The minimal number of homogeneous geodesics
depending on the signature of the Killing form

Zdeněk Dušek

Institute of Technology and Business in České Budějovice

Okružní 517/10, 370 01 České Budějovice, Czech Republic

<EMAIL_ADDRESS>

###### Abstract

The existence of at least two homogeneous geodesics in any homogeneous Finsler
manifold was proved in a previous paper by the author. The examples of
solvable Lie groups with invariant Finsler metric which admit just two
homogeneous geodesics were presented in another paper. In the present work, it
is shown that a homogeneous Finsler manifold with indefinite Killing form
admits at least four homogeneous geodesics. Examples of invariant Randers
metrics on Lie groups with definite Killing form admitting just two
homogeneous geodesics and examples with indefinite Killing form admitting just
four homogeneous geodesics are presented.

MSClassification: 53C22, 53C60, 53C30
Keywords: Homogeneous Finsler manifold, homogeneous geodesic

##  1 Introduction

The existence of at least one homogeneous geodesics in arbitrary homogeneous
Riemannian manifold was proved by O. Kowalski and J. Szenthe in [10], by an
algebraic construction in the Lie algebra. In the papers [9] and [11], it was
proved that this result is optimal, namely, examples of homogeneous Riemannian
metrics on solvable Lie groups were constructed which admit just one
homogeneous geodesic through any point. Generalization of this existence
result to pseuddo-Riemannian geometry was proved by the author in [7], in the
more general framework of affine geometry, using a purely affine approach and
differential topology.

Generalization of the above existence result to Finsler geometry was proved in
the series of papers [15] by Z. Yan and S. Deng for Randers metrics (by the
algebraic construction), [3] by the author for odd-dimensional Finsler
metrics, [5] by the author for Berwald or reversible Finsler metrics (in both
cases using the affine approach), [16] by Z. Yan and L. Huang in general
(using again the original idea by Kowalski and Szenthe and a purely Finslerian
construction). However, due to the nonreversibility of general Finsler
metrics, it was conjectured by the author that the result and its proofs in
the nonreversible situation are not optimal, namely that an arbitrary
homogeneous Finsler manifold admits at least two homogeneous geodesics through
arbitrary point.

In comparison with Riemannian geometry, the situation is rather delicate. In
the context of Finsler geometry, the trajectory of the unique homogeneous
geodesic in a Riemannian manifold should be regarded as two geodesics, with
initial vectors $X$ and $-X$, and only reparametrizations in the same
direction may be identified as a trajectory of a geodesic. For a general
homogeneous Finsler manifold, the initial vectors of the two homogeneous
geodesics may be non-opposite. In [4], examples of invariant Randers metrics
which admit just two homogeneous geodesics are constructed. The initial
vectors of these geodesics are $X+Y$ and $-X+Y$, for certain vectors $X,Y\in
T_{p}M$. For the construction, Randers metrics which are modifications of
Riemannian metrics of examples from [9] and [11] are used. These examples are
the solvable Lie groups and hence their Killing form vanishes identically. It
was also demonstrated with an example that general Randers metrics whose
underlying Riemannian metric admits just two homogeneous geodesics (with
initial vectors $X$ and $-X$) may admit more than two homogeneous geodesics.

The proof of the existence of at least two homogeneous geodesics in Finsler
geometry was given in [8], by the geometrical interpretation of the algebraic
condition from the original proof by Kowalski and Szenthe and used later by
Yan and Huang. The proof in [8] contained a small inaccuracy, namely, it was
implicitly assumed that the Killing form of the isometry group is negatively
semidefinite. If the restriction of the Killing form to ${\mathfrak{m}}$ is
indefinite, the proof works as well. Moreover, there are at least two further
solutions in this situation. In the present paper, in Section 4, we give a
full proof of the existence of four homogeneous geodesics for arbitrary
homogeneous Finsler manifold such that the restriction of the Killing form to
${\mathfrak{m}}$ is indefinite. Before this, in Section 3, we illustrate the
crucial geometrical idea with the examples of Randers metrics on Lie groups in
dimension $3$ and with definite Killing form admitting just two homogeneous
geodesics and Randers metrics with indefinite Killing form admitting just four
homogeneous geodesics.

##  2 Preliminaries

Let $(M,F)$ be a Finsler manifold. If there is a connected Lie group $G$ which
acts transitively on $M$ as a group of isometries, then $M$ is called a
homogeneous manifold. Homogeneous manifold $M$ can be naturally identified
with the homogeneous space $G/H$, where $H$ is the isotropy group of the
origin $p\in M$. A homogeneous Finsler space $(G/H,F)$ is always a reductive
homogeneous space: We denote by ${\mathfrak{g}}$ and ${\mathfrak{h}}$ the Lie
algebras of $G$ and $H$ respectively and consider the adjoint representation
${\mathrm{Ad}}\colon H\times{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ of $H$ on
${\mathfrak{g}}$. There exists a reductive decomposition of the form
${\mathfrak{g}}={\mathfrak{m}}+{\mathfrak{h}}$ where
${\mathfrak{m}}\subset{\mathfrak{g}}$ is a vector subspace such that ${\rm
Ad}(H)({\mathfrak{m}})\subset{\mathfrak{m}}$. For a fixed reductive
decomposition ${\mathfrak{g}}={\mathfrak{m}}+{\mathfrak{h}}$ there is the
natural identification of ${\mathfrak{m}}\subset{\mathfrak{g}}=T_{e}G$ with
the tangent space $T_{p}M$ via the projection $\pi\colon G\rightarrow G/H=M$.
Using this natural identification, from the Minkovski norm and its fundamental
tensor on $T_{p}M$, we obtain the ${\mathrm{Ad}}(H)$-invariant Minkowski norm
and the ${\mathrm{Ad}}(H)$-invariant fundamental tensor on ${\mathfrak{m}}$
and we denote these again by $F$ and $g$.

Special Minkowski norms (on a vector space ${\mathbb{V}}$) are the Randers
norms. They are determined by a symmetric positively definite bilinear form
$\alpha$ and a vector $V$ such that $\alpha(V,V)<1$, or, equivalently, its
$\alpha$-equivalent 1-form $\beta$ related with $V$ by the formula
$\beta(U)=\alpha(V,U)$ for all $U\in{\mathbb{V}}$. The Randers norm $F$ is
then defined by the formula

$\displaystyle F(U)=\sqrt{\alpha(U,U)}+\beta(U),\qquad\forall
U\in{\mathbb{V}}.$ (1)

If a Finsler metric $F$ on $M$ restricted to any tangent space $T_{p}M$ is a
Randers norm, it is called a Randers metric. Obviously, a Randers metric $F$
is determined by a Riemannian metric $\alpha$ and a smooth $1$-form $\beta$. A
homogeneous Randers metric $F$ is determined by a Randers norm on
${\mathfrak{m}}$, in other words by a symmetric positively definite $2$-form
and a $1$-form on ${\mathfrak{m}}$ and these forms are denoted again by
$\alpha$ and $\beta$. We remark that, in the literature, the letter $\alpha$
is sometimes used for the norm induced by the $2$-form $\alpha$ and then
formula (1) above is without the square root. We choose the notation above
because for $\beta=0$, $F$ is the Riemannian norm and components $g_{ij}$ of
the fundamental tensor are just the components of the Riemannian metric
$\alpha$.

We further recall that the slit tangent bundle $TM_{0}$ is defined as
$TM_{0}=TM\setminus\\{0\\}$. Using the restriction of the natural projection
$\pi\colon TM\rightarrow M$ to $TM_{0}$, we naturally construct the pullback
vector bundle $\pi^{*}TM$ over $TM_{0}$. The Chern connection is the unique
linear connection on the vector bundle $\pi^{*}TM$ which is torsion free and
almost $g$-compatible, see some monograph, for example [1] by D. Bao, S.-S.
Chern and Z. Shen or [2] by S. Deng for details. Using the Chern connection,
the derivative along a curve $\gamma(t)$ can be defined. A regular smooth
curve $\gamma$ with tangent vector field $T$ is a geodesic if
$D_{T}(\frac{T}{F(T)})=0$. In particular, a geodesic of constant speed
satisfies $D_{T}{T}=0$. A geodesic $\gamma(s)$ through the point $p$ is
homogeneous if it is an orbit of a one-parameter group of isometries. More
explicitly, if there exists a nonzero vector $X\in{\mathfrak{g}}$ such that
$\gamma(t)={\rm exp}(tX)(p)$ for all $t\in{\mathbb{R}}$. The vector $X$ is
called a geodesic vector. Geodesic vectors are characterized by the following
geodesic lemma.

######  Lemma 1 ([12])

Let $(G/H,F)$ be a homogeneous Finsler space with a reductive decomposition
${\mathfrak{g}}={\mathfrak{m}}+{\mathfrak{h}}$. A nonzero vector
$Y\in{{\mathfrak{g}}}$ is geodesic vector if and only if it holds

$g_{Y_{\mathfrak{m}}}(Y_{\mathfrak{m}},[Y,U]_{\mathfrak{m}})=0\qquad\forall
U\in{\mathfrak{m}},$ (2)

where the subscript ${\mathfrak{m}}$ indicates the projection of a vector from
${\mathfrak{g}}$ to ${\mathfrak{m}}$.

We shall use this lemma for Randers metrics $F=\sqrt{\alpha}+\beta$ and in the
situation with trivial algebra ${\mathfrak{h}}$. In this special situation,
the above statement has the following form.

######  Lemma 2 ([6])

Let $F=\sqrt{\alpha}+\beta$ be a homogeneous Randers metric on $G$, let
${\mathfrak{g}}$ be the Lie algebra of $G$ and $V\in{\mathfrak{m}}$ be the
vector $\alpha$-equivalent with $\beta$. The vector $X\in{\mathfrak{g}}$ is
geodesic if and only if

$\displaystyle\alpha\Bigl{(}X+{\sqrt{\alpha(X,X)}}\cdot
V,[X,U]\Bigr{)}=0\qquad\forall U\in{\mathfrak{m}}.$ (3)

In the proof of the existence of a homogeneous geodesic, see [10], [16] or
[8], the geometrical interpretation of the crucial idea is the following. With
respect to the Killing form $K$, the bracket $[Y,U]_{\mathfrak{m}}$ is always
orthogonal to $Y_{\mathfrak{m}}$. We shall denote the restriction of $K$ from
${\mathfrak{g}}$ to ${\mathfrak{m}}$ again by $K$ and we shall consider
orthogonal complements in the vector space ${\mathfrak{m}}$. If we find a
vector $Y\in{\mathfrak{g}}$ such that $K(Y_{\mathfrak{m}})\neq 0$ and the
orthogonal spaces to $Y_{\mathfrak{m}}$ with respect to $g_{Y_{\mathfrak{m}}}$
and with respect to $K$ are equal, then the bracket $[Y,U]_{\mathfrak{m}}$
will be also orthogonal to $Y_{\mathfrak{m}}$ with respect to
$g_{Y_{\mathfrak{m}}}$, we reach the equality

$\displaystyle
g_{Y_{\mathfrak{m}}}(Y_{\mathfrak{m}},[Y,U]_{\mathfrak{m}})=K(Y_{\mathfrak{m}},[Y,U]_{\mathfrak{m}})=0\qquad\forall
U\in{\mathfrak{m}}$ (4)

and, according to geodesic lemma, $Y$ will be geodesic vector. We denote by
$I_{F}$ the unit indicatrix in ${\mathfrak{m}}$ given by the condition
$F(X)=1$ and we denote by $S_{K}$ the unit (pseudo-)sphere in ${\mathfrak{m}}$
given by the condition $K(X,X)=\pm 1$. We shall use the geometrical property
that the orthogonal space to a vector $X\in I_{F}$ with respect to the scalar
product $g_{X}$ is the tangent space to the indicatrix $I_{F}$ at $X$. In the
same way, the orthogonal space to a vector $X\in S_{K}$ with respect to $K$ is
the tangent space to the (pseudo-)sphere $S_{K}$ at $X$. For each vector
$X\in{\mathfrak{m}}$ such that $K(X,X)\neq 0$, we put $X_{F}=X/F(X)\in I_{F}$
and $X_{K}=X/|K(X,X)|\in S_{K}$. We are looking for vectors
$X\in{\mathfrak{m}}$, such that the tangent space to $I_{F}$ at $X_{F}$ and
the tangent space to $S_{K}$ at $X_{K}$ are equal. If we interpret these
tangent spaces (vector subspaces of ${\mathfrak{m}}$) as affine subspaces of
${\mathfrak{m}}$ and put them into their origin points ($X_{F}$, or $X_{K}$,
respectively), these spaces will be parallel. Such vectors
$X\in{\mathfrak{m}}$ will satisfy the above conditon (4).

We shall illustrate the situation with examples of Lie groups with invariant
Randers metrics and whose ${\mathrm{rad}}(K)$ is trivial. In the first
example, $G={\mathrm{SO}}(3)$, the Killing form is definite and $G$ admits
just two homogeneous geodesics, in generic situation. In the second example,
$G={\mathrm{SL}}(2)$, the Killing form is indefinite and $G$ admits just four
homogeneous geodesics, in generic situation. Further, we shall generalize the
procedure of finding at least two homogeneous geodesics in any homogeneous
Finsler manifold described in [8]. We show that any homogeneous Finsler
manifold such that the restriction of the Killing form of the isometry group
$G$ from ${\mathfrak{g}}$ to ${\mathfrak{m}}$ is indefinite admits at least
four homogeneous geodesics.

##  3 Examples

###  3.1 Example 1, ${\mathfrak{g}}_{1}\simeq{\mathfrak{so}}(3)$

Consider the Lie algebra
${\mathfrak{g}}_{1}={\mathrm{span}}\\{E_{i}\\}_{i=1}^{3}$ generated by the Lie
brackets

$\displaystyle[E_{1},E_{2}]=aE_{3},\qquad[E_{1},E_{3}]=-bE_{2},\qquad[E_{2},E_{3}]=cE_{1}.$

In the matrix form, for the special choice $a=b=c=1$, we can identify the
generators $E_{i}$ with the matrices

$\displaystyle E_{1}=\left(\begin{array}[]{ccc}0&1&0\cr-1&0&0\cr
0&0&0\end{array}\right),\quad E_{2}=\left(\begin{array}[]{ccc}0&0&0\cr
0&0&1\cr 0&-1&0\end{array}\right),\quad
E_{3}=\left(\begin{array}[]{ccc}0&0&1\cr 0&0&0\cr-1&0&0\end{array}\right).$

It is easy to check that ${\mathfrak{g}}_{1}\simeq{\mathfrak{so}}(3)$ for
$a,b,c>0$. By the direct calculations, we also easily check that the Killing
form, with respect to the basis $B=\\{E_{1},E_{2},E_{3}\\}$, is

$\displaystyle K=-2ab\,x_{1}^{2}-2ac\,x_{2}^{2}-2bc\,x_{3}^{2}.$ (6)

We now put $X=x_{1}E_{1}+x_{2}E_{2}+x_{3}E_{3}$ and we write down the Lie
brackets

$\displaystyle[X,E_{1}]$ $\displaystyle=$ $\displaystyle-
ax_{2}E_{3}+bx_{3}E_{2},$ $\displaystyle[X,E_{2}]$ $\displaystyle=$
$\displaystyle ax_{1}E_{3}-cx_{3}E_{1},$ (7) $\displaystyle[X,E_{3}]$
$\displaystyle=$ $\displaystyle-bx_{1}E_{2}+cx_{2}E_{1}.$ (8)

From Lemma 2 and the equation (3) with $\alpha$ given by the identity matrix
with respect to the basis $B$ above and with
$V=v_{1}E_{1}+v_{2}E_{2}+v_{3}E_{3}$, we obtain the system of equations

$\displaystyle
bx_{3}\left(x_{2}+{\sqrt{\alpha(X,X)}v_{2}}\right)-ax_{2}\left(x_{3}+{\sqrt{\alpha(X,X)}v_{3}}\right)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle-
cx_{3}\left(x_{1}+{\sqrt{\alpha(X,X)}v_{1}}\right)+ax_{1}\left(x_{3}+{\sqrt{\alpha(X,X)}v_{3}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (9) $\displaystyle
cx_{2}\left(x_{1}+{\sqrt{\alpha(X,X)}v_{1}}\right)-bx_{1}\left(x_{2}+{\sqrt{\alpha(X,X)}v_{2}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (10)

which simplifies into the form

$\displaystyle\left(b-a\right)x_{2}x_{3}+\left(bx_{3}v_{2}-ax_{2}v_{3}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0,$ (11)
$\displaystyle\left(a-c\right)x_{1}x_{3}+\left(ax_{1}v_{3}-cx_{3}v_{1}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle\left(c-b\right)x_{1}x_{2}+\left(cx_{2}v_{1}-bx_{1}v_{2}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0.$ (13)

Let us first investigate the situation with Riemannian metrics determined by
the orthonormal basis $B$, which means $v_{i}=0$. The system of equations
simplifies further into the form

$\displaystyle\left(b-a\right)x_{2}x_{3}$ $\displaystyle=$ $\displaystyle 0,$
(14) $\displaystyle\left(a-c\right)x_{1}x_{3}$ $\displaystyle=$ $\displaystyle
0,$ (15) $\displaystyle\left(c-b\right)x_{1}x_{2}$ $\displaystyle=$
$\displaystyle 0.$ (16)

For $a=b=c$, any vector $X=(x_{1},x_{2},x_{3})$ is obviously a solution of
this system. From the geometrical point of view, we observe that the unit
indicatrix $I_{F}$ given by the (Riemannian) Finsler function
$F=\sqrt{x_{1}^{2}+x_{2}^{2}}$ is just the coordinate sphere with radius $1$.
The hypersurface $S_{K}$ is a coordinate sphere with radius
$\frac{\sqrt{2}}{2a}$. Hence, obviously, for each $0\neq X\in{\mathfrak{m}}$,
the tangent plane to $I_{F}$ at $X_{F}$ and the tangent plane to $S_{K}$ at
$X_{K}$ are parallel.

For different values of parameters $a,b,c$, the unit indicatrix $I_{F}$ is
still the coordinate unit sphere (green in the picture which follows), but the
hypersurface $S_{K}$ is the coordinate elipsoid (blue in the picture). Each
nonzero vector $X\in{\mathfrak{m}}$ determines vectors $X_{F}\in I_{F}$ and
$X_{K}\in S_{K}$. The vectors $X$ for which the tangent space to the
indicatrix $I_{F}$ at $X_{F}$ is parallel with the tangent space to the
hypersurface $S_{K}$ at $X_{K}$ are the positive multiples of coordinate
vectors (red in the picture). If two of the parameters $a,b,c$ are equal, then
also positive multiples of all vectors in the corresponding coordinate plane
have this property. We illustrate the situation with the picture in the
coordinate plane $x_{3}=0$ with $a=b=\frac{1}{2}$, $c=3$. For the Finsler
function $F=\sqrt{x_{1}^{2}+x_{2}^{2}}$, the indicatrix $I_{F}$ is just the
coordinate unit sphere, the Killing form is

$\displaystyle K=-\frac{1}{2}\,x_{1}^{2}-3\,x_{2}^{2}$ (17)

and the hypersurface $S_{K}$ is the blue ellipse in the picture. The
nontrivial solutions of the system (14), with the restriction $x_{3}=0$, are
just the positive multiples of vectors $X_{1}=(1,0)$, $X_{2}=(-1,0)$,
$X_{3}=(0,1)$, $X_{4}=(0,-1)$.

Let us now turn to general Randers metrics. For the simplicity, let us
consider $V=v_{1}E_{1}$, hence $v_{2}=v_{3}=0$. The system of equations (11)
simplifies into the form

$\displaystyle\left(b-a\right)x_{2}x_{3}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Bigl{(}\left(a-c\right)x_{1}-cv_{1}{\sqrt{\alpha(X,X)}}\Bigr{)}x_{3}$
$\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle\Bigl{(}\left(c-b\right)x_{1}+cv_{1}{\sqrt{\alpha(X,X)}}\Bigr{)}x_{2}$
$\displaystyle=$ $\displaystyle 0.$ (19)

If $a\neq b$, the solutions are:
1) $x_{2}=x_{3}=0$,
2) $x_{2}=0$, $x_{3}\neq 0$ and possible solutions of the equation

$\displaystyle(a-c)x_{1}={cv_{1}}{\sqrt{\alpha(X,X)}},$ (20)

3) $x_{3}=0$, $x_{2}\neq 0$ and possible solutions of the equation

$\displaystyle(b-c)x_{1}={cv_{1}}{\sqrt{\alpha(X,X)}}.$ (21)

Let us illustrate the situation for the particular values of the parameters,
which we set for example $v_{1}=\frac{1}{2}$, $a=c=1$, $b=2$. The equation
(20) has no solutions. We look for solutions of equation (21) such that
$\alpha(X,X)=1$ and we obtain

$\displaystyle x_{1}=\frac{cv_{1}}{b-c}=\frac{1}{2},\qquad
x_{2}=\pm\frac{\sqrt{3}}{2}.$

Altogether, we have the four solutions, up to positive scalar multiples. With
respect to the basis $B$, they are

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle(1,0,0),$ $\displaystyle
X_{2}$ $\displaystyle=$ $\displaystyle(-1,0,0),$ (22) $\displaystyle X_{3}$
$\displaystyle=$ $\displaystyle(\frac{1}{2},\frac{\sqrt{3}}{2},0),$ (23)
$\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle(\frac{1}{2},-\frac{\sqrt{3}}{2},0).$ (24)

Again, we illustrate these solutions in the plane $x_{3}=0$ with a picture.
The Finsler function (with the restriction $x_{3}=0$) is

$\displaystyle F=\sqrt{x_{1}^{2}+x_{2}^{2}}+\frac{1}{2}x_{1}$ (25)

and the hypersurface $I_{F}$ is given by the equation $F(X)=1$. The Killing
form is given by the formula (6). For our values of parameters and in the
plane $x_{3}=0$, the hypersurface $S_{K}$ is given by the equation

$\displaystyle 2x_{1}^{2}+x_{2}^{2}=1.$ (26)

In the picture, directions given by the positive multiples of solutions
$X_{i}$ above are in red. We can see geometrically that the tangent space to
$I_{F}$ at ${X_{i}^{F}}$ and the tangent space to $S_{K}$ at ${X_{i}^{K}}$ are
parallel, for each $i=1,\dots,4$.

An important observation shows that the situation may be different for another
values of the parameters. For example, let us choose
$v_{1}=\frac{1}{2},a=b=2,c=3$. Because $|x_{1}|<\alpha(X,X)$ and ${cv_{1}}>1$,
none of the equations (20) and (21) have any nonzero solution and we are left
with solutions $X_{1}=(1,0,0)$ and $X_{2}=(-1,0,0)$. Again, we illustrate the
situation with a picture in the plane $x_{3}=0$. The hypersufrace $I_{F}$ is
the same as before, the hypersufrace $S_{K}$ is given by the equation

$\displaystyle 4x_{1}^{2}+6x_{2}^{2}=1.$ (27)

We see also geometrically from the picture that there are no nontrivial
solutions $X$ (other than those in the direction of the $x_{1}$-axis) in this
plane such that the tangent space to $I_{F}$ at ${X^{F}}$ and the tangent
space to $S_{K}$ at ${X^{K}}$ are parallel.

###  3.2 Example 2, ${\mathfrak{g}}_{2}\simeq{\mathfrak{sl}}(2)$

Consider the Lie algebra ${\mathfrak{g}}_{2}$ generated by the Lie brackets

$\displaystyle[E_{1},E_{2}]=aE_{3},\qquad[E_{1},E_{3}]=bE_{2},\qquad[E_{2},E_{3}]=cE_{1}.$

In the matrix form, for the special choice $a=b=c=1$, we can identify the
generators $E_{i}$ with the matrices

$\displaystyle
E_{1}=\left(\begin{array}[]{cc}0&1\cr-1&0\end{array}\right),\quad
E_{2}=\left(\begin{array}[]{cc}0&1\cr 1&0\end{array}\right),\quad
E_{3}=\left(\begin{array}[]{cc}1&0\cr 0&-1\end{array}\right).$

It is easy to check that ${\mathfrak{g}}_{2}\simeq{\mathfrak{sl}}(2)$ for
$a,b,c>0$. By the direct calculations we obtain that the Killing form, with
respect to the basis $B=\\{E_{1},E_{2},E_{3}\\}$, is

$\displaystyle K=2ab\,x_{1}^{2}-2ac\,x_{2}^{2}+2bc\,x_{3}^{2}.$ (29)

We put again $X=x_{1}E_{1}+x_{2}E_{2}+x_{3}E_{3}$ and we write down the Lie
brackets

$\displaystyle[X,E_{1}]$ $\displaystyle=$ $\displaystyle-
ax_{2}E_{3}-bx_{3}E_{2},$ $\displaystyle[X,E_{2}]$ $\displaystyle=$
$\displaystyle ax_{1}E_{3}-cx_{3}E_{1},$ (30) $\displaystyle[X,E_{3}]$
$\displaystyle=$ $\displaystyle bx_{1}E_{2}+cx_{2}E_{1}.$ (31)

From Lemma 2 and the equation (3) with $\alpha$ given by the identity matrix
with respect to the basis $B$ above and with
$V=v_{1}E_{1}+v_{2}E_{2}+v_{3}E_{3}$, we obtain the system of equations

$\displaystyle
bx_{3}\left(x_{2}+{\sqrt{\alpha(X,X)}v_{2}}\right)+ax_{2}\left(x_{3}+{\sqrt{\alpha(X,X)}v_{3}}\right)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
cx_{3}\left(x_{1}+{\sqrt{\alpha(X,X)}v_{1}}\right)-ax_{1}\left(x_{3}+{\sqrt{\alpha(X,X)}v_{3}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (32) $\displaystyle
cx_{2}\left(x_{1}+{\sqrt{\alpha(X,X)}v_{1}}\right)+bx_{1}\left(x_{2}+{\sqrt{\alpha(X,X)}v_{2}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (33)

which simplifies into the form

$\displaystyle\left(a+b\right)x_{2}x_{3}+\left(bx_{3}v_{2}+ax_{2}v_{3}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(a-c\right)x_{1}x_{3}+\left(ax_{1}v_{3}-cx_{3}v_{1}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0,$ (34)
$\displaystyle\left(b+c\right)x_{1}x_{2}+\left(cx_{2}v_{1}+bx_{1}v_{2}\right){\sqrt{\alpha(X,X)}}$
$\displaystyle=$ $\displaystyle 0.$ (35)

For the simplicity, let us consider again just the case $V=v_{1}E_{1}$, hence
$v_{2}=v_{3}=0$. This system of equations simplifies further into the form

$\displaystyle\left(a+b\right)x_{2}x_{3}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\Bigl{(}\left(a-c\right)x_{1}-cv_{1}{\sqrt{\alpha(X,X)}}\Bigr{)}x_{3}$
$\displaystyle=$ $\displaystyle 0,$ (36)
$\displaystyle\Bigl{(}\left(b+c\right)x_{1}+cv_{1}{\sqrt{\alpha(X,X)}}\Bigr{)}x_{2}$
$\displaystyle=$ $\displaystyle 0.$ (37)

The solutions are:
1) $x_{2}=x_{3}=0$,
2) $x_{2}=0$, $x_{3}\neq 0$ and solutions of the equation

$\displaystyle(a-c)x_{1}=cv_{1}{\sqrt{\alpha(X,X)}},$ (38)

3) $x_{3}=0$, $x_{2}\neq 0$ and

$\displaystyle x_{1}=\frac{-cv_{1}}{b+c}{\sqrt{\alpha(X,X)}}.$ (39)

Let us notice that $|v_{1}|<1,|\frac{c}{b+c}|<1$ and hence, with the
assumption $\alpha(X,X)=1$, there is always a solution of the equation (39)
with $|x_{1}|<1$ and $x_{2}$ determined from the assumption $\alpha(X,X)=1$.
On the other hand, the equation (38) may have no nontrivial solutions. Let us
set the particular values of the parameters, for example $v_{1}=\frac{1}{2}$,
$a=b=c=1$. The equation (38) has no nontrivial solution and the solution of
the equation (39) is (up to a positive multiple)

$\displaystyle x_{1}=\frac{-cv_{1}}{b+c}=-\frac{1}{4},\qquad
x_{2}=\pm\frac{\sqrt{15}}{4}.$

Altogether, we have the four solutions, up to positive scalar multiples. With
respect to the basis $B$, they are

$\displaystyle X_{1}$ $\displaystyle=$ $\displaystyle(1,0,0),$ $\displaystyle
X_{2}$ $\displaystyle=$ $\displaystyle(-1,0,0),$ (40) $\displaystyle X_{3}$
$\displaystyle=$ $\displaystyle(-\frac{1}{4},\frac{\sqrt{15}}{4},0),$ (41)
$\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle(-\frac{1}{4},-\frac{\sqrt{15}}{4},0).$ (42)

Again, we illustrate these solutions, which are all in the plane $x_{3}=0$,
with a picture. The Finsler function (with the restriction $x_{3}=0$) is still
given by the equation (25) and the hypersurface $I_{F}$ is given by the
equation $F(X)=1$. For our values of the parameter and in the plane $x_{3}=0$,
the hypersurface $S_{K}$ is given by the equation

$\displaystyle 2x_{1}^{2}-2x_{2}^{2}=1.$ (43)

In the picture, directions given by the positive multiples of solutions
$X_{i}$ above are again in red. Again, we can see that, for each
$i=1,\dots,4$, the tangent space to $I_{F}$ at ${X_{i}^{F}}$ and the tangent
space to $S_{K}$ at ${X_{i}^{K}}$ are parallel.

##  4 The existence

We state the main theorem about the existence of homogeneous geodesics,
depending on the signature of the Killing form. We recall that the first part
of theorem was stated in [8], however, the proof was given with the assumption
that the Killing form is semidefinite. We present here the complete proof
which shows that the same idea for finding two geodesic vectors works on each
component $K=1$ and $K=-1$ of the unit (pseudo-)sphere of the Killing form.
Let us remark that the present result is optimal. The examples given above
illustrate that this result cannot be improved in general. The examples of
solvable Lie groups with invariant Finsler metrics which admit just two
homogeneous geodesics were given in [4].

As we will not express vectors in components now, it is more convenient to
change notation and use lowercase letters for vectors.

###### Theorem 3

Let $(M,F)$ be a homogeneous Finsler manifold. There exist at least two
homogeneous geodesics through arbitrary point $p\in M$. Let $K$ be the Killing
form of a transitive isometry group $G$ of $M$ and let
${\mathfrak{g}}={\mathfrak{h}}+{\mathfrak{m}}$ be a reductive dedomposition.
If the restriction of $K$ to ${\mathfrak{m}}$ is indefinite, then there exist
at least four homogeneous geodesics through arbitrary point $p\in M$.

Proof. Let $G$ be a transitive isometry group of $M$ and let $H$ be the
isotropy group of a fixed point $p\in M$. We express $M$ as the homogeneous
space $M=G/H$. Let $K$ be the Killing form of $G$ and let ${\mathrm{rad}}(K)$
be the null space of $K$. We choose ${\mathfrak{m}}={\mathfrak{h}}^{\perp}$
with respect to $K$. The decomposition in ${\mathrm{Ad}}(H)$-invariant and the
Finsler metric induces the invariant Minkowski norm and its fundamental tensor
on ${\mathfrak{m}}$. We shall denote these again by $F$ and $g$. The Killing
form $K$ negatively definite on ${\mathfrak{h}}$, because $H$ is compact.
Hence, ${\mathrm{rad}}(K)\subseteq{\mathfrak{m}}$. We shall distinguish the
two cases:

Case 1) ${\mathrm{rad}}(K)={\mathfrak{m}}$: We choose a hyperplane
$W\subset{\mathfrak{m}}$ such that $[{\mathfrak{m}},{\mathfrak{m}}]\subset W$.
There exist two vectors $n_{1},n_{2}\in{\mathfrak{m}}$ which are normal to
$W$, which means

$\displaystyle g_{n_{i}}(n_{i},w)$ $\displaystyle=$ $\displaystyle
0\qquad\forall w\in W.$

Vectors $n_{1},n_{2}$ are on the different sides of the hyperplane $W$ and
they are in general not opposite to each other, unless $F$ is reversible. See
[14] or [8] for details of the construction. We shall now write $n$ for any of
the two vectors $n_{1},n_{2}$. For an arbitrary fixed vector $w\in W$, the
function $F^{2}(n+tw)$ attains its minimum at $t=0$ and hence, using the
standard formula, we obtain

$\displaystyle
0=\frac{1}{2}\frac{d}{dt}F^{2}(n+tw)\big{|}_{t=0}=g_{n}(n,w),\qquad\forall
w\in W,$

which is the desired property. In particular, it is satisfied for any
$w\in[{\mathfrak{m}},{\mathfrak{m}}]\subset W$. We obtain immediately, using
Lemma 1, that $n_{1}$ and $n_{2}$ are geodesic vectors.

Case 2) ${\mathrm{rad}}(K)\subsetneq{\mathfrak{m}}$: We start with the
construction and notation as in [16]. We shall investigate the function

$\displaystyle f(z)$ $\displaystyle=$ $\displaystyle\frac{K(z,z)}{F^{2}(z)}.$

This function is positively homogeneous and it is reasonable to restrict the
definition domain to the indicatrix

$\displaystyle I_{F}=\\{z\in{\mathfrak{m}};F(z)=1\\}.$

Since the group $H$ is compact and ${\mathrm{rad}}(K)$ is an
${\mathrm{Ad}}(H)$-invariant subspace, there exist an
${\mathrm{Ad}}(H)$-invariant $K$-orthogonal complement $W$ of
${\mathrm{rad}}(K)$ in ${\mathfrak{m}}$. Each vector $z\in{\mathfrak{m}}$ can
be uniquely decomposed as $z=z_{1}+z_{2}$, where $z_{1}\in{\mathrm{rad}}(K)$
and $z_{2}\in W$. Denote $k=\dim({\mathrm{rad}}(K))$ and let

$\displaystyle D_{k}=\\{z_{1}\in{\mathrm{rad}}(K),F(z_{1})<1\\}$

be the open unit disc in ${\mathrm{rad}}(K)$. For each fixed $z_{1}\in D_{k}$,
consider the submanifold

$\displaystyle S_{z_{1}}=\\{z_{2}\in W,F(z_{1}+z_{2})=1\\}\subset W,$

which has the topology of the sphere ${\mathbb{S}}^{m-1}$, where
$m={\mathrm{dim}}(W)={\mathrm{dim}}(M)-k$. From now on, $z_{1}+z_{2}$ means
$z_{1}\in D_{k}$, $z_{2}\in S_{z_{1}}$ and $z_{1}+z_{2}\in I_{F}$. Each sphere
$S_{z_{1}}\subset W$ is split by the nullcone of $K|_{W}$ into open
submanifolds $S_{z_{1}}^{+}=\\{z_{2}\in S_{z_{1}};K(z_{2},z_{2})>0\\}$ and
$S_{z_{1}}^{-}=\\{z_{2}\in S_{z_{1}};K(z_{2},z_{2})<0\\}$. Both boundaries
$\partial\bar{S}_{z_{1}}^{+}$ and $\partial\bar{S}_{z_{1}}^{-}$ of closures
$\bar{S}_{z_{1}}^{+}$ and $\bar{S}_{z_{1}}^{-}$ are the intersection of
$S_{z_{1}}$ with the nullcone of $K|_{W}$. It is easily seen that the function
$f(z)$ defined above is positive on $S_{z_{1}}^{+}$ and negative on
$S_{z_{1}}^{-}$. Limits of $f(z_{1}+z_{2})$ for $z_{2}\in S_{z_{1}}^{+}$,
$z_{2}\rightarrow\partial\bar{S}_{z_{1}}^{+}$ and for $z_{2}\in
S_{z_{1}}^{-}$, $z_{2}\rightarrow\partial\bar{S}_{z_{1}}^{-}$ and are all
zero. For the later use, we define already now the distinguished open
submanifolds of $I_{F}$, namely

$\displaystyle S^{+}=\\{z_{1}+z_{2}\in
I_{F};K(z_{2},z_{2})>0\\}=\cup_{z_{1}\in D_{k}}(z_{1}+S_{z_{1}}^{+}),$
$\displaystyle S^{-}=\\{z_{1}+z_{2}\in
I_{F};K(z_{2},z_{2})<0\\}=\cup_{z_{1}\in D_{k}}(z_{1}+S_{z_{1}}^{-}).$ (44)

We further denote by $K_{1}$ and $K_{-1}$ the standard (pseudo-)spheres with
respect to the Killing form $K|_{W}$, namely $K_{1}=\\{w\in W;K(w,w)=1\\}$ and
$K_{-1}=\\{w\in W;K(w,w)=-1\\}$. It is easy to observe that, for fixed
$z_{1}$, for each vector $z_{2}\in S_{z_{1}}^{+}$, there is a positive real
number $p$ such that $pz_{2}\in K_{1}$. In the same way, for each vector
$z_{2}\in S_{z_{1}}^{-}$ there is a positive real number $p$ such that
$pz_{2}\in K_{-1}$. This correspondence gives homeomorphisms
$\psi^{+}_{z_{1}}$ of each $S_{z_{1}}^{+}$ with $K_{1}$ and homeomorphisms
$\psi^{-}_{z_{1}}$ of each $S_{z_{1}}^{-}$ with $K_{-1}$.

Let the signature of $K$ be $(p,q,k)$, where $p$ is the number of positive
signs in the diagonal form of $K$, $q$ is the number of negative signs and
$p+q=m$. It is well known (see for example [13]) that the topology of $K_{1}$,
or $K_{-1}$, respectively, (and hence also topology of each $S_{z_{1}}^{+}$,
or each $S_{z_{1}}^{-}$, respectively) is the topology of
${\mathbb{S}}^{p-1}\times{\mathbb{R}}^{q}$, or
${\mathbb{R}}^{p}\times{\mathbb{S}}^{q-1}$, respectively. In the special case
$q=0$, or $p=0$, respectively, it reduces to the topology of the sphere
${\mathbb{S}}^{m-1}$. In the special case $p=1$, or $q=1$, respectively, it
reduces to the topology of the two copies of ${\mathbb{R}}^{q}$, or the two
copies of ${\mathbb{R}}^{p}$, respectively. We continue with the general case
$p>1,q>1$. We investigate the manifold $S^{+}$, which is homeomorphic to
$D_{k}\times{\mathbb{S}}^{p-1}\times{\mathbb{R}}^{q}$ and the function $f(z)$
defined above is positive on it. We have observed that, on each
$S_{z_{1}}^{+}$,
$\lim_{z_{2}\rightarrow\partial\bar{S}_{z_{1}}^{+}}f(z_{1}+z_{2})=0$. It is
also easy to see that for $z_{1}\in D_{k}$, it holds
$\lim_{z_{1}\rightarrow\partial\bar{D}_{k}}f(z_{1}+z_{2})=0$. Obviously, the
function $f(z)$ reaches its maximum on $S^{+}$ for some vector $y_{1}\in
S^{+}$. Now we are going to show that there exist a vector $y_{2}\in S^{+}$,
where the function $f(z)$ on $S^{+}$ reaches the saddle point.

We identify $W$ with ${\mathbb{R}}^{p}\times{\mathbb{R}}^{q}$ and we fix the
homeomorphism $\phi\colon{\mathbb{S}}^{p-1}\times{\mathbb{R}}^{q}\rightarrow
K_{1}\subset{\mathbb{R}}^{p}\times{\mathbb{R}}^{q}$ by the formula
$(s,x)\mapsto(\sqrt{1+|x|^{2}}\cdot s,x)$. We define, for fixed
$z_{1}\in{\mathrm{rad}}(K)$ and fixed $x\in{\mathbb{R}}^{q}$, the submanifolds
$C_{z_{1},x}$ of $S_{z_{1}}^{+}$ as
$((\psi^{+}_{z_{1}})^{-1}\circ\phi)({\mathbb{S}}^{p-1}\times x)$. Each
$C_{z_{1},x}$ is homeomorphic to the sphere ${\mathbb{S}}^{p-1}$. For fixed
$z_{1}$ and $x$, and with compact definition domain $C_{z_{1},x}$, the
function $f(z_{1}+z_{2})$ restricted to $C_{z_{1},x}$ attains its minimum
$\varepsilon(z_{1},x)>0$ at some $z_{1}+\bar{z}_{2}(z_{1},x)\in C_{z_{1},x}$.
For each $z_{1}\in D_{k}$ and $x\in{\mathbb{R}}^{q}$, we choose one such
$\bar{z}_{2}$ and consider the mapping $\varphi\colon
D_{k}\times{\mathbb{R}}^{q}\rightarrow I_{F}$, $(z_{1},x)\mapsto
z_{1}+\bar{z}_{2}(z_{1},x)$. The function
$f(\varphi(z_{1},x))=\varepsilon(z_{1},x)$ is smooth on
$D_{k}\times{\mathbb{R}}^{q}$ and it attains its maximum $\lambda_{2}$ at
$(\bar{z}_{1},\bar{x})$. Here $(\bar{z}_{1},\bar{x})$ can be choosen and the
map $\varphi$ can be defined in a way that there is a neighbourhood $U\subset
D_{k}\times{\mathbb{R}}^{q}$ of $(\bar{z}_{1},\bar{x})$ such that the mapping
$\varphi\big{|}_{U}$ is smooth. We put $y_{2}=\varphi(\bar{z}_{1},\bar{x})\in
S^{+}$. The definition of $y_{2}$ does not depend on the identification of $W$
with ${\mathbb{R}}^{p}\times{\mathbb{R}}^{q}$. From the construction, it
follows the existence of a basis
$B=\\{u_{1},\dots,u_{p-1},v_{1},\dots,v_{k},w_{1},\dots,w_{q}\\}$ of
$T_{y_{2}}S^{+}$ with the following property: Vectors $u_{i}$ form a basis of
the tangent space of $C_{\bar{z}_{1},\bar{x}}$ at
$y_{2}=\varphi(\bar{z}_{1},\bar{x})$, vectors $v_{i}$ are images in the
tangent mapping to $\varphi$ of tangent vectors to $D_{k}$ at $(z_{1},x)$ and
vectors $w_{i}$ are images in the tangent mapping to $\varphi$ of tangent
vectors to ${\mathbb{R}}^{q}$ at $(z_{1},x)$. The function $f(z)$ attains its
local minimum along any curve $\gamma(t)$ in $S^{+}$ with $\gamma(0)=y_{2}$
and whose tangent vector at $t=0$ is any of the vectors $u_{i}$. And the
function $f(z)$ attains its local maximum along any curve $\gamma(t)$ in
$S^{+}$ with $\gamma(0)=y_{2}$ and whose tangent vector at $t=0$ is any of the
vectors $v_{i}$ or $w_{i}$.

It remains to show that $y_{1}$ and $y_{2}$ are geodesic vectors. As to
$y_{1}$, the function

$\displaystyle\tilde{f}(z)=K(z,z)-\lambda_{1}F^{2}(z)$

attains its minimum $0$ at $y_{1}$. For any fixed $w\in{\mathfrak{m}}$, the
function $\hat{f}(t)=\tilde{f}(y_{1}+tw)$ attains its minimum $0$ at $t=0$ and
hence $\hat{f}^{\prime}(0)=0$. It follows that

$\displaystyle K(y_{1},w)$ $\displaystyle=$ $\displaystyle\lambda_{1}\cdot
g_{y_{1}}(y_{1},w),\qquad\forall w\in{\mathfrak{m}}.$ (45)

As to $y_{2}$, the function

$\displaystyle\tilde{f}(z)=K(z,z)-\lambda_{2}F^{2}(z)$

attains value $0$ at $y_{2}$. For any vector $u_{i}$ defined above, the
function $\hat{f}(t)=\tilde{f}(y_{2}+tu_{i})$ attains its maximum $0$ at $t=0$
and hence $\hat{f}^{\prime}(0)=0$. For any of the vectors $v_{i}$ or $w_{i}$
defined above, the function $\hat{f}(t)=\tilde{f}(y_{2}+tu_{i})$ attains its
minimum $0$ at $t=0$ and hence also $\hat{f}^{\prime}(0)=0$. It follows that

$\displaystyle K(y_{2},w)$ $\displaystyle=$ $\displaystyle\lambda_{2}\cdot
g_{y_{2}}(y_{2},w)$ (46)

holds for any vector $w$ from the above basis $B$ of $T_{y_{2}}S^{+}$. It is
obvious that this equality holds also for $w=y_{2}$ and consequently formula
(46) holds for any $w\in{\mathfrak{m}}$. Formulas (45) and (46) lead to
formula

$\displaystyle
g_{y_{i}}({y_{i}},[y_{i},z]_{\mathfrak{m}})=\frac{1}{\lambda_{i}}K(y_{i},[y_{i},z]_{\mathfrak{m}})=\frac{1}{\lambda_{i}}K([y_{i},y_{i}],z)=0,\qquad\forall
z\in{\mathfrak{m}},\,i=1,2,$

which shows that $y_{1}$ and $y_{2}$ are geodesic vectors.

Finally, similar construction with $S^{-}$ and with the function $-f(z)$ leads
to geodesic vectors $y_{3},y_{4}\in S^{-}$. If $p=1$, the manifold $S^{+}$ has
two connected components and vectors $y_{1},y_{2}$ are chosen as vectors where
the function $f(z)$ reaches its maximum on each of these components. The
saddle point may not exist in this situation, as the examples above
illustrate. If $q=1$, the manifold $S^{-}$ has two connected components and
vectors $y_{3},y_{4}$ are chosen in similar way. If $p=0$ or $q=0$, then
either $S^{+}$ or $S^{-}$ is trivial and the procedure leads only to two
geodesic vectors. $\hfill\square$

## Acknowledgements

The research was supported by the grant IGS 8210-017/2020 of the Internal
Grant Agency of Institute of Technology and Business in České Budějovice.

## References

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# Weighted tensorized fractional Brownian textures

Céline Esser  Département de Mathématique
Université de Liège
Liège, Belgium
<EMAIL_ADDRESS>Claire Launay  UMR CNRS 6205, LMBA
Université Bretagne Sud
Vannes, France
<EMAIL_ADDRESS>Laurent Loosveldt  Département de Mathématique
Université de Liège
Liège, Belgium
<EMAIL_ADDRESS>Béatrice Vedel  UMR CNRS 6205, LMBA
Université Bretagne Sud
Vannes, France
<EMAIL_ADDRESS>
###### Abstract

This paper presents a new model of textures, obtained as realizations of a new
class of fractional Brownian fields. These fields, called weighted tensorized
fractional Brownian fields, are obtained by a relaxation of the tensor-product
structure that appears in the definition of fractional Brownian sheets.
Statistical properties such as self-similarity, stationarity of rectangular
increments and regularity properties are obtained. An operator scaling
extension is defined and we provide simulations of the fields using their
spectral representation.

###### Index Terms:

Brownian fields, texture synthesis, stationary rectangular increments, self-
similarity, anisotropy.

##  I Introduction

Classical extensions of the fractional Brownian motion in higher dimension.
The modelization of phenomena, in particular textures, by random objects has
led to the introduction of numerous stochastic processes and fields. The most
famous and historically the first one is the well-known Brownian motion, which
has been extended to fractional Brownian motions by Kolmogorov in the famous
paper [1] from 1940, to define “Gaussian spirals” in Hilbert spaces. The first
systematic study of fractional Brownian motion goes back to the article [2]
from Mandelbrot and Van Ness. Given a Hurst parameter $H\in(0,1)$, the
fractional Brownian motion $B^{H}$ is the unique Gaussian process with
stationary increments satisfying the self-similarity relation
$B^{H}_{at}\stackrel{{\scriptstyle(d)}}{{=}}a^{H}B_{t}$ for any $a,t>0$, where
$\stackrel{{\scriptstyle(d)}}{{=}}$ means that the equality holds in the sense
of finite-dimensional distributions. The fractional Brownian motion $B^{H}$
can be characterized as the unique centered Gaussian process with covariance
given by

$\forall
s,t\in\mathbb{R}_{+},\quad\mathbb{E}(B^{H}_{s}B^{H}_{t})=\frac{1}{2}(|s|^{2H}+|t|^{2H}-|s-t|^{2H}).$

The later process can also be defined via a moving average formula or via its
harmonizable representation :

$\forall
t\in\mathbb{R}_{+},\,B_{t}^{H}=\int_{\mathbb{R}}\frac{e^{it\xi}-1}{|\xi|^{H+1/2}}d\hat{\bf{W}}(\xi),$
(1)

see e.g. the monograph [3] for a precise definition of this last stochastic
integral.

Several extensions have been proposed in higher dimensions. In particular two
natural generalizations are provided. The first one is the Levy fractional
Brownian motion (LFBM) of Hurst index $H\in(0,1)$, also called fractional
Brownian field (see e.g. [4]). It is the unique real-valued isotropic Gaussian
field $Y^{H}$ with stationary increments satisfying the similarity property
$Y^{H}_{a\bf{x}}\stackrel{{\scriptstyle(d)}}{{=}}a^{H}Y^{H}_{\bf{x}}$, where
“isotropic” means that the field is invariant in law by rotation. Again, it
can be defined using its harmonizable representation:

$Y^{H}_{\bf{x}}=\int_{\mathbb{R}^{N}}\frac{e^{i\langle\bf{x},\boldsymbol{\xi}\rangle}-1}{\|\boldsymbol{\xi}\|^{H+\frac{N}{2}}}d\hat{\bf{W}}({\boldsymbol{\xi}})$
(2)

where $\langle\cdot,\cdot\rangle$ denotes the standard scalar product in
$\mathbb{R}^{N}$.

A second famous extension is given by the fractional Brownian sheet (fBs)
studied in [5, 6]. For a given vector ${\bf{H}}=(H_{1},...H_{N})\in(0,1)^{N}$,
the fBs of Hurst index ${\bf{H}}$ is a real-valued centered Gaussian random
field $S^{\bf{H}}$ with covariance function given by

$\mathbb{E}(S^{\bf{H}}_{\bf{x}}S^{\bf{H}}_{\bf{y}})\\!\\!=\\!\\!\\!\prod_{m=1}^{N}\frac{1}{2}\left(|x_{m}|^{2H_{m}}+|y_{m}|^{2H_{m}}-|x_{m}-y_{m}|^{2H_{m}}\right).$

for all ${\bf{x}},{\bf{y}}\in\mathbb{R}_{+}^{N}$. The fBs has the following
harmonizable representation

$S^{\bf{H}}_{\bf{x}}=\int_{\mathbb{R}^{N}}\prod_{m=1}^{N}\frac{e^{ix_{m}\xi_{m}}-1}{|\xi_{m}|^{H_{m}+\frac{1}{2}}}d\hat{\bf{W}}({\boldsymbol{\xi}}).$
(3)

Setting $H_{m}=\frac{1}{2}$ for each $m\in\\{1,\dots,N\\}$ yields to the
standard Brownian sheet. While LFBMs are isotropic, fBs exhibit a strong
“tensor-product” structure even when $H_{m}=H$ for all $m\in\\{1,\dots,N\\}$
and no longer have stationary increments but only rectangular stationary
increments (see Definition III.2 below). Nevertheless, this field has been
widely studied for its interesting mathematical aspects, including asymptotic
properties [7], fractal dimensions [7, 8], geometric properties [9], local
times [10, 11], stochastic differential equations [12], and many more.

Further developments. Focusing on the class of Gaussian fields, the two
previously mentioned extensions of the fractional Brownian motions can be seen
as particular cases of more general models. Important properties have been
introduced in these models, such as anisotropy with different properties along
directions [13, 14].

The Operator Scaling Gaussian Random Fields (OSGRF, also defined for
$\alpha$-stable fields) introduced in [15, 16] satisfy a matricial self-
similarity condition, which is given by

$\forall a>0,\quad
Z_{a^{E}{\bf{x}}}\stackrel{{\scriptstyle(d)}}{{=}}a^{H}Z_{\bf{x}}$

for some $H>0$, where $E$ is a $N\times N$ matrix with eigenvalues having
positive real parts, and where $\displaystyle{a^{E}=\exp(E\ln(a))=\sum_{k\geq
0}\frac{\ln^{k}(a)E^{k}}{k!}.}$ These fields have been shown to exhibit
anisotropic regularity properties [17, 18], which offer strategies for
numerical estimations of the parameters of the model [19, 20].

Other extensions provide models with local changes in the Hurst parameter
[21], [22], or changes in the local direction of anisotropy [23, 24], for
example. For an overview of the various models, we refer to [24].

Goals, contribution and outline.  The contribution of the paper is to provide
a new class of fields called _Weighted tensorized fractional Brownian fields_
(WTFBFs). They are defined through their harmonizable representation in (4).
The importance of the “tensor-product” effect emerges as the parameter
$\alpha$ goes from $1$ to $0$, yielding the fractional Brownian sheet for
$\alpha=0$ and a field closer to an LFBM in terms of regularity for
$\alpha=1$. The introduction of the WTFBFs is detailed in Section 2, and the
fundamental properties of self-similarity and rectangular stationary
increments are established in Section 3. Section 4 is dedicated to estimating
the variance of the rectangular increments, leading to deductions regarding
regularity properties. An operator-scaling extension is presented in Section
5, along with simulations of the fields obtained via the spectral
representation.

##  II Definition of the fields

Let $\alpha\in[0,1]$ and $H\in(0,1)$, we set

$H_{\alpha}^{+}:=(1+\alpha)H\text{ and }H_{\alpha}^{-}:=(1-\alpha)H$

and we define the Gaussian field
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ by

$X^{\alpha,H}_{(x_{1},x_{2})}:=\int_{\mathbb{R}^{2}}\frac{(e^{ix_{1}\xi_{1}}-1)(e^{ix_{2}\xi_{2}}-1)}{\phi_{\alpha,H}(\xi_{1},\xi_{2})}d\hat{\bf{W}}({\boldsymbol{\xi}})$
(4)

where the function

$\phi_{\alpha,H}(\xi_{1},\xi_{2})=\min(|\xi_{1}|,|\xi_{2}|)^{H_{\alpha}^{-}+\frac{1}{2}}\max(|\xi_{1}|,|\xi_{2}|)^{H_{\alpha}^{+}+\frac{1}{2}}$

denotes the square root of the inverse of the spectral density of the field.
In the sequel, we also use the notation

$\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}(\xi_{1},\xi_{2}):=\frac{(e^{ix_{1}\xi_{1}}-1)(e^{ix_{2}\xi_{2}}-1)}{\phi_{\alpha,H}(\xi_{1},\xi_{2})}$

for the kernel in the stochastic integral (4). Note that the field (4) is
well-defined since this last kernel belongs to $L^{2}(\mathbb{R}^{2})$. Note
furthermore that the Fourier transform of
$\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}$ is real. Indeed, one has

$\displaystyle\Im\left(e^{-i(t_{1}\xi_{1}+t_{2}\xi_{2})}(e^{ix_{1}\xi_{1}}-1)(e^{ix_{2}\xi_{2}}-1)\right)$
$\displaystyle=$
$\displaystyle\sin((x_{1}-t_{1})\xi_{1}+(x_{2}-t_{2})\xi_{2})-\sin(-t_{1}\xi_{1}+(x_{2}-t_{2})\xi_{2})$
$\displaystyle-\sin((x_{1}-t_{1})\xi_{1}-t_{2}\xi_{2})+\sin(-t_{1}\xi_{1}-t_{2}\xi_{2})$

which is an odd function in $(\xi_{1},\xi_{2})$. It follows that

$\int_{\mathbb{R}^{2}}\frac{\Im\left(e^{-i(t_{1}\xi_{1}+t_{2}\xi_{2})}(e^{ix_{1}\xi_{1}}-1)(e^{ix_{2}\xi_{2}}-1)\right)}{\phi_{\alpha,H}(\xi_{1},\xi_{2})}d\boldsymbol{\xi}=0.$

It implies that $\widehat{\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}}$ is real,
hence so is the field

$\displaystyle X^{\alpha,H}_{(x_{1},x_{2})}$
$\displaystyle=\int_{\mathbb{R}^{2}}\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}(\xi_{1},\xi_{2})d\hat{\bf{W}}({\boldsymbol{\xi}})$
$\displaystyle=\int_{\mathbb{R}^{2}}\widehat{\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}}(\xi_{1},\xi_{2})d{\bf{W}}({\boldsymbol{\xi}}).$

|  |  |  |  |
---|---|---|---|---|---
(a) $\alpha=0$ | (b) $\alpha=0.5$ | (c) $\alpha=1$ | (d) $\alpha=0$ | (e) $\alpha=0.5$ | (f) $\alpha=1$
Figure 1: Weighted tensorized fractional Brownian fields simulated using a
spectral representation approximation method, with parameters (a-c)
$H~{}=~{}0.3$ or (d-f) $H~{}=~{}0.7$ and (a,d) $\alpha~{}=~{}0$, (b,e)
$\alpha~{}=~{}0.5$ or (c,f) $\alpha~{}=~{}1$.

##  III Basic properties

######  Proposition III.1.

For all $\alpha\in[0,1]$ and $H\in(0,1)$, the process $X^{\alpha,H}$ is self-
similar : for all $a>0$,
$\\{X^{\alpha,H}_{(ax_{1},ax_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}\stackrel{{\scriptstyle(d)}}{{=}}\\{a^{2H}X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$

###### Proof.

For all $(x_{1},x_{2})\in\mathbb{R}^{2}$ and $a>0$, we have

$\displaystyle X^{\alpha,H}_{(ax_{1},ax_{2})}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{(e^{iax_{1}\xi_{1}}-1)(e^{iax_{2}\xi_{2}}-1)}{\phi_{\alpha,H}(\xi_{1},\xi_{2})}d\hat{\bf{W}}({\boldsymbol{\xi}})$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\int_{\mathbb{R}^{2}}\frac{(e^{ix_{1}\eta_{1}}-1)(e^{ix_{2}\eta_{2}}-1)}{\phi_{\alpha,H}(\frac{\eta_{1}}{a},\frac{\eta_{2}}{a})}a^{-1}d\hat{\bf{W}}({\boldsymbol{\eta}})$
$\displaystyle=a^{2H}\int_{\mathbb{R}^{2}}\frac{(e^{ix_{1}\eta_{1}}-1)(e^{ix_{2}\eta_{2}}-1)}{\phi_{\alpha,H}(\eta_{1},\eta_{2})}d\hat{\bf{W}}({\boldsymbol{\eta}})$
$\displaystyle=a^{2H}X^{\alpha,H}_{(x_{1},x_{2})},$

where we used the change of variables
$(\eta_{1},\eta_{2})=(a\xi_{1},a\xi_{2})$ in the stochastic integral. ∎

Classically, the stationarity of increments is a too strong property for
stochastic fields, and it is preferable to use the following property of
stationarity for rectangular increments [25, 22, 26, 27].

######  Definition III.2.

If $\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ is a field and
if $(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{R}^{2}$, we set

$\displaystyle\Delta X_{(x_{1},x_{2});(y_{1},y_{2})}$
$\displaystyle:=X_{(x_{1}+y_{1},x_{2}+y_{2})}-X_{(y_{1},x_{2}+y_{2})}-X_{(x_{1}+y_{1},y_{2})}+X_{(y_{1},y_{2})}.$

We say that $\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ has
stationary rectangular increments if, for any
$(y_{1},y_{2})\in\mathbb{R}^{2}$, we have

$\displaystyle\\{\Delta
X_{(x_{1},x_{2});(y_{1},y_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}\stackrel{{\scriptstyle(d)}}{{=}}\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}.$

######  Proposition III.3.

For all $\alpha\in[0,1]$ and $H\in(0,1)$, the field
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ has
stationary rectangular increments.

###### Proof.

First, we remark that for any $(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{R}^{2}$,
we get from (4)

$\displaystyle\Delta
X_{(x_{1},x_{2});(y_{1},y_{2})}^{\alpha,H}\\!=\\!\int_{\mathbb{R}^{2}}e^{i(y_{1}\xi_{1}+y_{2}\xi_{2})}\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}(\xi_{1},\xi_{2})d\hat{\bf{W}}({\boldsymbol{\xi}}).$

Thus, recalling [28, Corollary 6.3.2], we have

$\displaystyle\mathbb{E}\left(\exp\left(i\sum_{j=1}^{n}t^{(j)}\Delta
X_{(x_{1}^{(j)},x_{2}^{(j)});(y_{1},y_{2})}^{\alpha,H}\right)\right)$
$\displaystyle\\!=\exp\\!\\!\left(\\!\\!-c_{0}\\!\\!\int_{\mathbb{R}^{2}}\\!\left|\sum_{j=1}^{n}t^{(j)}e^{i(y_{1}\xi_{1}+y_{2}\xi_{2})}\mathcal{K}^{\alpha,H}_{(x_{1}^{(j)},x_{2}^{(j)})}(\xi_{1},\xi_{2})\right|^{2}\\!\\!\\!\\!d{\boldsymbol{\xi}}\\!\right)$
$\displaystyle\\!=\mathbb{E}\left(\exp\left(i\sum_{j=1}^{n}t^{(j)}X^{\alpha,H}_{(x_{1}^{(j)},x_{2}^{(j)})}\right)\right),$

for any
$(x_{1}^{(1)},x_{2}^{(1)}),\dots,(x_{1}^{(n)},x_{2}^{(n)}),(y_{1},y_{2})\in\mathbb{R}^{2}$
and any $t^{(1)},\dots,t^{(n)}\in\mathbb{R}$, with

$c_{0}:=\frac{1}{2\pi}\int_{0}^{\pi}\cos(\theta)^{2}\,d\theta.$

The conclusion follows directly. ∎

It is also noteworthy that, for all $\alpha\in[0,1]$ and $H\in(0,1)$, the
field $\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$
does have some stationary increments, namely the horizontal and vertical ones.

######  Definition III.4.

A field $\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ has
stationary horizontal increments if, for any $y_{1}\in\mathbb{R}$,

$\displaystyle\\{X_{(x_{1}+y_{1},x_{2})}-X_{(y_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$
$\displaystyle\quad\stackrel{{\scriptstyle(d)}}{{=}}\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}.$

Similarly, $\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ has
stationary vertical increments if, for any $y_{2}\in\mathbb{R}$,

$\displaystyle\\{X_{(x_{1},x_{2}+y_{2})}-X_{(x_{1},y_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$
$\displaystyle\quad\stackrel{{\scriptstyle(d)}}{{=}}\\{X_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}.$

######  Proposition III.5.

For all $\alpha\in[0,1]$ and $H\in(0,1)$, the field
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ has
stationary horizontal and vertical increments.

###### Proof.

For any $x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}$, we have, on one hand,

$X^{\alpha,H}_{(x_{1}+y_{1},x_{2})}-X^{\alpha,H}_{(y_{1},x_{2})}\\!=\\!\int_{\mathbb{R}^{2}}e^{iy_{1}\xi_{1}}\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}(\xi_{1},\xi_{2})d\hat{\bf{W}}({\boldsymbol{\xi}})$

and, on the other hand,

$X^{\alpha,H}_{(x_{1},x_{2}+y_{2})}-X^{\alpha,H}_{(x_{1},y_{2})}\\!=\\!\int_{\mathbb{R}^{2}}e^{iy_{2}\xi_{2}}\mathcal{K}^{\alpha,H}_{(x_{1},x_{2})}(\xi_{1},\xi_{2})d\hat{\bf{W}}({\boldsymbol{\xi}}).$

Then, we conclude the proof in exactly the same way as in the proof of
Proposition III.3. ∎

##  IV Variance of rectangular increments and Regularity properties

######  Proposition IV.1.

For all $\alpha\in[0,1]$ and $H\in(0,1)$, there is a constant $c_{1}>0$ such
that the rectangular increments of
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ satisfy

$\displaystyle\mathbb{E}(|\Delta
X^{\alpha,H}_{(h_{1},h_{2});(x_{1},x_{2})}|^{2})$ $\displaystyle\,\,\leq
c_{1}\left(\max\\{|h_{1}|,|h_{2}|\\}^{1-\alpha}\min\\{|h_{1}|,|h_{2}|\\}^{1+\alpha}\right)^{2H}$

for all $(x_{1},x_{2}),(h_{1},h_{2})\in\mathbb{R}^{2}$.

###### Proof.

The isometry property of the stochastic integral gives

$\displaystyle\mathbb{E}\big{(}|\Delta
X^{\alpha,H}_{(h_{1},h_{2});(x_{1},x_{2})}|^{2}\big{)}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{|e^{i(x_{1}+h_{1})\xi_{1}}-e^{ix_{1}\xi_{1}}|^{2}|e^{i(x_{2}+h_{2})\xi_{2}}-e^{ix_{2}\xi_{2}}|^{2}}{(\phi_{\alpha,H}(\xi_{1},\xi_{2}))^{2}}d{\boldsymbol{\xi}}$
$\displaystyle=\frac{1}{|h_{1}|\,|h_{2}|}\int_{\mathbb{R}^{2}}\frac{|e^{i\eta_{1}}-1|^{2}|e^{i\eta_{2}}-1|^{2}}{(\phi_{\alpha,H}(\frac{\eta_{1}}{h_{1}},\frac{\eta_{2}}{h_{2}}))^{2}}d{\boldsymbol{\eta}}$

using the change of variables
$(\eta_{1},\eta_{2})=(h_{1}\xi_{1},h_{2}\xi_{2})$. Notice now that if
$|h_{1}|\geq|h_{2}|$, one has

$\displaystyle(\phi_{\alpha,H}(\tfrac{\eta_{1}}{h_{1}},\tfrac{\eta_{2}}{h_{2}}))^{2}$
$\displaystyle\geq\frac{\min(|\eta_{1}|,|\eta_{2}|)^{2H_{\alpha}^{-}+1}|\eta_{2}|^{2H_{\alpha}^{+}+1}}{|h_{1}|^{(2H_{\alpha}^{-}+1)}|h_{2}|^{(2H_{\alpha}^{+}+1)}}.$

This implies $\displaystyle{\mathbb{E}\big{(}|\Delta
X^{\alpha,H}_{(h_{1},h_{2});(x_{1},x_{2})}|^{2}\big{)}\leq
c_{1}|h_{1}|^{2H_{\alpha}^{-}}|h_{2}|^{2H_{\alpha}^{+}}}$ where

$c_{1}=\int_{\mathbb{R}^{2}}\frac{|e^{i\eta_{1}}-1|^{2}|e^{i\eta_{2}}-1|^{2}}{\min(|\eta_{1}|,|\eta_{2}|)^{2H_{\alpha}^{-}+1}|\eta_{2}|^{2H_{\alpha}^{+}+1}}d{\boldsymbol{\eta}}$
(5)

if $|h_{1}|\geq|h_{2}|$. The same argument for $|h_{1}|<|h_{2}|$ leads to the
conclusion. ∎

Similarly, we obtain the following result regarding the horizontal and
vertical increments.

######  Proposition IV.2.

For all $\alpha\in[0,1]$, $H\in(0,1)$ and every compact subset
$K\subset\mathbb{R}$, there is a constant $c_{K}>0$ such that the horizontal
and vertical increments of the field
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ satisfy

$\mathbb{E}\big{(}|X^{\alpha,H}_{(x_{1}+h_{1},x_{2})}-X^{\alpha,H}_{(x_{1},x_{2})}|^{2}\big{)}\leq
c_{K}|h_{1}|^{2H^{+}_{\alpha}}$

for all $x_{1}\in\mathbb{R}$, $x_{2}\in K$ and $h_{1}\in\mathbb{R}$ such that
$|h_{1}|\leq|x_{2}|$, and

$\mathbb{E}\big{(}|X^{\alpha,H}_{(x_{1},x_{2}+h_{2})}-X^{\alpha,H}_{(x_{1},x_{2})}|^{2}\big{)}\leq
c_{K}|h_{2}|^{2H^{+}_{\alpha}}$

for all $x_{1}\in K$, $x_{2}\in\mathbb{R}$ and $h_{2}\in\mathbb{R}$ such that
$|h_{2}|\leq|x_{1}|$.

###### Proof.

It suffices to proceed as in the proof of Proposition IV.1 to write

$\displaystyle\mathbb{E}\big{(}|X^{\alpha,H}_{(x_{1}+h_{1},x_{2})}-X^{\alpha,H}_{(x_{1},x_{2})}|^{2}\big{)}$
$\displaystyle=\int_{\mathbb{R}^{2}}\frac{|e^{i(x_{1}+h_{1})\xi_{1}}-e^{ix_{1}\xi_{1}}|^{2}|e^{ix_{2}\xi_{2}}-1|^{2}}{(\phi_{\alpha,H}(\xi_{1},\xi_{2}))^{2}}d{\boldsymbol{\xi}}$
$\displaystyle\leq
c_{1}\max\\{|h_{1}|,|x_{2}|\\}^{2H^{-}_{\alpha}}\min\\{|h_{1}|,|x_{2}|\\}^{2H^{+}_{\alpha}}$
$\displaystyle\leq c_{K}|h_{1}|^{2H^{+}_{\alpha}}$

where $c_{1}$ is the constant given in Equation (5) and where we set
$c_{K}=c_{1}\sup_{y_{2}\in K}|y_{2}|^{2H^{-}_{\alpha}}.$ ∎

The result indicates that the regularity along the axes is governed by
$H^{+}_{\alpha}$. Regarding the rectangular increments, a generalization of
Kolmogorov’s continuity theorem allows us to assert that there exists a
modification of the field
$\\{X^{\alpha,H}_{(x_{1},x_{2})}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$ which is
_locally $(H^{+}_{\alpha},H^{-}_{\alpha})$-rectangular Hölder_. This means
that for every bounded intervals $I,J$ of $\mathbb{R}$, every $x_{1}\in I$,
$x_{2}\in J$ and every $\varepsilon>0$, there exists a positive finite random
variable $C>0$ such that almost surely

$\displaystyle|\Delta X^{\alpha,H}_{(h_{1},h_{2});(x_{1},x_{2})}|$
$\displaystyle\leq
C\left(\max\\{|h_{1}|,|h_{2}|\\}^{(1-\alpha)}\min\\{|h_{1}|,|h_{2}|\\}^{(1+\alpha)}\right)^{H-\varepsilon}$

for all $h_{1},h_{2}\in\mathbb{R}$ such that $x_{1}+h_{1}\in I$ and
$x_{2}+h_{2}\in J$. This relation will be further investigated in an upcoming
paper.

##  V Anisotropic extension and Simulations

###  V-A Anisotropic extension

The model introduced in the previous sections can be extended to provide
anisotropic textures by imposing an operator scaling property. These fields
then satisfy anisotropic properties of regularities that will be explored in a
forthcoming work.

We consider $\beta_{1},\beta_{2}\in(1/2,3/2)$ such that
$\beta_{1}+\beta_{2}=2$, and we set

$\displaystyle
X^{\alpha,H,\beta_{1},\beta_{2}}_{(x_{1},x_{2})}:=\int_{\mathbb{R}^{2}}\frac{(e^{ix_{1}\xi_{1}}-1)(e^{ix_{2}\xi_{2}}-1)}{\phi_{\alpha,H,\beta_{1},\beta_{2}}(\xi_{1},\xi_{2})}d\hat{\bf{W}}({\boldsymbol{\xi}})$
(6)

where

$\phi_{\alpha,H,\beta_{1},\beta_{2}}(\xi_{1},\xi_{2})=\phi_{\alpha,H}\big{(}|\xi_{1}|^{\frac{1}{\beta_{1}}},|\xi_{2}|^{\frac{1}{\beta_{2}}}\big{)}.$

If $\max(\beta_{1},\beta_{2})-1<2H<3\min(\beta_{1},\beta_{2})-1$, the
corresponding field is well-defined and satisfies

$X^{\alpha,H,\beta_{1},\beta_{2}}_{a^{D}\bf{x}}\stackrel{{\scriptstyle(d)}}{{=}}a^{2H}X^{\alpha,H,\beta_{1},\beta_{2}}_{\bf
x}$ (7)

with $D=\mbox{diag}(\beta_{1},\beta_{2})$ and
$a^{D}{\bf{x}}=(a^{\beta_{1}}x_{1},a^{\beta_{2}}x_{2})$.

|  |  |  |  |
---|---|---|---|---|---
(a) $\alpha=0$ | (b) $\alpha=0.5$ | (c) $\alpha=1$ | (d) $\alpha=0$ | (e) $\alpha=0.5$ | (f) $\alpha=1$
Figure 2: Anisotropic weighted tensorized fractional Brownian fields simulated
using a spectral representation approximation method, with parameters (a-c)
$H=0.4$, $\beta_{1}=0.7$, $\beta_{2}=1.3$ and $\alpha=0$, $\alpha=0.5$ or
$\alpha=1$, and (d-f) $H=0.6$, $\beta_{1}=0.85$, $\beta_{2}=1.15$ and
$\alpha=0$, $\alpha=0.5$ or $\alpha=1$.

###  V-B Simulation

Several strategies have been developed to simulate Gaussian random fields, as
reviewed in [24]. Methods based on an explicit expression for the covariance
of the field allow for exact simulations that preserve statistical properties
such as stationarity. These methods have been used in [29] and [30]. When the
covariance is not explicitly known but is known along radial directions, the
turning-bands method introduced in [31, 32] can be employed, as done in [33]
to simulate some anisotropic fields.

Using the spectral density of the field, approximations of AFBF have been
obtained in [34, 35]. Since we only have an integral expression of the
covariance, we employ the later strategy to generate a WTFBF. It relies on a
discretization of the field in the Fourier domain. Its main drawbacks include
the difficulty in obtaining results on the convergence of the approximation.
There is also a possibility that the inverse Fourier transform “breaks” the
statistical properties of the field. However, it is very fast and easy to
perform as it involves fast Fourier transforms. Approximations based on
wavelet methods could be used, but they are known to be quite slow in practice
even if they provide the best approximation rate by a series in the case of
FBF [36].

The results presented in Figures 1 and 2 are generated using a spectral
representation approximation on a discrete grid of size $(M+1)\times(M+1)$,
with $M=512$. For a given WTFBF
$\\{X_{(x_{1},x_{2})}^{\alpha,H}\\}_{(x_{1},x_{2})\in\mathbb{R}^{2}}$, the
strategy involves generating $W$, a collection of independent standard complex
Gaussian variables of size $(2M\times 2M)$. These variables are then
multiplied by a function $g$. Next, in both directions successively, a 1D
Fourier transform is applied, followed by subtracting the value of the field
at the origin.

If we set $g(x,y)=(\phi_{\alpha,H}(x,y))^{-1}\textbf{1}_{\\{x\neq 0,y\neq
0\\}}$ for $(x,y)\in\mathbb{R}^{2}$, the generated field $x^{\alpha,H}$ is
given, for all $k_{1},k_{2}\in\\{0,\dots,M\\}$, by

${x^{\alpha,H}\left(\tfrac{k_{1}}{M},\tfrac{k_{2}}{M}\right)=\mathcal{R}\left(y_{2}\left(\tfrac{k_{1}}{M},\tfrac{k_{2}}{M}\right)-y_{2}\left(0,\tfrac{k_{2}}{M}\right)\right)},$

where for any $n_{1}\in\\{-M+1,\dots,M\\}$

$\displaystyle{y_{1}\left(n_{1},\tfrac{k_{2}}{M}\right)=\sum_{n_{2}=-M+1}^{M}W(n_{1},n_{2})g\left(\pi
n_{1},\pi n_{2}\right)e^{-\frac{2i\pi n_{2}k_{2}}{2M}}},$

and

$\displaystyle{y_{2}\left(\tfrac{k_{1}}{M},\tfrac{k_{2}}{M}\right)=\pi\sum_{n_{1}=-M+1}^{M}\left(y_{1}\left(n_{1},\tfrac{k_{2}}{M}\right)-y_{1}(n_{1},0)\right)e^{-\frac{2i\pi
n_{1}k_{1}}{2M}}}.$

The same method is used to simulate the anisotropic extension. Figure 1
presents synthesized WTFBFs with various parameters $H$ and $\alpha$. Figure 2
shows anisotropic WTFBF. These fields $X^{\alpha,H,\beta_{1},\beta_{2}}$
produce anisotropic textures, where the highest $\beta$ determines the
dominant direction. The images (a-f) illustrate the effects of the parameters
$\alpha$, $\beta$ and $H$ on the fields.

##  VI Conclusion and perspectives

A new class of fractional Brownian fields has been defined using a tensor-
product structure. These fields are indexed by two parameters: a Hurst index
$H\in(0,1)$ and a parameter $\alpha\in[0,1]$ which measures the “deviation” of
the corresponding field from the fractional Brownian sheet with the same Hurst
index, obtained with $\alpha=0$. These fields are demonstrated to be self-
similar and to possess stationary rectangular, horizontal and vertical
increments. The variance of these increments has been bounded to have a
glimpse of the regularity of these fields. An anistropic extension has been
proposed. Finally, simulations have been provided, for various values of the
parameters $H$ and $\alpha$ as well as for the anisotropic extension. In a
forthcoming work, we aim at providing function spaces which are well-suited to
consider the rectangular regularity of such fields. An analysis of these
spaces, based on hyperbolic wavelet transform should be performed. Further
development concerning the anisotropic extension could also be considered, as
well as a $N$-dimensional definition of the weighted tensorized fractional
Brownian textures.

## Acknowledgment

This work was conducted within the France 2030 framework programs, Centre
Henri Lebesgue ANR-11-LABX-0020-01, and the ANR Mistic project
(ANR-19-CE40-005).

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