
A}]\ ,$

and

$\langle
A_{ij}(t^{\prime})g[t,\mathbf{A}]\rangle=\sum_{n=0}^{\infty}\frac{1}{n!}{w^{(n+1)}_{i\,j\,i_{1}j_{1}\dots
i_{n}j_{n}}}\langle G^{(n)}_{i_{1}j_{1}\dots
i_{n}j_{n}}[t,t^{\prime},...,t^{\prime};{\bf A}]\rangle\ \ ,\ t^{\prime}<t$
(70) $\langle A_{ij}(t^{\prime})g[t,\mathbf{A}]\rangle=\langle
A_{ij}(t^{\prime})\rangle\langle g[t,{\bf A}]\rangle\ \ ,\ t^{\prime}>t$ (71)

But (69) is inapplicable for $t=t^{\prime}$ because it contains undefined
values ${\rm I}_{[0,\infty)}(0)$. So, to calculate the simultaneous
correlator, we have to return to (67) and consider some sequence of (formal)
processes with cumulants converging to (68). Thus, we choose the sequence of
cumulants

$\frac{w^{(n)}_{i_{1}j_{1}\dots
i_{n}j_{n}}}{n}\,\sum\limits_{k=1}^{n}\prod\limits_{\begin{smallmatrix}{l=1}\\\
l\neq k\end{smallmatrix}}^{n}\delta_{\epsilon}(t_{k}-t_{l}),$

where $\delta_{\epsilon}(t)$ are even regular functions,
$\delta_{\epsilon}(t)\stackrel{{\scriptstyle\epsilon\to
0}}{{\rightarrow}}\delta(t)$, $\int\delta_{\epsilon}dt=1$. With account of

$\int^{t}\mathrm{d}t_{1}\dots\int^{t}\mathrm{d}t_{n}\prod\limits_{l=1}^{n}\delta_{\epsilon}(t-t_{l})=\frac{1}{2^{n}},$
$\int^{t}\mathrm{d}t_{1}\dots\int^{t}\mathrm{d}t_{n}\,\delta_{\epsilon}(t-t_{k})\prod\limits_{\begin{smallmatrix}{l=1}\\\
l\neq
k\end{smallmatrix}}^{n}\delta_{\epsilon}(t_{k}-t_{l})=\frac{1}{n}\left(1-\frac{1}{2^{n}}\right),$

we arrive at

$\langle
A_{ij}(t)g[t,\mathbf{A}]\rangle=\sum_{n=0}^{\infty}\frac{1}{(n+1)!}{w^{(n+1)}_{i\,j\,i_{1}j_{1}\dots
i_{n}j_{n}}}\langle G^{(n)}_{i_{1}j_{1}\dots i_{n}j_{n}}[t,...,t;{\bf
A}]\rangle$ (72)

This relation was presented without derivation in Klyackin ; here we derive it
taking the formal limit.

Comparing (72) with (70) we see that the correlations of ’causal’ functionals
are discontinuous at $t^{\prime}=t$ for $\delta$-processes, and the ’naive’
convention $I_{[0,\infty)}(0)=1/2$ is valid only for the Gaussian processes.

## References

  * (1) G. Falkovich, K. Gawedzki, & M. Vergassola, Rev. Mod. Phys. 73, 913 (2001)
  * (2) V. Sabel’nikov & O. Soulard, Phys. Rev. E 72, 1, 016301 (2005)
  * (3) R.H. Kraichnan, Phys. Fluids 11 945 (1968)
  * (4) A.P. Kazantsev, Sov. Phys. JETP 26, 1031 (1968)
  * (5) F.R. Gantmacher, Matrix theory, 2nd edn (AMS Providence, Rhode Island, 1990)
  * (6) J.M. Ottino, W.E. Ranz, & C.W. Macosko, AIChE Journ. 27, 4, 565 (1981)
  * (7) S.B. Pope, Int. J. Engng Sci. 26, 5, 445 (1988)
  * (8) E. Balkovsky & A. Fouxon, Phys. Rev. E 60,4164 (1999)
  * (9) A.S. Il’yn, V.A. Sirota, & K.P. Zybin, Phys. Rev. E 96, 013117 (2017)
  * (10) Ya.B. Zeldovich, A.A. Ruzmaikin, S.A. Molchanov, & D.D. Sokoloff, J. Fluid Mech. 144, 1 (1984)
  * (11) M. Chertkov, G. Falkovich, I. Kolokolov, & M. Vergassola, Phys. Rev. Lett. 83, 4065 (1999)
  * (12) A.S. Il’yn, A.V. Kopyev, V.A. Sirota, & K.P. Zybin, Phys. Fluids 33, 075105 (2021)
  * (13) V.I. Oseledets, Trans. Moscow Math. Soc. 19, 197 (1968)
  * (14) A. Crisanti, G. Paladin, & A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer, 1993)
  * (15) Y. Le Jan, Zeit. f$\ddot{u}$r Warsch. 70, 4, 609 (1985)
  * (16) P.H. Baxendale, in Lyapunov Exponents, Bremen 1984, eds. Arnold, L., and V. Wihstutz, Lecture Notes in Math. vol. 1186, p. 322 (1986)
  * (17) N. Newman, Commun. Math. Phys. 103, 121 (1986)
  * (18) A. Gamba, J. Stat. Phys. 112, 1, 193 (2003)
  * (19) M. Liao, Ann. Probab. 25, 3, 1241 (1997)
  * (20) N.G. Van Kampen, Physica, 74, 215 (1974); Physica, 74, 239 (1974)
  * (21) A.S. Il’yn, V.A. Sirota, & K.P. Zybin, Journ. Stat. Phys. 163, 765 (2016)
  * (22) P. Dittrich, S.A. Molchanov, D.D. Sokolov, & A.A. Ruzmaikin, AN, 305, 119 (1984)
  * (23) E.A. Illarionov, & D.D. Sokoloff, Phys. Rev. E, 104, 015214 (2021)
  * (24) S.R.S. Varadhan, Ann. Probab. 36, 2, 397 (2008)
  * (25) J. Zinn-Justin, Field Theory and Critical Phenomena, (Clarendon Press, Oxford, 1989)
  * (26) A.V. Kopyev, A.M. Kiselev, A.S. Il’yn, V.A. Sirota, & K.P. Zybin, Astrophys. Journ., 927(2), 172 (2022)
  * (27) A.V. Kopyev, A.S. Il’yn, V.A. Sirota, & K.P. Zybin, Phys. Fluids, 34(3), 035126 (2022)
  * (28) A.S. Il’yn, A.V. Kopyev, V.A. Sirota, & K.P. Zybin, Phys. Fluids, 32, 12, 125114
  * (29) E. Wong & M. Zakai, Ann Math. Stat. 36, 5, 1560 (1965)
  * (30) K. Furutsu., J. Res. NBS D67, 1, 39 (1963)
  * (31) E.A. Novikov, Soviet Phys. JETP 20, 5, 1290 (1965)
  * (32) V.I. Klyatskin, Dynamics of Stochastic Systems, (Elsevier, 2005)