# On the Navier-Stokes equations and the Hamilton-Jacobi-Bellman equation on
the group of volume preserving diffeomorphisms

Xiang-Dong Li, Guoping Liu Research supported by National Key R$\&$D Program
of China (No. 2020YF0712700), NSFC No. 12171458, and Key Laboratory RCSDS,
CAS, No. 2008DP173182.Research supported by NSFC No. 11801196 and the
Fundamental Research Funds for Central Universities, No. 2018KFYYXJJ043.

###### Abstract

In this paper, we give a new derivation of the incompressible Navier-Stokes
equations on a compact Riemannian manifold $M$ via the Bellman dynamic
programming principle on the infinite dimensional group of volume preserving
diffeomorphisms $G={\rm SDiff}(M)$. In particular, when the viscosity
vanishes, we give a new derivation of the incompressible Euler equation on a
compact Riemannian manifold. The main result of this paper indicates an
interesting relationship between the incompressible Navier-Stokes equations on
$M$ and the Hamilton-Jacobi-Bellman equation on $G={\rm SDiff}(M)$.

MSC2010 Classification: primary 35Q30, 49L20，secondary 58J65, 60H30

Keywords: Navier-Stokes equations, Bellman dynamic programming principle,
Hamilton-Jacobi-Bellman equation, the group of volume preserving
diffeomorphisms

##  1 Introduction

Hydrodynamics is one of the fundamental areas in mathematics and physics. In
[4] , Arnold gave a geometric interpretation of the incompressible Euler
equation on a Riemannian manifold $M$ as a geodesic on the group $G={\rm
SDiff}(M)$ of volume preserving diffeomorphisms equipped with the
$L^{2}$-right invariant Riemannian metric. See also [5]. In [14] (see also
Taylor [35]), the local existence and uniqueness of the Cauchy problem to the
incompressible Euler equation (i.e., Arnold’s geodesic) within suitable
Sobolev space was well-established. When ${\rm{dim}\hskip 0.85358pt}M=2$, the
global existence and uniqueness is known. See e.g. [35]. When ${\rm{dim}\hskip
0.85358pt}M\geq 3$, it remains a major outstanding problem whether the Euler
equation has global smooth solutions. On the other hand, concerning the
minimal geodesic linking two points in $G$, Ebin and Marsden [14] proved that
if $h$ belongs to a sufficiently small neighborhood of the identity map for a
suitable Sobolev norm, there is a unique minimal geodesic connecting $h$ and
the identity map. However, a striking result of Shnirelman [34] shows that,
for $M=[0,1]^{3}$ and for a large class of data, there is no minimal geodesic.
This shows the difficulty of applying the variational method to construct
nonstationary incompressible flows in the three-dimensional case. In recent
years, there have been some new investigations on the incompressible Euler
equation. For instance, Brenier [6] introduced the notion of generalized
solutions to the imcompressible Euler equation and proved the existence of the
generalized solutions. He also proved a regularity estimate on the pressure of
the generalized solution to the Euler equation. In [1], Ambrosio and Figalli
improved Brenier’s regularity estimates for the gradient of the pressure.

When the fluid is viscid, the incompressible Navier-Stokes equation on
$\mathbb{R}^{d}$ reads

$\displaystyle\begin{cases}$$\partial_{t}u+\left(u\cdot\nabla\right)u-\nu\Delta
u=-\nabla p,$$&\mbox{$$}\\\ $$\rm{div}\hskip
0.85358ptu=0,$$&\mbox{$$}\end{cases}$ (1.1)

where $\nu>0$ is the viscosity constant. In this situation, the system is no
longer conservative because of the friction term. Local existence with initial
data $u_{0}\in L^{n}(\mathbb{R}^{d})$ was established in Kato [24]. Ebin and
Marsden [14] proved the local existence and uniqueness of smooth solutions to
the incompressible Navier-Stokes equation on Riemannian manifolds. See also
Taylor [35]. When ${\rm{dim}\hskip 0.85358pt}M=2$, global existence of smooth
solutions is also known. When ${\rm{dim}\hskip 0.85358pt}M\geq 3$, it is a
long time outstanding open problem (one of the Millennium Prize Problems)
whether the incompressible Navier-Stokes equation has global smooth solutions.

It is interesting to ask the question whether one can extend Arnold’s point of
view to give a new derivation of the incompressible Navier-Stokes equation
$(\ref{NS1})$. It is well-known that the Laplacian $\Delta$ is the
infinitesimal generator of the Brownian motion. So, it is reasonable to derive
the incompressible Navier-Stokes equations by adding the effect of the
Brownian motion to the Euler equation. In some sense, this gives an
explanation of the physical term “internal friction” (Landau-Lifshitz [26],
see also Serrin [33]) by a probabilistic point of view. To our knowledge, the
connections between Navier-Stokes equations and stochastic evolution traced
back to Chorin [8], and a stochastic Hamiltonian approach was given by Inoue
and Funaki [23]. Since then, many people have studied the incompressible
Navier-Stokes equation using a stochastic Lagrangian variational calculus
together with Nelson’s stochastic mechanics [32]. For this, we mention the
works by Yasue [36, 37], Nakagomi-Yasue-Zambrini[31] and Esposito et al [16].

In recent years, more probabilistic works have been developed for the
incompressible Navier-Stokes equations. In [28], Le Jan and Sznitman used a
backward-in-time branching process in Fourier space to express the velocity
field of a three-dimensional viscous fluid as the average of a stochastic
process, which then leads to a new existence theorem. In [7], Busnello
introduced a probabilistic approach to the existence of a unique global
solution for two dimensional Navier-Stokes equations. In [8], Busnello,
Flandoli and Remito gave the probabilistic representation formulas for the
vorticity and the velocity of three dimensional Navier-Stokes equations and
gave a new proof of the local existence of solution. In [10], Cipriano and
Cruzeiro gave a stochastic variational principle for two dimensional
incompressible Navier-Stokes equations by using the Brownian motions on the
group of homeomorphisms on the torus. In [13], Cruzeiro and Shamarova
established a connection between the strong solution to the spatially periodic
Navier-Stokes equations and a solution to a system of forward-backward
stochastic differential equations on the group of volume preserving
diffeomorphisms of a flat torus. They also constructed representations of the
strong solution to the Navier-Stokes equations in terms of diffusion
processes. In [11], Constantin and Iyer derived a probabilistic representation
of the three-dimensional Navier-Stokes equations based on stochastic
Lagrangian paths. As they proved, the particle trajectories obey SDEs driven
by a uniform Wiener process, the inviscid Weber formula for the Euler
equations of ideal fluids is used to recover the velocity field. This method
admits a self-contained proof of local existence for the nonlinear stochastic
system and can be extended to formulate stochastic representations of related
hydrodynamic-type equations, including viscous Burgers equations and
Lagrangian-averaged Navier-Stokes alpha models. By reversing the time
variable, Zhang [40] derived a stochastic representation for backward
incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths
and gave a self-contained proof of local existence of solutions in Sobolev
spaces in the whole space. He also gave an alternative proof to the global
existence for the two-dimensional incompressible Navier-Stokes equations with
large viscosity. For extension of the above results to compact Riemannian
manifolds, see Fang-Zhang[20], Arnaudon-Cruzeiro[2], Arnaudon-Cruzeiro-Fang
[3], Fang-Luo[19], Luo[29] and Fang [18]. Due to the limit of the paper, we
will not try to give a full description and citation of all previous works on
the probabilistic studies for incompressible Navier-Stokes equations.

The purpose of this paper is to use the Bellman dynamic programming principle
on the infinite dimensional group of volume prerserving diffeomorphisms on a
compact Riemannian manifold to derive the incompressible Navier-Stokes
equation on a compact Riemannian manifold. To the best knowledge of the
authors, this is a new point of view even in the case where the viscosity
coefficient vanishes. That is to say, the incompressible Euler equation can be
derived from the deterministic dynamic programming principle on the infinite
dimensional group of volume preserving diffeomorphisms over a compact
Riemannian manifold.

Our results suggest a deep and close connection between the incompressible
Navier-Stokes equations with the Bellman dynamic programming principle on the
group of volume preserving diffeomorphisms. We hope that our work may bring
some new point of views for the further study of the incompressible Navier-
Stokes and Euler equations.

The rest of the paper is organized as follows. In Section 2, we extend the
Bellman dynamic programming principle and derive the Hamilton-Jacobi-Bellman
equation on compact Riemannian manifolds. In Section 3, we prove the Bellman
dynamic programming principle on the infinite dimensional group of volume
preserving diffeomorphisms $G={\rm SDiff}(M)$. In Section 4, we derive the
incompressible Navier-Stokes equations on a compact Riemannian manifold using
the Bellman dynamic programming principle and the Hamilton-Jacobi-Bellman
equation on $G={\rm SDiff}(M)$. In Section 5, we give a new derivation of the
incompressible Euler equation via the deterministic dynamic programming
principle and the Hamilton-Jacobi equation on $G={\rm SDiff}(M)$. In Section
6, we formulate our main result which indicates an interesting relationship
between the incompressible Navier-Stokes equations on $M$ and the Hamilton-
Jacobi-Bellman equation on the group of volume preserving diffeomorphisms
$G={\rm SDiff}(M)$, and we raise some problems for further research work.

##  2 Dynamic programming principle and HJB equation on Riemannian manifolds

In this section, we extend the classical Bellman dynamic programming principle
from the Euclidean space to compact Riemannian manifolds, and derive the
Hamilton-Jacobi-Bellman and Burgers equations on Riemannian manifolds.

First, for the convenience of the reader, we briefly review the standard
results on the Bellman dynamic programming principle and the Hamilton-Jacobi-
Bellman equation in stochastic control theory.

Consider the following controlled SDE on $\mathbb{R}^{d}$

$\displaystyle\begin{cases}$$dx^{u}(t)=\sqrt{2\nu}dB_{t}+u(t,x^{u}(t))dt,$$&\mbox{$$}\\\
$$x^{u}(0)=x,$$&\mbox{$$}\end{cases}$ (2.1)

where $B_{t}$ is a $\mathbb{R}^{d}$-valued standard Brownian motion. The
corresponding Lagrangian action on $\mathbb{R}^{d}$ is given by

$\displaystyle
S[x^{u}]=\mathbb{E}\left[\int_{0}^{T}\frac{1}{2}\|D_{t}x^{u}(t)\|_{\mathbb{R}^{d}}dt\right].$
(2.2)

Let $\Psi\in C(\mathbb{R}^{d},\mathbb{R})$, and define the value function

$\displaystyle
W(t,x)=\mathbb{E}\left[\left.\int_{t}^{T}\frac{1}{2}\|D_{s}x^{u}(s)\|_{\mathbb{R}^{d}}^{2}ds+\Psi(x^{u}(T))\right|x^{u}(t)=x\right].$
(2.3)

By Bellman’s principle of stochastic dynamic programming on Euclidean space,
it is well-known that $W$ satisfies the Hamilton-Jacobi-Bellman equation

$\displaystyle\partial_{t}W+\nu\Delta W-\frac{1}{2}\|\nabla
W\|_{\mathbb{R}^{d}}^{2}=0,$ (2.4)

and the optimal Markov control is given by

$\displaystyle u(t,x)=-\nabla W(t,x).$ (2.5)

See e.g. Zambrini [39], Fleming and Soner [21]. Suppose that $W\in
C^{1,3}([0,T]\times\mathbb{R}^{d},\mathbb{R})$. Differentiating the both sides
of $(\ref{HJB-1})$ with respect to spatial variable $x\in\mathbb{R}^{d}$, we
derive the Burgers equation on $\mathbb{R}^{d}$

$\displaystyle\partial_{t}u+\nu\Delta u+u\cdot\nabla u=0.$ (2.6)

###  2.1 Dynamic programming principle on Riemannian manifolds

Let $M$ be a $d$-dimensional compact Riemannian manifold without boundary,
$(\Omega,\mathcal{F},(\mathcal{F}_{s}),\mathbb{P})$ be a complete probability
space endowed with an increasing filtration $(\mathcal{F}_{s})$ which
satisfies the usual condition.

Let $u\in C^{1,2}([0,T]\times M,TM)$. Let $x_{0}\in M$ be fixed, and $B_{t}$
be a standard Brownian motion on $T_{x_{0}}M$. By Itô’s SDE theory, the
following controlled stochastic differential equation on $M$

$\displaystyle\begin{cases}$$dx(s)=\sqrt{2\nu}U_{t\rightarrow s}\circ
dB_{s}+u(s,x(s))ds,$$&\mbox{$t\leq s\leq T$}\\\ $$x(t)=x,$$&\mbox{$x\in
M$,}\end{cases}$ (2.7)

admits a unique strong solution, where $U_{t\rightarrow
s}:T_{x(t)}M\rightarrow T_{x(s)}M$ denotes the stochastic parallel transport
along the trajectory of the diffusion process $\\{x(r):r\in[t,s]\\}$, which
satisfies the covariant SDE on $M$

$\displaystyle\nabla_{\circ dx(s)}U_{t\rightarrow s}=0,\ \ \ U_{t\rightarrow
t}={\rm id}_{T_{x(t)}M},\ \ \forall\ t\leq s\leq T.$ (2.8)

where $\circ$ denotes the Stratonovich differentiation. See [30, 15].

Let $\mathcal{U}_{ad}$ be the set of all time dependent adapted smooth vector
fields $X$ on $M$ which satisfies the $L^{2}$-integrability condition

$\displaystyle\|X\|_{L^{2}}^{2}=\mathbb{E}\left[\int_{0}^{T}|X(s,x(s))|_{T_{x(s)}M}^{2}ds\right]<+\infty.$
(2.9)

Given a Lagrangian function $L\in C^{1}([0,T]\times TM,\mathbb{R})$ and
$\Psi\in C(M,\mathbb{R})$, we consider the following minimization problem on a
finite time interval $t\leq s\leq T$

$\displaystyle{\rm minimizes}\
J(t,x;u)=\mathbb{E}_{t,x}\left[\int_{t}^{T}L(s,x(s),u(s))ds+\Psi(x(T))\right],$
(2.10)

subject to
$u\in\mathcal{U}_{ad}=\left\\{u{\rm{~{}is~{}(\mathcal{F}_{s})~{}adapted~{}}}:\mathbb{E}\left[\int_{0}^{T}|u(s,x(s))|_{T_{x(s)}M}^{2}ds\right]<\infty\right\\}$.
For simplicity, we write $\mathbb{E}_{t,x}[\cdots]$ instead of
$\mathbb{E}\left[\left.\cdots\right|x(t)=x\right]$ throughout this paper.

By the same argument as in the proof of the dynamic programming equation on
the Euclidean spaces, see e.g. Fleming and Soner[21], we can prove the
following

###### Theorem 2.1

Let $M$ be a compact Riemannian manifold without boundary. Let $W$ be the
value function defined by

$\displaystyle W(t,x)=\inf\limits_{u\in\mathcal{U}_{ad}}J(t,x;u).$ (2.11)

Assume that $L\in C^{1}([0,T]\times TM,\mathbb{R})$ and $W\in
C^{1,2}([0,T]\times M,\mathbb{R})$. Then the dynamic programming equation
holds

$\displaystyle
0=\min\limits_{v\in\mathcal{U}}\left[A^{v}W(t,x)+L(t,x,v)\right].$ (2.12)

Proof. Taking the constant control $u(s)=v$ in a small time interval
$s\in[t,t+h]$, we have

$\displaystyle
W(t,x)\leq\mathbb{E}_{t,x}\int_{t}^{t+h}L(s,x(s),v)ds+\mathbb{E}_{t,x}W(t+h,x(t+h)).$
(2.13)

Subtracting first $W(t,x)$ from both sides of $(\ref{lopt})$, then dividing by
positive constant $h$ and letting $h$ tend to $0$. Using the continuity of
$L$, we have

$\displaystyle\lim\limits_{h\rightarrow
0^{+}}h^{-1}\mathbb{E}_{t,x}\int_{t}^{t+h}L(s,x(s),v)ds=L(t,x,v).$

Using the covariant Itô-Dynkin formula (see [15]) and similar to the proof on
the Euclidean space as in [21], if $W\in C^{1,2}([0,T]\times M,\mathbb{R})$,
we have

$\displaystyle\lim\limits_{h\rightarrow
0^{+}}h^{-1}\left[\mathbb{E}_{tx}W(t+h,x(t+h))-W(t,x)\right]$ $\displaystyle=$
$\displaystyle\lim\limits_{h\rightarrow
0^{+}}h^{-1}\mathbb{E}_{tx}\int_{t}^{t+h}A^{v}W(s,x(s))ds=A^{v}W(t,x),$

where $A^{v}=\partial_{t}+\nu\Delta+v\cdot\nabla$. Therefore, for all constant
control $v\in\mathcal{U}_{ad}$, we obtain

$\displaystyle 0\leq A^{v}W(t,x)+L(t,x,v).$ (2.14)

On the other hand, if $u^{\ast}$ is the optimal Markov control, we have

$\displaystyle
W(t,x)=\mathbb{E}_{t,x}\int_{t}^{t+h}L(s,x^{\ast}(s),u^{\ast}(s,x^{\ast}(s)))ds+\mathbb{E}_{t,x}W(t+h,x^{\ast}(t+h)),$

where $x^{\ast}(s)$ is the diffusion process generated by $A^{u^{\ast}}$. If
$L\in C^{1}([0,T]\times TM,\mathbb{R})$ and $W\in C^{1,2}([0,T]\times
M,\mathbb{R})$, a similar argument as for the proof on the Euclidean space
(see e.g. [21]) gives

$\displaystyle 0=A^{u^{\ast}}W(t,x)+L(t,x,u^{\ast}(t,x)).$ (2.15)

Combining (2.14) and (2.15), we derive the dynamic programming equation on $M$

$\displaystyle
0=\min\limits_{v\in\mathcal{U}}\left[A^{v}W(t,x)+L(t,x,v)\right].$

The value function $W$ satisfies the terminal condition

$\displaystyle W(T,x)=\Psi(x).$

The above argument suggests that an optimal Markov control satisfies

$\displaystyle u^{\ast}(t,x)\in\arg\min\left[A^{v}W(t,x)+L(t,x,v)\right],$

where $\arg\min f:=\left\\{u\in\mathcal{U}_{ad}:~{}f(u)\leq f(v),~{}\forall
v\in\mathcal{U}_{ad}\right\\}$.

###  2.2 The Hamilton-Jacobi-Bellman equation on Riemannian manifolds

In this subsection, we derive the Hamilton-Jacobi-Bellman equation from the
Bellman dynamic programming principle on compact Riemannian manifolds.

Let $(M,g)$ be a compact Riemannian manifold, $g$ the Riemannian metric, and
$d\mu=\sqrt{{\rm det}g(x)}dx$ the volume measure on $(M,g)$. Let $\nabla^{TM}$
be the Levi-Civita covariant derivative operator on $TM$, and $\nabla$
(respectively, $\Delta$) the Riemannian gradient (respectively, the Laplace-
Beltrami) operator on $(M,g)$.

Following [32], see also [10, 2, 3], the Nelson derivative of the diffusion
process $x(t)$ on $M$ is defined as follows

$\displaystyle D_{t}x(t):=\lim\limits_{\varepsilon\rightarrow
0}\mathbb{E}\left[\left.{U_{t+\varepsilon\rightarrow
t}dx(t+\varepsilon)-dx(t)\over\varepsilon}\right|\mathcal{F}_{t}\right].$
(2.16)

Note that, by [32, 3], for $x(t)$ defined by SDE $(\ref{SDEM})$, the Nelson
derivative of $x(t)$ is given by

$D_{t}x(t)=u(t,x(t)).$

We now state the main result of this subsection.

###### Theorem 2.2

Given $V\in C^{1}(M,\mathbb{R})$ and $\Psi\in C(M,\mathbb{R})$. Let

$\displaystyle L(s,x(s),u(s))=\frac{1}{2}|u(s,x(s))|_{T_{x(s)}M}^{2}-V(x(s))$
(2.17)

be the Lagrangian function, and let

$\displaystyle
W(t,x)=\mathbb{E}_{t,x}\left[\int_{t}^{T}L(s,x(s),u(s))ds+\Psi(x(T))\right]$

be the value function. Suppose that $W\in C^{1,3}([0,T]\times M,\mathbb{R})$.
Then $W$ satisfies the Hamilton-Jacobi-Bellman equation on $M$

$\displaystyle\partial_{t}W+\nu\Delta W-\frac{1}{2}|\nabla
W|^{2}_{T_{x}M}-V(x)=0,$ (2.18)

with the terminal condition $W(T,x)=\Psi(x)$. Moreover, the optimal control is
given by

$\displaystyle u^{*}(t,x)=-\nabla W(t,x).$ (2.19)

Define

$\displaystyle v(t,x)=\nabla W(T-t,x).$ (2.20)

Then $v$ satisfies the Burgers equation on $M$

$\displaystyle\partial_{t}v+\nu\square v+\nabla_{v}^{TM}v+\nabla V=0,$ (2.21)

where $\square=-\Delta+{\rm Ric}$ is the Hodge Laplacian, $\Delta={\rm
Tr}\nabla^{2}$ is the covariant Laplacian, ${\rm Ric}$ is the Ricci curvature
on $M$, and $\nabla^{TM}$ denotes the Levi-Civita covariant derivative
operator on $TM$.

Proof. The Bellman stochastic dynamic programming principle on $M$ says that
the value function $W$ satisfies the corresponding dynamic programming
equation

$\displaystyle 0=\inf_{v\in T_{x}M}\left\\{\partial_{t}W+\nu\Delta W+\langle
v,\nabla W\rangle+\frac{1}{2}|v|_{T_{x}M}^{2}-V(x)\right\\}.$ (2.22)

Using the Riemannian inner product on the tangent space, we have

$\displaystyle\inf_{v\in\mathcal{U}}\left\\{\partial_{t}W+\nu\Delta
W+v\cdot\nabla W+\frac{1}{2}|v|^{2}_{T_{x}M}-V(x)\right\\}$ $\displaystyle=$
$\displaystyle\partial_{t}W+\nu\Delta
W+\inf_{v\in\mathcal{U}}\left\\{v\cdot\nabla
W+\frac{1}{2}|v|^{2}_{T_{x}M}\right\\}-V(x)$ $\displaystyle=$
$\displaystyle\partial_{t}W+\nu\Delta W-\frac{1}{2}|\nabla
W|^{2}_{T_{x}M}-V(x).$

This proves that $W$ satisfies the Hamilton-Jacobi-Bellman equation
$(\ref{ch2-eq:9})$ on $M$. Moreover, from the above argument, we see that the
optimal Markov control $u^{*}$ is given by

$\displaystyle u^{*}=-\nabla W.$

Assuming that $W\in C^{1,3}([0,T]\times M)$, and taking the covariant
differentiation on both sides of the Hamilton-Jacobi-Bellman equation
$(\ref{ch2-eq:9})$, we have

$\displaystyle\partial_{t}\nabla W+\nu\nabla\Delta W-\frac{1}{2}\nabla|\nabla
W|^{2}-\nabla V(x)=0.$ (2.23)

Using the commutation formula in differential geometry

$\displaystyle\Delta\nabla W=\nabla\Delta W+{\rm Ric}(\nabla W),$ (2.24)

and the geometric formula

$\displaystyle{1\over 2}\nabla|\nabla W|^{2}=\langle\nabla\nabla W,\nabla
W\rangle=\nabla^{TM}_{v}v,$ (2.25)

we derive that $v(t,x)=\nabla W(T-t,x)$ satisfies the Burgers equation
$(\ref{M-Burgers})$.

###  2.3 Regularity of solution to HJB equation on $M$

Note that $(\ref{M-Burgers})$ is a system of quasi-linear parabolic partial
differential equations on compact Riemannian manifolds. By standard argument
as used in [35] on quasi-linear parabolic PDEs, we can prove the local
existence and uniqueness of the Cauchy problem to the Burgers equation within
suitable Sobolev space on a compact Riemannian manifold $M$.

Under the Cole-Hopf transformation

$\displaystyle\Phi(t,x)=\exp{\left[-{W(t,x)\over{2\nu}}\right]},~{}~{}\varphi(x)=\exp[-\Psi(x)],$
(2.26)

the Hamilton-Jacobi-Bellman equation $(\ref{ch2-eq:9})$ becomes the heat
equation

$\displaystyle\partial_{t}\Phi+\nu\Delta\Phi+\frac{1}{2\nu}V(x)\Phi=0,\ \ \
\Phi(T,x)=\varphi(x),$ (2.27)

which is the backward heat equation with a potential term
$\frac{1}{2\nu}V\Phi$.

Conversely, we can use the inverse of the Cole-Hopf transformation
$(\ref{Hopf})$ to derive the Hamilton-Jacobi-Bellman equation
$(\ref{ch2-eq:9})$ from the heat equation $(\ref{HeatV})$ provided that $\Phi$
is regular. Using the Feynman-Kac formula for Eq. $(\ref{HeatV})$, it is well-
known that $\Phi\in C^{1,2}([0,T]\times M)$ provided that $\varphi\in
C^{2}(M)$. Indeed, we have the following

###### Theorem 2.3

Suppose that the terminal value function $x\mapsto\Psi(x)$ is $C^{k}$-smooth.
Then the value function $x\mapsto W(t,x)$ is also $C^{k}$-smooth on $M$.

Proof. The proof is standard and is omitted.

##  3 Dynamic programming principle on $G={\rm SDiff}(M)$

In this section, we prove the Bellman dynamic programming principle on the
group of volume preserving diffeomorphisms.

Let $(\Omega,\mathcal{F},(\mathcal{F}_{s}),\mathbb{P})$ be a filtered
probability space satisfying the usual condition. Let $M$ be a $d$-dimensional
compact Riemannian manifold, $G={\rm SDiff}(M)$ the group of volume preserving
diffeomorphisms on $M$, and $\mathcal{G}$ its Lie algebra, i.e.,

$\mathcal{G}=\left\\{X\in\mathcal{X}(M):{\rm div}\hskip 0.56905ptX=0\right\\}$

is set of divergence free smooth vector fields on $M$.

Let $Q$ be a symmetric non-negative Hilbert-Schmidt operator on $\mathcal{G}$
equipped with the $L^{2}$-Riemannian metric defined in Section $2$. Recall the
following

###### Definition 3.1

A continuous $\mathcal{G}$-valued stochastic process $B^{\mathcal{G}}_{s}$ is
called a $Q$-Wiener process or $Q$-Brownian motion if

(1) $B^{\mathcal{G}}_{s}=id$ a.s.,

(2) $B^{\mathcal{G}}_{s}$ has independent increments
$B^{\mathcal{G}}_{st}=B^{\mathcal{G}}_{t}-B^{\mathcal{G}}_{s}$ for all $0\leq
s<t$,

(3) the increments have the following Gaussian laws:
$\mathbb{P}\circ(B^{\mathcal{G}}_{st})^{-1}=\mathcal{N}(0,(t-s)Q)$ for all
$0\leqslant s\leqslant t$.

Let $\\{g(s),s\in[t,T]\\}$ be defined as the solution to the following
Stratonovich stochastic differential equation on $G$

$\displaystyle\begin{cases}$$dg(s)=b(g(s),u(s))ds+\sigma(g(s),u(s))\circ
dB^{\mathcal{G}}_{s},$$&\mbox{}\\\ $$g(t)=g,$$&\mbox{}\end{cases}$ (3.1)

where $b:TG\rightarrow TG$, $\sigma:TG\rightarrow\mathcal{L}(TG;TG)$ satisfy
the Lipschitz condition, $TG$ is the tangent bundle on $G$,
$\mathcal{L}(TG;TG)$ is the bounded linear operator, and $u(s)$ is a
$T_{g(s)}G$-valued $\mathcal{F}_{s}$-adapted process along the diffusion
process $g(s)$ on $G$. We call such $u$ an admissible control on SDE
$(\ref{SDEG-1})$ on $G$.

Following [32], see also [10, 2, 3], the Nelson derivative of the diffusion
process $g(t)$ on $G$ is defined as follows

$\displaystyle D_{t}g(t):=\lim\limits_{\varepsilon\rightarrow
0}\mathbb{E}\left[\left.{//^{g}_{t+\varepsilon\rightarrow
t}dg(t+\varepsilon)-dg(t)\over\varepsilon}\right|\mathcal{F}_{t}\right]$

where $//^{g}_{t+\varepsilon\rightarrow t}:T_{g(t+\varepsilon)}G\rightarrow
T_{g(t)}G$ denotes the stochastic parallel transport from
$T_{g(t+\varepsilon)}G$ to $T_{g(t)}G$. Indeed, it is well-known that

$D_{t}g(t)=b(g(t),u(t)).$

Let

$\mathcal{U}=\\{u:[0,T]\times G\rightarrow TG\ \ {\rm adapted\ to}\
(\mathcal{F}_{s}):u(s,g(s))\in T_{g(s)}G\\}$

be the set of admissible controls on $G$. For simplicity, we use
$\mathbb{E}_{t,g}[\cdots]$ instead of
$\mathbb{E}\left[\left.\cdots\right|g(t)=g\right].$

We introduce the value function on $[t,T]\times G$ as follows

$\displaystyle
W(t,g)=\inf\limits_{u\in\mathcal{U}}\mathbb{E}_{t,g}\left[\int_{t}^{T}L(s,g(s),u(s))ds+\Psi(g(T))\right].$
(3.2)

The following result extends the Bellman dynamic programming principle from
Euclidean spaces to the infinite dimensional group of volume preserving
diffeomorphisms.

###### Theorem 3.2

(Bellman’s principle of dynamic programming on $G$) For $h$ small enough such
that $0\leq h\leq T-t$, we have

$\displaystyle
W(t,g)=\inf\limits_{u\in\mathcal{U}}\mathbb{E}_{t,g}\left[\int_{t}^{t+h}L(s,g(s),u(s))ds+W(t+h,g(t+h))\right].$
(3.3)

Proof. We first prove the “$\leq$” part. Choose any control
$\tilde{u}_{1}(\cdot)\in\mathcal{U}$, and let $g_{1}(\cdot)$ be the strong
solution of the stochastic differential equation

$\displaystyle\begin{cases}$$dg_{1}(s)=b(g_{1}(s),u_{1}(s))ds+\sigma(g_{1}(s),u_{1}(s))dB^{G}(s),$$&\mbox{}\\\
$$g_{1}(t)=g.$$&\mbox{}\end{cases}$ (3.4)

By definition of the infimum, there exists $u_{2}(\cdot)\in\mathcal{U}$ such
that

$W(t+h,g_{1}(t+h))+\varepsilon\geq\mathbb{E}_{t,g}\left[\int_{t+h}^{T}L(s,g_{2}(s),u_{2}(s))ds+\Psi(g_{2}(T))\right],$

where

$\displaystyle\begin{cases}$$dg_{2}(s)=b(g_{2}(s),u_{2}(s))ds+\sigma(g_{2}(s),u_{2}(s))dB^{G}(s),$$&\mbox{}\\\
$$g_{2}(t+h)=g_{1}(t+h).$$&\mbox{}\end{cases}$ (3.5)

Now define the control

$\displaystyle u_{3}(s):=\begin{cases}$$u_{1}(s),$$&\mbox{$t\leq s\leq
t+h$},\\\ $$u_{2}(s),$$&\mbox{$t+h\leq s\leq T$},\end{cases}$ (3.6)

and let

$\displaystyle\begin{cases}$$dg_{3}(s)=b(g_{3}(s),u_{3}(s))ds+\sigma(g_{3}(s),u_{3}(s))dB^{G}(s),$$&\mbox{}\\\
$$g_{3}(t)=g.$$&\mbox{}\end{cases}$ (3.7)

By the pathwise uniqueness of strong solutions to SDE (3.1), we have

$\displaystyle g_{3}(s)=\begin{cases}$$g_{1}(s),$$&\mbox{$t\leq s\leq
t+h$},\\\ $$g_{2}(s),$$&\mbox{$t+h\leq s\leq T$}.\end{cases}$ (3.8)

By definition of the value function

$\displaystyle W(t,g)$ $\displaystyle\leq$
$\displaystyle\mathbb{E}_{t,g}\left[\int_{t}^{T}L(g_{3}(s),u_{3}(s))ds+\Psi(g_{3}(T))\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{t,g}\left[\int_{t}^{t+h}L(g_{1}(s),u_{1}(s))ds+\int_{t+h}^{T}L(g_{2}(s),u_{2}(s))+\Psi(g_{2}(T))\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{t,g}\int_{t}^{t+h}L(g_{1}(s),u_{1}(s))ds+W(t+h,g_{1}(t+h))+\varepsilon.$

Due to arbitrary of $\tilde{u}_{1}(\cdot)$, we obtain

$W(t,g)\leq\inf\limits_{\mathcal{U}}\mathbb{E}_{t,g}\int_{t}^{t+h}L(g(s),u(s))ds+W(t+h,g(t+h))+\varepsilon,$

where $g(\cdot)=g^{\tilde{u}(\cdot)}(\cdot)$. Since $\varepsilon>0$ is
arbitrary, we prove the part “$\leq$”.

Next, we prove the part “$\geq$”. Fix $\varepsilon>0$, and select
$u_{4}\in\mathcal{U}$ be such that

$\displaystyle
W(t,g)+\varepsilon\geq\mathbb{E}_{t,g}\left[\int_{t}^{T}L(g_{4}(s),u_{4}(s))ds+\Psi(g_{4}(T))\right],$

where

$\displaystyle\begin{cases}$$dg_{4}(s)=b(g_{4}(s),u_{4}(s))ds+\sigma(g_{4}(s),u_{4}(s))dB^{G}(s),$$&\mbox{}\\\
$$g_{4}(t)=g.$$&\mbox{}\end{cases}$

By definition of value function

$\displaystyle
W(t+h,g_{4}(t+h))\leq\mathbb{E}_{t+h,g_{4}(t+h)}\left[\int_{t+h}^{T}L(g_{4}(s),u_{4}(s))ds+\Psi(g_{4}(T))\right].$

Therefore

$\displaystyle W(t,g)+\varepsilon$ $\displaystyle\geq$
$\displaystyle\mathbb{E}_{t,g}\left[\int_{t}^{t+h}L(g_{4}(s),u_{4}(s))ds+\int_{t+h}^{T}L(g_{4}(s),u_{4}(s))+\Psi(g_{4}(T))\right]$
$\displaystyle\geq$
$\displaystyle\mathbb{E}_{t,g}\left[\int_{t}^{t+h}L(g_{4}(s),u_{4}(s))ds\right]+\mathbb{E}_{t,g}W(t+h,g_{4}(t+h)).$

Hence

$\displaystyle
W(t,g)+\varepsilon\geq\inf\limits_{\mathcal{U}}\mathbb{E}_{t,g}\left[\int_{t}^{t+h}L(g(s),u(s))ds+W(t+h,g(t+h))\right],$

where $g(\cdot)=g^{u}(\cdot)$ is the strong solution of (3.4). The proof is
completed.

Now we derive the dynamic programming equation from (3.3) following standard
argument as used in [21, 38].

###### Theorem 3.3

Let $\Psi\in C(G,\mathbb{R})$ and $W(T,g)=\Psi(g)$, $g\in G$. Assume that
$W\in C^{1,2}([0,T]\times G,\mathbb{R})$ and satisfies Bellman’s principle of
dynamic programming, that is, for any $0<t<\eta<T$ and $g\in G$,

$\displaystyle
W(t,g)=\inf\limits_{u\in\mathcal{U}}\mathbb{E}\left[\left.\int_{t}^{\eta}L(s,g(s),u(s))ds+W(\eta,g(\eta))\right|g(t)=g\right].$

Then $W$ is a solution of the Bellman dynamic programming equation

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\inf\limits_{v\in\mathcal{U}}\left\\{\partial_{t}W(t,g)+\langle\langle\nabla
W(t,g),b(t,g,v)\rangle\rangle\right.$ $\displaystyle\left.+\frac{1}{2}{\rm
Tr}\left[(\sigma(t,g,v)Q^{\frac{1}{2}})^{*}(\sigma(t,g,v)Q^{\frac{1}{2}})\nabla^{2}W(t,g)\right]+L(t,g,v)\right.\\}.$

Proof. The proof consists of two steps.

First step, we prove that $W$ is a sub solution. Choose $v\in\mathcal{U}$, we
consider the constant control $u(s)\equiv v\in\mathcal{U},~{}t\leq s\leq t+h$.
Let $\\{g(s)\\}$ be a strong solution of $(\ref{SDEG-1})$ driven by $u$. Then
for all $h\in(0,T-t)$,

$\displaystyle
W(t,g)\leq\mathbb{E}\left[\int_{t}^{t+h}L(s,g(s),v)ds+W(t+h,g(t+h))|g(t)=g\right].$

Subtracting $W(t,g)$ from both sides, dividing by $h$, and according to Itô-
Dynkin formula 111Here we need to use the condition $W\in C^{1,2}([0,T]\times
G,\mathbb{R})$., we obtain

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\frac{1}{h}\mathbb{E}\left[W(t+h,g(t+h))-W(t,g)|g(t)=g\right]$
$\displaystyle+\frac{1}{h}\mathbb{E}\left[\left.\int_{t}^{t+h}L(s,g(s),v)ds\right|g(t)=g\right]$
$\displaystyle\leq$
$\displaystyle\frac{1}{h}\mathbb{E}\left[\int_{t}^{t+h}[\partial_{s}W(s,g(s))+\langle\nabla
W(s,g(s)),b(s,g(s),v)\rangle\right.$ $\displaystyle+\frac{1}{2}{\rm
Tr}[(\sigma(s,g(s),v)Q^{\frac{1}{2}})(\sigma(s,g(s),v)Q^{\frac{1}{2}})^{*}\nabla^{2}W(s,g(s))]$
$\displaystyle\left.\left.+L(s,g(s),v)ds]\right|g(t)=g\right].$

Let $h\rightarrow 0$, for $v\in\mathcal{U}$, $W\in C^{1,2}([0,T]\times
G;\mathbb{R})$, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\partial_{t}W(t,g)+\langle\nabla W(t,g),b(t,g,v)\rangle$
$\displaystyle+\frac{1}{2}{\rm
Tr}[(\sigma(t,g,v)Q^{\frac{1}{2}})(\sigma(t,g,v)Q^{\frac{1}{2}})^{*}\nabla^{2}W(t,g)]+L(t,g,v).$

Then $W(t,g)$ is a sub solution of $(\ref{HJB-3})$ with terminal condition
$W(T,g)=\Psi(g)$.

Second step, we prove that $W$ is a super solution.

For $\varepsilon\in(0,T-t)$, by definition of infimum, we can choose the
minimizing sequences $u_{\varepsilon}\in\mathcal{U}$ such that

$\displaystyle
W(t,g)+\varepsilon^{2}\geq\mathbb{E}\left[\left.\int_{t}^{t+h}L(s,g^{\varepsilon}(s),u_{\varepsilon})ds+W(t+h,g^{\varepsilon}(t+h))\right|g^{\varepsilon}(t)=g\right].$

We subtract $W(t,g)$ from bothsides, then divide by $\varepsilon$,

$\displaystyle\varepsilon$ $\displaystyle\geq$
$\displaystyle\frac{1}{\varepsilon}\mathbb{E}\left[\left.W(t+\varepsilon,g^{\varepsilon}(t+\varepsilon))-W(t,g)\right|g^{\varepsilon}(t)=g\right]$
$\displaystyle+\frac{1}{\varepsilon}\mathbb{E}\left[\left.\int_{t}^{t+\varepsilon}L(s,g^{\varepsilon}(s),u^{\varepsilon})ds\right|g^{\varepsilon}(t)=g\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\varepsilon}\mathbb{E}\left[\int_{t}^{t+\varepsilon}[\partial_{s}W(s,g^{\varepsilon}(s))+\langle\nabla
W(s,g^{\varepsilon}(s)),b(s,g^{\varepsilon}(s),u^{\varepsilon})\rangle\right.$
$\displaystyle+\frac{1}{2}{\rm
Tr}[(\sigma(s,g^{\varepsilon}(s),u^{\varepsilon})Q^{\frac{1}{2}})(\sigma(s,g^{\varepsilon}(s),u^{\varepsilon})Q^{\frac{1}{2}})^{*}\nabla^{2}W(s,g^{\varepsilon}(s))]$
$\displaystyle\left.\left.+L(s,g^{\varepsilon}(s),g^{\varepsilon})ds]\right|g^{\varepsilon}(t)=g\right].$

Let $\varepsilon\rightarrow 0$. Since $W\in C^{1,2}([0,T]\times
G,\mathbb{R})$, noticing that $g^{\varepsilon}$ is the strong solution driven
by the minimizing sequences, we obtain

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\partial_{t}W(t,g)+\inf\limits_{v\in\mathcal{U}}[\langle\langle\nabla
W(t,g),b(t,g,v)\rangle\rangle$ $\displaystyle+\frac{1}{2}{\rm
Tr}[(\sigma(t,g,v)Q^{\frac{1}{2}})(\sigma(t,g,v)Q^{\frac{1}{2}})^{*}\nabla^{2}W(t,g)]+L(t,g,v)].$

Hence $W(t,g)$ is a super solution of $(\ref{HJB-3})$ with terminal condition
$W(T,g)=\Psi(g)$. This finishes the proof.

###### Remark 3.4

Notice that the dynamic programming equation is equivalent to

$\displaystyle
0=\min\limits_{v\in\mathcal{U}}\left[A^{v}W(t,g)+L(t,g,v)\right],$ (3.10)

with the terminal condition

$\displaystyle W(T,g)=\Psi(g),$ (3.11)

where

$\displaystyle A^{v}=\partial_{t}+b(t,g,v)\cdot\nabla+\frac{1}{2}{\rm
Tr}[(\sigma(t,g,v)Q^{\frac{1}{2}})(\sigma(t,g,v)Q^{\frac{1}{2}})^{*}\nabla^{2}].$
(3.12)

To simplify the notation, for fixed $Q$, we introduce the $Q$-Laplacian on $G$
as follows

$\displaystyle\Delta_{Q}^{G}=\frac{1}{2}{\rm
Tr}[(\sigma(t,g,v)Q^{\frac{1}{2}})^{*}(\sigma(t,g,v)Q^{\frac{1}{2}})\nabla^{2}].$
(3.13)

Let $V\in C^{1}(G,\mathbb{R})$. Consider the Lagrangian action functional

$\displaystyle L(s,g(s),u(s))=\frac{1}{2}\|u(s)\|_{T_{g(s)}G}^{2}-V(g(s)),$

where $\|\cdot\|_{T_{g(s)}G}$ is the right-invariant $L^{2}$-Riemannian metric
on $G={\rm SDiff}(M)$. By the dynamic programming equation $(\ref{HJB-4})$
derived from Theorem 3.3, we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\inf_{v\in\mathcal{U}}\left\\{\partial_{t}W+\nu\Delta_{Q}^{G}W+v\cdot\nabla^{G}W+\frac{1}{2}\|v\|_{\mathcal{G}}^{2}-V(g)\right\\}$
$\displaystyle=$
$\displaystyle\partial_{t}W+\nu\Delta_{Q}^{G}W-\frac{1}{2}\|\nabla^{G}W\|_{\mathcal{G}}^{2}-V(g),$

with terminal condition (3.11). Moreover, the optimal control $u$ is given by

$\displaystyle u(t,g)=-\nabla^{G}W(t,g).$ (3.15)

where $\nabla^{G}$ is the gradient on $G$, and $\Delta_{Q}^{G}$ is the
$Q$-Laplacian on $G$.

Finally, let us mention that, without any difficulty we can extend the main
result in this section to any $M$-type $2$ Banach manifold.

##  4 New derivation of the incompressible Navier-Stokes equations

In this section, we first derive the Hamilton-Jacobi-Bellman and the Burgers
equations on the infinite dimensional group of volume preserving
diffeomorphisms. By pushing forward to the underlying manifold, and using the
Hodge decomposition theorem, we then derive the incompressible Navier-Stokes
equations on compact Riemannian manifolds.

###  4.1 The idea for the derivation of incompressible Navier-Stokes equations

In Section $2$, we derive the Burgers equation $(\ref{M-Burgers})$ on compact
Riemannian manifolds from the Bellman dynamic programming principle, which, at
least in the case of bounded domain in Euclidean space, has the similar form
as the incompressible Navier-Stokes equations except the divergence-free
constraint condition ${\rm div}\hskip 0.85358ptu=0$. In a series works of
Cruzeiro and her collaborators [10, 13], they developed a stochastic
variational principle for the action

$\displaystyle\mathcal{A}(x^{u},u)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\frac{1}{2}\|D_{t}x^{u}(t)\|_{L^{2}(M)}^{2}dt\right]$
(4.1) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\int_{M}\frac{1}{2}|D_{t}x^{u}(t)(x)|^{2}d\mu
dt\right]$

among the diffusion processes $\\{x^{u}(t),t\in[0,T]\\}$ defined by SDE
$(\ref{SDEM})$ with control $u\in\mathcal{U}_{ad}$. More precisely, Cipriano
and Cruzeiro [10] proved that the optimal control $u\in C^{1,2}([0,T]\times
M,\Gamma(TM))$ for the Euler-Lagrangian equation of the Lagrangian action
$\mathcal{A}$ on the two-dimensional torus $\mathbb{T}^{2}$ satisfies the
incompressible Navier-Stokes equation

$\displaystyle\partial_{t}u+u\cdot\nabla u{\color[rgb]{0,0,1}-}\nu\Delta u$
$\displaystyle=$ $\displaystyle-\nabla p,$ $\displaystyle{\rm div}\hskip
0.85358ptu$ $\displaystyle=$ $\displaystyle 0.￼$

See further works in [2, 3, 19, 29] for the extension of this idea to compact
Riemannian manifolds.

Let $\mathcal{V}:G\rightarrow\mathbb{R}$ be a $C^{1}$-continuous function. We
introduce

$\displaystyle\mathcal{L}(g^{u},u)=\mathbb{E}\left[\int_{0}^{T}\left(\int_{M}\frac{1}{2}|D_{s}g^{u}(s)(x)|^{2}d\mu-\mathcal{V}(g^{u}(s))\right)ds\right],$
(4.2)

which is the Lagrangian action function on $G={\rm SDiff}(M)$, where
$g^{u}(s)$ is the horizontal lift of the solution $x^{u}(s)$ of the Controlled
SDE $(\ref{SDEM})$ on $G={\rm SDiff}(M)$.

Let $\Psi\in C(G,\mathbb{R})$. We introduce the value function $W$ on $G={\rm
SDiff}(M)$

$\displaystyle
W(t,g)=\inf\limits_{\mathcal{U}}\mathbb{E}_{t,g}\left[\int_{t}^{T}\left(\int_{M}\frac{1}{2}|D_{s}g^{u}(s)(x)|^{2}d\mu-\mathcal{V}(g^{u}(s))\right)ds+\Psi(g^{u}(T))\right].$
(4.3)

Here and throughout this paper,
$\mathbb{E}_{t,g}[\ldots]=\mathbb{E}\left[\left.\ldots\right|g^{u}(t)=g\right]$.

By the Bellman dynamic programming principle on $G={\rm SDiff}(M)$ which has
been proved in Section $3$, we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\inf_{v\in
T_{g}G}\left\\{\partial_{t}W+\nu\Delta^{G}W+\langle
v,\nabla^{G}W\rangle_{T_{g}G}+\frac{1}{2}\|v\|_{T_{g}G}^{2}-\mathcal{V}(g)\right\\}$
$\displaystyle=$ $\displaystyle\partial_{t}W+\nu\Delta^{G}W+\inf_{v\in
T_{g}G}\left\\{\langle
v,\nabla^{G}W\rangle_{T_{g}G}+\frac{1}{2}\|v\|_{T_{g}G}^{2}\right\\}-\mathcal{V}(g).$

where $\nabla^{G}$ and $\Delta^{G}$ denote the Riemannian gradient operator
and the weighted Laplace-Beltrami (i.e., the Ornstein-Uhlenbeck) type operator
on the infinite dimensional Riemannian manifold on $G={\rm SDiff}(M)$.

Then, by a similar argument as used in the derivation of the Burgers equation
$(\ref{M-Burgers})$ on $M$, we can prove that the value function $W$ satisfies
the infinite dimensional Hamilton-Jacobi-Bellman equation on $G={\rm
SDiff}(M)$

$\displaystyle\partial_{t}W+\nu\Delta^{G}W-\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}-\mathcal{V}(g)=0,$
(4.5)

with the terminal condition $W(T,g)=\Psi(g)$. Moreover, the optimal Markov
control $U$ is given by

$\displaystyle U=-\nabla^{G}W.$

By the same argument as explained in the derivation of the Burgers equation
$(\ref{M-Burgers})$, we will prove that $U$ satisfies the Burgers type
equation on $G$.

Furthermore, by the Hodge decomposition theorem, and by pushing forward
$(\ref{HJBG})$ to $M$ via the evaluation map $e_{x}:G\rightarrow M,g\mapsto
g(x)$, we can derive that $u=(de_{x})U$ satisfies the incompressible Navier-
Stokes equation on the underlying Riemannian manifold $(M,g)$.

This gives a new derivation of the incompressible Navier-Stokes equation via
the Bellman dynamic programming principle on the infinite dimensional group
$G={\rm SDiff}(M)$. In particular, when the viscosity coefficient $\nu$
vanishes, we give a new derivation of the incompressible Euler equation via
the deterministic dynamic programming principle on the infinite dimensional
group $G={\rm SDiff}(M)$.

###  4.2 The Hamilton-Jacobi-Bellman equation on $G={\rm SDiff}(M)$

In this subsection, we introduce a controlled SDE on the group of volume
preserving diffeomorphisms $G={\rm SDiff}(M)$ and derive the Hamilton-Jacobi-
Bellman equation on $G={\rm SDiff}(M)$.

Let $\\{g^{u}(t),t\in[0,T]\\}$ be a diffusion process with values in $G={\rm
SDiff}(M)$ and satisfies the following Stratonovich stochastic differential
equation (briefly, SDE) with right action on $G={\rm SDiff}(M)$

$\displaystyle\begin{cases}$$dg^{u}(t)=\left(\sqrt{2\nu}\circ
dW_{t}\right)g^{u}(t)+u(t,g^{u}(t))dt,$$&\mbox{$$}\\\
$$g^{u}(0)=e,$$&\mbox{$$}\end{cases}$ (4.6)

where $W_{t}$ is a $\mathcal{G}$-valued $Q$-Brownian motion (see Definition
3.1), $u$ is a horizontal lift of a smooth divergence-free vector field on $M$
to $\mathcal{G}$, and $e=id$ is the identity map of $G={\rm{SDiff}}(M)$. The
short time existence and uniqueness of solution to SDE $(\ref{SDEG})$ follows
from standard Picard iteration as in Itô’s SDE theory on Euclidean spaces.

Let $c_{i}\in\mathbb{R}$, $\\{W_{i},i=1,2,...\\}$ be i.i.d. Brownian motions,
$\\{\widetilde{A}_{i},i=1,2,...\\}$ and $U$ be smooth vector fields on $G={\rm
SDiff(M)}$. Let $g(s)$ be the solution to the following SDE on $G$

$\begin{cases}$$dg(s)=\sqrt{2\nu}\sum\limits_{i=1}^{\infty}c_{i}\tilde{A}_{i}(g(s))\circ
dW_{i}(s)+U(s,g(s))ds,$$&\mbox{$s\in(t,T]$}\\\ $$g(t)=g.$$&\mbox{}\end{cases}$
(4.7)

Since $\\{\widetilde{A}_{i},i=1,2,...\\}$ and $U$ are smooth vector fields on
$G$, SDE $(\ref{ch2-SDEG})$ has a unique solution $\\{g(s),s\in[t,T]\\}$.

Let $e_{x}$ be the evaluation map, $e_{x}:G\rightarrow M,~{}~{}e_{x}(g)=g(x)$,
$de_{x}$ be the differential of $e_{x}$ at the point $g$

$de_{x}:T_{g}G\rightarrow T_{g(x)}M.$

By the chain rule of differential, we have

$d(e_{x}\circ R_{g})_{e}=(de_{x})_{g}(dR_{g})_{e}.$

###### Lemma 4.1

Suppose that there exists a family of vector fields
$\\{A_{i}\\}_{i\in\mathcal{I}}$ on $M$ such that for all $g\in G$, it holds

$A_{i}(g(x))=d(e_{x}\circ R_{g})_{e}\widetilde{A_{i}}(e),$

where $\widetilde{A_{i}}(e)\in\mathcal{G}$ is the horizontal lift of
$A_{i}(x)$ by $e_{x}$, and is right-invariant, i.e.,

$\widetilde{A_{i}}(g)=(dR_{g})_{e}\widetilde{A_{i}}(e).$

Then for any $f\in C^{\infty}(M,\mathbb{R})$, and $F_{x}=f\circ e_{x}$, it
holds

$(\widetilde{\mathcal{L}}F_{x})(g)=(\mathcal{L}f)(g(x)),$

where $\widetilde{\mathcal{L}}$ is the infinitesimal generator of SDE on $G$

$dg_{t}=\sum c_{i}\widetilde{A}_{i}(g_{t})\circ
dW_{t}^{i}+\widetilde{A}_{0}(g_{t})dt,~{}~{}g_{0}=g,$

and $\mathcal{L}$ is the infinitesimal generator of SDE on $M$

$dg_{t}(x)=\sum c_{i}A_{i}(g_{t}(x))\circ
dW_{t}^{i}+A_{0}(g_{t}(x))dt,~{}~{}g_{0}(x)=g(x)$

Proof. By the fact $A_{i}(g(x))=d(e_{x}\circ
R_{g})_{e}\widetilde{A_{i}}(e)=(de_{x})_{g}(dR_{g})_{e}\widetilde{A_{i}}(e)=(de_{x})_{g}\widetilde{A_{i}}(g)$,
we have
$\widetilde{A_{i}}F_{x}(g)=(A_{i}f)(g(x))=(A_{i}f)(e_{x}(g))=(A_{i}f)\circ
e_{x}(g)$. Hence

$\displaystyle\widetilde{A_{i}}F_{x}$ $\displaystyle=$
$\displaystyle(A_{i}f)\circ e_{x},$
$\displaystyle\widetilde{A_{i}}(\widetilde{A_{i}}F_{x})$ $\displaystyle=$
$\displaystyle A_{i}(A_{i}f)\circ e_{x}.$

This implies

$\displaystyle(\widetilde{\mathcal{L}}F_{x})(g)$ $\displaystyle=$
$\displaystyle\left.\frac{\partial}{\partial
t}\mathbb{E}_{g}[F_{x}(g_{t})]\right|_{t=0}$ $\displaystyle=$
$\displaystyle\sum
c_{i}^{2}\widetilde{A_{i}}^{2}F_{x}(g)+\widetilde{A_{0}}F_{x}(g)\cdot\nabla$
$\displaystyle=$ $\displaystyle\sum
c_{i}^{2}A_{i}^{2}f(g(x))+A_{0}f(g(x))\cdot\nabla$ $\displaystyle=$
$\displaystyle\left.\frac{\partial}{\partial
t}\mathbb{E}_{g(x)}[f(g_{t}(x))]\right|_{t=0}$ $\displaystyle=$
$\displaystyle(\mathcal{L}f)(g(x)).$

Fix $x\in M$. Let $e_{x}:G\rightarrow M$ be the evaluation map, and
$g(s)(x)=e_{x}(g(s))$. By the Itô formula, we can derive the SDE for $g(s)(x)$
on $M$ from SDE $(\ref{ch2-SDEG})$ for $g(s)$ on $G$ as follows

$\begin{cases}$$dg(s)(x)=\sqrt{2\nu}\sum\limits_{i=1}^{\infty}c_{i}A_{i}(g(s)(x))\circ
dW_{i}(s)+U(s,g(s)(x))ds,$$&\mbox{$s\in(t,T]$}\\\
$$g(t)(x)=g(x).$$&\mbox{}\end{cases}$ (4.8)

Here the vector fields $\tilde{A}_{i}$ and $U$ on $G={\rm SDiff(M)}$ and
$A_{i}$ and $u$ on $M$ satisfy the relationship

$\displaystyle(de_{x})(g)U(g)=u(g(x)),$
$\displaystyle(de_{x})(g)\tilde{A}_{i}(g)=A_{i}(g(x)),i=1,2,\cdots,$

which is the case when $\widetilde{A}_{i}$ and $U$ are right-invariant on $G$
in the sense that

$\displaystyle U(g)=dR_{g}U(e),$
$\displaystyle\tilde{A}_{i}(g)=dR_{g}\tilde{A}_{i}(e),\ \ i=1,2,\ldots$
$\displaystyle(de_{x})U(e)=u(x),$
$\displaystyle(de_{x})\widetilde{A}_{i}(e)=A_{i}(x),\ \ i=1,2,\ldots$

Since $A_{i}$ and $u$ are $C^{\infty}$ vector field on $M$, then SDE
$(\ref{ch2-SDED})$ has a unique solution $g(s)(x)$, and $x\mapsto g(s)(x)$ is
a diffeomorphism on $M$ using the theory of stochastic flows. See [22] and
[25]. The infinitesimal generator of $g(s)(x)$ of SDE $(\ref{ch2-SDED})$ is
given by

$\displaystyle L=\nu\sum\limits_{i=1}^{\infty}c_{i}^{2}A_{i}^{2}+u.$

We need a family of vector fields $\\{A_{i},i\in\mathbb{N}\\}$ and
renormalization coefficients $\\{c_{i},.i\in\mathbb{N}\\}$ such that

$\displaystyle\sum\limits_{i=1}^{\infty}c_{i}^{2}A_{i}^{2}$ $\displaystyle=$
$\displaystyle c\Delta^{M},$ (4.9) $\displaystyle{\rm div}A_{i}$
$\displaystyle=$ $\displaystyle 0,\ \ \forall i\in\mathbb{N}$ (4.10)

where $c>0$ is a constant, $\Delta^{M}$ is the Laplace-Beltrami operator on
the Riemannian manifold $(M,g)$. When $M=\mathbb{T}^{d}$, this was proved by
Cipriano-Cruzeiro [10]. When $M=\mathbb{S}^{d}$, this was proved by Le Jan-
Raimond [27], see also Fang-Zhang [20], Fang-Luo [19] and Luo [29]. In [15],
Elworthy, Le Jan and X.-M. Li proved the existence of such vector fields on
compact Riemannian manifolds equipped with a Le Jan-Watanabe connection, in
particular, for gradient Brownian system on compact Riemannian manifolds
equipped with a torsion-skew symmetric metric connection. More precisely,
assuming that there exist a family of vector fields
$\\{A_{i},i\in\mathbb{N}\\}$ such that

$\displaystyle\|v\|^{2}=\sum\limits_{i=1}^{\infty}\langle
A_{i}(x),v\rangle^{2},\ \ \forall x\in M,v\in T_{x}M,$ (4.11)
$\displaystyle\sum\limits_{i=1}^{\infty}\nabla_{A_{i}}A_{i}=0,$ (4.12)
$\displaystyle\sum\limits_{i=1}^{\infty}A_{i}\wedge\nabla_{X}A_{i}=0,\ \
\forall X\in\Gamma(TM),$ (4.13)

then for any differential form $\omega$ on $M$, it holds (see Remark 2.3.1 in
[15])

$\displaystyle\square^{M}\omega=-\sum\limits_{i=1}^{\infty}L_{A_{i}}^{2}\omega.$
(4.14)

Moreover, we have the following

###### Lemma 4.2

([27, 10, 20, 19]) Assume that there exist a family of vector fields
$\\{A_{i},i\in\mathbb{N}\\}$ and $\\{c_{i},i\in\mathbb{N}\\}$ satisfying
$(\ref{cA1})$ and $(\ref{cA2})$. Fix $x\in M$. Let $e_{x}:G\rightarrow M$ be
the evaluation map $e_{x}(g)=g(x)$. Then, for any $f\in C^{\infty}(M)$, it
holds

$\displaystyle L_{G}(f\circ e_{x})=(L_{M}f)\circ e_{x}.$ (4.15)

Equivalently, letting $F_{x}:G\rightarrow\mathbb{R}$ be defined by

$F_{x}(g)=(f\circ e_{x})(g)=f(g(x)),\ \ \forall g\in G.$

Then

$\displaystyle(L_{G}F_{x})(g)=(L_{M}f)(g(x)).$ (4.16)

The main result of this subsection is the following Hamilton-Jacobi-Bellman
equation on $G=\rm{SDiff}(M)$.

###### Theorem 4.3

Let $g(s)$ be the solution to the controlled SDE $(\ref{SDEG})$. Let

$\displaystyle
W(t,g):=\inf\limits_{U\in\mathcal{U}}\mathbb{E}_{t,g}\left[\int_{t}^{T}L(s,g(s),U(s))ds+\Psi(g(T))\right]$
(4.17)

be the value function of the related stochastic control problem, where the
Lagrangian is given by

$L(s,g(s),U(s))=\frac{1}{2}\|U\|_{T_{g(s)}G}^{2}-V(g(s)).$

Then $W$ satisfies the Hamilton-Jacobi-Bellman equation on
$G=\operatorname{SDiff}(M)$

$\partial_{t}W-\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}+\nu\Delta^{G}W=V(g),$
(4.18)

with the terminal condition $W(T,g)=\Psi(g)$.

Proof. By the extended Bellman dynamic programming principle on $G$ (see Eq.
$(\ref{HJB-4})$ in Section $3$), we have

$\displaystyle\inf_{U\in\mathcal{G}}\\{\partial_{t}W+U\cdot\nabla^{G}W+\nu\Delta^{G}W+L(t,g,U)\\}=0.$

By the same argument as used in Section 2 for the proof of the HJB equation on
$M$, see also the description in Subsection 2.1, we conclude that the optimal
Markov control $U(t,g)$ is given by

$U(t,g)=-\nabla^{G}W(t,g),$

and $W$ satisfies the Hamilton-Jacobi-Bellman equation $(\ref{HJB-W})$ with
the terminal condition $W(T,g)=\Psi(g)$.

###  4.3 New derivation of the incompressible Navier-Stokes equation

In this subsection, we use the Hamilton-Jacobi-Bellman equation on
$G=\operatorname{SDiff}(M)$ to give a new derivation of the incompressible
Navier-Stokes equation on a compact Riemannian manifold $M$.

Let $d^{G}$ be the exterior differential on $G$. Following [15], we introduce

$\displaystyle\delta^{G}=\sum\limits_{i=1}^{\infty}c_{i}^{2}{\rm
int}_{\widetilde{A}_{i}}\nabla_{\widetilde{A}_{i}},$

and define the Hodge like Laplacian acting on $1$-form on $G$ as follows

$\displaystyle\square^{G}=d^{G}\delta^{G}+\delta^{G}d^{G}.$

Since $(d^{G})^{2}=0$, it holds

$\displaystyle d^{G}\square^{G}$ $\displaystyle=$ $\displaystyle
d^{G}d^{G}\delta^{G}+d^{G}\delta^{G}d^{G}$ $\displaystyle=$ $\displaystyle
d^{G}\delta^{G}d^{G}$ $\displaystyle=$ $\displaystyle
d^{G}\delta^{G}d^{G}+\delta^{G}d^{G}d^{G}$ $\displaystyle=$
$\displaystyle\square^{G}d^{G}.$

Now we state the main result of this paper as follows.

###### Theorem 4.4

Let $W\in C^{1,3}([0,T]\times G)$ be a solution to the Hamilton-Jacobi-Bellman
equation $(\ref{HJB-W})$ on $G={\rm SDiff}(M)$. Let

$\displaystyle U(t,g)=-\nabla^{G}W(t,g)$

be the optimal control. Then

$\displaystyle\partial_{t}U+\frac{1}{2}\nabla^{G}\|U\|^{2}-\nu\Delta^{G}U=-\nabla^{G}V.$

Let

$u(t,x)=(de_{x})U(t,e),~{}v(x)=(de_{x})V(e).$

Then there exists a function $p\in C^{1}([0,T]\times M,\mathbb{R})$ such that
$(u,p)$ satisfies the backward incompressible Navier-Stokes equation on $M$

$\displaystyle\partial_{t}u+\nabla^{TM}_{u}u-\nu\square^{M}u$ $\displaystyle=$
$\displaystyle-\nabla^{M}p,$ (4.19) $\displaystyle\rm{div}\hskip 0.85358ptu$
$\displaystyle=$ $\displaystyle 0,$ (4.20)

and $(-u(T-t,\cdot),p(T-t,\cdot))$ satisfies the incompressible Navier-Stokes
equation on $M$

$\displaystyle\partial_{t}u+\nabla^{TM}_{u}u+\nu\square^{M}u$ $\displaystyle=$
$\displaystyle-\nabla^{M}p,$ (4.21) $\displaystyle\rm{div}\hskip 0.85358ptu$
$\displaystyle=$ $\displaystyle 0.$ (4.22)

Note that the $\nabla^{M}v$ has been absorbed into $\nabla^{M}p$.

Proof. Taking differentiation on the Hamilton-Jacobi-Bellman equation
$(\ref{HJB-W})$, we have

$\displaystyle
d^{G}(\partial_{t}W)-d^{G}\left(\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}\right)+\nu
d^{G}(\Delta^{G}W)=d^{G}V(t,g).$

Using the commutative formulae

$d^{G}\partial_{t}W=\partial_{t}d^{G}W,\ d^{G}\Delta^{G}W=-\square^{G}d^{G}W,$

we have

$\displaystyle\partial_{t}\nabla^{G}W-\frac{1}{2}\nabla^{G}\|\nabla^{G}W\|^{2}_{T_{g}G}-\nu\square^{G}\nabla^{G}W=\nabla^{G}V(t,g).$
(4.23)

Thus $U(t,g)=-\nabla^{G}W(t,g)$ satisfies the Burgers equation on $G$

$\displaystyle\partial_{t}U+\frac{1}{2}\nabla^{G}\|U\|^{2}_{T_{g}G}-\nu\square^{G}U=-\nabla^{G}V(t,g).$

Pushing forward the Burgers equation on $G$ to $M$ by the evaluation map

$e_{g^{-1}(x)}:G=\operatorname{SDiff}(M)\rightarrow M,\quad g\mapsto x.$

By definition of the right invariant vector fields on $G$, there is a one to
one correspondence between vector fields $u,~{}v$ on $M$ and right invariant
vector fields $U,~{}V$ on $G=\operatorname{SDiff}(M)$ such that

$u(t,x)=\left(de_{g^{-1}(x)}\right)U(t,g),~{}v(x)=\left(de_{g^{-1}(x)}\right)V(g).$

By the commutative formula $d\Delta^{G}W=-\square^{G}dW$ and the duality
between the Riemmanian gradient and exterior differential operator on
$\mathcal{G}=\\{X\in C^{\infty}(M,TM):\rm{div}\hskip 0.85358ptX=0\\}$, for any
$X\in\mathcal{G}=T_{e}G$, we have

$\displaystyle\langle\langle\nabla^{G}(\Delta^{G}W),X\rangle\rangle$
$\displaystyle=$
$\displaystyle\langle\langle\Box^{G}U,X\rangle\rangle=\int_{M}\langle\Box^{M}u(t,x),X(x)\rangle
d\mu(x).$ (4.24)

On the other hand, it holds

$\displaystyle\left\langle\left\langle\nabla^{G}(\partial_{t}W),X\right\rangle\right\rangle$
$\displaystyle=$ $\displaystyle-\int_{M}\langle\partial_{t}u(t,x),X(x)\rangle
d\mu(x),$ (4.25)
$\displaystyle\left\langle\left\langle\nabla^{G}\left(\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}\right),X\right\rangle\right\rangle$
$\displaystyle=$
$\displaystyle\int_{M}\left\langle\nabla^{M}\left(\frac{1}{2}|u(t,x)|^{2}\right),X(x)\right\rangle
d\mu(x),$ (4.26)
$\displaystyle\left\langle\left\langle\nabla^{G}V,X\right\rangle\right\rangle$
$\displaystyle=$
$\displaystyle\int_{M}\left\langle\nabla^{M}v(t,x),X(x)\right\rangle d\mu(x).$
(4.27)

Combining $(\ref{DSW2})$, $(\ref{DSW3})$, $(\ref{DSW4})$ and $(\ref{DSW4v})$
with $(\ref{HJBW})$, we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left\langle\left\langle\nabla^{G}(\partial_{t}W)-\nabla^{G}\left(\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}\right)+\nu\nabla^{G}(\Delta^{G}W)-\nabla^{G}V(t,g),X\right\rangle\right\rangle$
$\displaystyle=$
$\displaystyle\int_{M}\left\langle-\partial_{t}u(t,x)-\frac{1}{2}\nabla^{M}|u(t,x)|^{2}+\nu\Box^{M}u(t,x)-\nabla^{M}v(t,x),X(x)\right\rangle
d\mu(x).$

By the Hodge decomposition theorem, there exists a function $p\in
C^{1}([0,T]\times M)$ such that $u=u(t,x)$ satisfies the backward
incompressible Navier-Stokes equation on M

$\begin{cases}$$\partial_{t}u+\nabla^{TM}_{u}u-\nu\Box^{M}u=-\nabla^{M}p,$$&\mbox{}\\\
$$\rm{div}\hskip 0.85358ptu=0,$$&\mbox{}\end{cases}$

where we use

$\nabla^{M}|u|^{2}=2\nabla^{TM}_{u}u.$

##  5 The incompressible Euler equation

In this section, we consider the particular case where the viscosity vanishes,
i.e., $\nu=0$, and derive the incompressible Euler equation from the
deterministic dynamic programming principle on the group $G={\rm SDiff}(M)$
over a compact Riemannian manifold.

Consider the deterministic control problem on $G={\rm SDiff}(M)$

$\displaystyle
W(t,g):=\inf\limits_{U\in\mathcal{U}}\left[\left.\int_{t}^{T}L(s,g(s),U(s))ds+\Psi(g_{T})\right|g(t)=g\right],$

where $g(s)$ is the solution to the ODE on $G$

$\begin{cases}$${d\over ds}g(s)=U(s,g(s)),$$&\mbox{$s\in(t,T],$}\\\
$$g(t)=g.$$&\mbox{}\end{cases}$ (5.1)

Consider the Lagrangian

$L(s,g(s),U(s))=\frac{1}{2}\|U\|_{T_{g(s)}G}^{2}-V(g(s)).$

By the extended deterministic dynamic programming principle on $G={\rm
SDiff}(M)$, whose proof can be given similarly to the proof of Theorem 3.2, we
have

$\displaystyle\inf_{U\in
T_{g}G}\\{\partial_{t}W+U\cdot\nabla^{G}W+L(t,g,U)\\}=0.$

The optimal Markov control $U(t,g)$ is of the form

$U(t,g)=-\nabla^{G}W(t,g),$

and the value function $W$ satisfies the Hamilton-Jacobi equation

$\partial_{t}W-\frac{1}{2}\|\nabla^{G}W\|^{2}_{T_{g}G}=V(t,g),$ (5.2)

with the terminal value

$W(T,g)=\Psi(g).$

The following result gives a new derivation of the incompressible Euler
equation from the Hamilton-Jacobi equation on $G={\rm SDiff}(M)$.

###### Theorem 5.1

Let $W\in C^{1,2}([0,T]\times G)$ be a solution to the Hamilton-Jacobi
equation $(\ref{HJ-W})$. Let

$\displaystyle U(t,g)=-\nabla^{G}W(t,g)$

be the optimal control and

$\displaystyle u(t,x)=(de_{x})_{e}U(t,e).$

Then there exists a function $p\in C^{1}(M,\mathbb{R})$ such that $(u,p)$
satisfies the incompressible Euler equation on $M$

$\displaystyle\partial_{t}u+\nabla^{TM}_{u}u$ $\displaystyle=$
$\displaystyle-\nabla^{M}p,$ (5.3) $\displaystyle{\rm div}\hskip 0.85358ptu$
$\displaystyle=$ $\displaystyle 0.$ (5.4)

Proof. The proof is similar to the one of Theorem 4.4 by taking $\nu=0$.

##  6 Conclusion and some problems for further work

In conclusion, the main result of this paper can be formulated as follows.

###### Theorem 6.1

Let $M$ be a compact Riemannian manifold. Let $W$ be a $C^{1,3}$-smooth
solution to the Hamilton-Jacobi-Bellman equation $(\ref{HJB-W})$ on $G={\rm
SDiff}(M)$. Then

$u(t,x)=(de_{x})\nabla^{G}W(t,e)$

is a $C^{1,2}$-smooth solution to the backward incompressible Navier-Stokes
equation $(\ref{NSM-u1})$. Conversely, let $u$ be a smooth solution to the
backward incompressible Navier-Stokes equation $(\ref{NSM-u1})$. Let
$\\{g_{t},t\in[0,T]\\}$ be the unique solution to SDE $(\ref{SDEG})$ with
$g(0)=e$ on $G={\rm SDiff}(M)$. Then the value function $W$ for the optimal
stochastic control problem $(\ref{W-Gvaluefunction})$ is a solution to the
Hamilton-Jacobi-Bellman equation $(\ref{HJB-W})$ on $G={\rm SDiff}(M)$.

Proof. This is indeed the restatement of our main results in Section 2 and
Section 4.

To end this paper, we raise some problems for further research work.

###### Remark 6.2

Our main result indicates an interesting relationship between the
incompressible Navier-Stokes equation on a compact Riemannian manifold and the
Hamilton-Jacobi-Bellman equation on the group of volume preserving
diffeomorphisms. By [14, 35], if the initial value $u_{0}\in C^{\infty}(M)$,
then there exists some $T>0$ such that the incompressible Navier-Stokes
equation $(\ref{NSMF-u1})$ has a local smooth solution $u\in
C^{\infty}([0,T]\times M)$. This yields that the vector fields in SDE
$(\ref{SDEG})$ are smooth on $G$ and hence SDE $(\ref{SDEG})$ admits a unique
strong solution on $[0,T]\times G$. It is interesting to ask the question
whether we can prove the regularity of the solution to the Hamilton-Jacobi-
Bellman equation $(\ref{HJB-W})$ on the group of volume preserving
diffeomorphisms.

By the Cole-Hopf transformation

$\Phi(t,g)=\exp{\left[-{W(t,g)\over{2\nu}}\right]}$

and its inverse transform, it is well-known that, $W$ is a solution to the
Hamilton-Jacobi-Bellman equation $(\ref{HJB-W})$ on $G={\rm SDiff}(M)$ if and
only if $\Phi$ is a solution to the heat equation

$\displaystyle\partial_{t}\Phi+\nu\Delta^{G}\Phi+\frac{1}{2\nu}V\Phi=0.$ (6.1)

It remains a challenge problem whether we can prove the existence and
uniqueness of local or global smooth solution of the Hamilton-Jacobi-Bellman
equation $(\ref{HJB-W})$ and the heat equation $(\ref{HeatVG})$ on $G={\rm
SDiff}(M)$. Instead of smooth solution, it is also interesting to study
viscosity solution or solution in suitable Sobolev space in the sense of the
Malliavin calculus [30] over $G={\rm SDiff}(M)$. This will be our future
research project.

Acknowledgement. The authors would like to express their gratitudes to
Professors A. B. Cruzeiro, F. Flandoli and T. Funaki for their interests and
stimulating discussions in the beginning step of this work. We also thank Dr.
S. Li for helpful discussions in the preparation of this work.

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Guoping Liu, School of Mathematics and Statistics, Huazhong University of
Science and Technology, Wuhan, No.1037, Luoyu Road, Wuhan, 430074, China

E-mail<EMAIL_ADDRESS>
Xiang-Dong Li, Academy of Mathematics and Systems Science, Chinese Academy of
Sciences, No. 55, Zhongguancun East Road, Beijing, 100190, China

E-mail<EMAIL_ADDRESS>
and

School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing, 100049, China


# Two Criteria for Quasihomogeneity

Sarasij Maitra<EMAIL_ADDRESS>Department of Mathematics, Indian
Institute of Technology Delhi, Hauz Khas, India.  and  Vivek Mukundan
<EMAIL_ADDRESS>
###### Abstract.

Let $(R,\mathfrak{m}_{R},\mathbb{k})$ be a one-dimensional complete local
reduced $\mathbb{k}$-algebra over a field of characteristic zero. The ring $R$
is said to be quasihomogeneous if there exists a surjection
$\Omega_{R}\twoheadrightarrow\mathfrak{m}$ where $\Omega_{R}$ denotes the
module of differentials. We present two characterizations of quasihomogeniety
of $R$. The first one on the valuation semigroup of $R$ and the other on the
trace ideal of the module $\Omega_{R}$.

###### Key words and phrases:

module of differentials, Berger Conjecture, reduced curves

###### 2010 Mathematics Subject Classification:

Primary: 13A15. Secondary: 13H05

##  1\. Introduction

Let $(R,\mathfrak{m},\mathbb{k})$ be a one dimensional complete local domain
where $\mathbb{k}$ is algebraically closed field of characteristic zero. We
can identify $R$ with $\mathbb{k}\llbracket{X_{1},\dots,X_{n}}\rrbracket/I$
where $I$ is called the defining ideal of $R$. Scheja in [11] defines $R$ to
be quasihomogeneous if there exists a surjection
$\Omega_{R}\twoheadrightarrow\mathfrak{m}$ where $\Omega_{R}$ is the module of
differentials. When $I$ is generated by polynomials, this definition coincides
with the standard definition of quasihomogeneity found in literature: i.e.,
$I$ is generated by quasihomogeneous polynomials ([6, Satz 2.1]). A polynomial
$f$ is said to be quasihomogeneous of degree $d$, if there exists weights
$w_{1},\dots,w_{n}\in\mathbb{N}$ such that
$f(\lambda^{w_{1}}x_{1},\dots,\lambda^{w_{n}}x_{n})=\lambda^{d}f(x_{1},\dots,x_{n})$
for $\lambda\in\mathbb{k}$. Scheja proved Berger’s conjecture ([1]) for
quasihomogeneous rings in [11]. A more general version of this result appeared
recently in [3, Theorem 4.11]. Thus identifying when the ring $R$ is
quasihomogeneous is indeed important and it is desirable to get easy methods
to recognize the quasihomogeneous property for $R$. One of the first results
in this direction is due to Zariski ([14]) who showed that for irreducible
plane algebroid curves, $R$ is quasihomogeneous when the module of
differentials $\Omega_{R}$ has maximal torsion. Saito (in [10]) showed that if
$R$ has an isolated hypersurface singularity ($I=(f)$) then $R$ is
quasihomogeneous when $f$ is in the ideal generated by the partial derivatives
$\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}}$.
The most often used characterization of quasihomogeneous rings defined by
polynomials, is due to Kunz and Ruppert [6, Satz 3.1]. They showed that $R$ is
quasihomogeneous if and only if $R$ is isomorphic to a numerical semigroup
ring $\mathbb{k}\llbracket{H}\rrbracket$ where $H$ is a numerical semigroup.

In this article, we present two new characterizations of quasihomogeneity of
$R$. For the first characterization we make use of the conductor ideal
$\mathfrak{C}_{R}=\overline{R}:_{\operatorname{Quot}(R)}R$ where
$\operatorname{Quot}(R)$ is the quotient field of $R$ and $\overline{R}$ is
the integral closure of $R$ in $\operatorname{Quot}(R)$. Since
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ we can identify $R$ with
$\mathbb{k}\llbracket{t^{a_{1}},\dots,t^{a_{n}}}\rrbracket$. We can define an
order valuation $v(\sum\alpha_{i}t^{i})=\min\\{j~{}|~{}a_{j}\neq 0\\}$. A
subsequent valuation can be defined as
$o(\sum_{i=0}^{\infty}\alpha_{i}t^{i})=v\left(\sum_{i=0}^{\infty}\alpha_{i}t^{i}-\alpha_{0}\right)$.
We use this valuation as a check for quasihomogeneity.

######  Theorem A.

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular, complete, local one
dimensional domain which is a $\mathbb{k}$-algebra with $\mathbb{k}$
algebraically closed of characteristic $0$. Let
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ with the conductor of $R$
given by $\mathfrak{C}_{R}=(t^{c_{R}})\overline{R}$. Writing
$R=\mathbb{k}\llbracket{\alpha_{1}t^{a_{1}},\ldots,\alpha_{n}t^{a_{n}}}\rrbracket$
with $a_{1}<a_{2}<\cdots<a_{n}$, set

$\displaystyle o(\alpha_{r})=\min_{1\leqslant i\leqslant
n}\\{o(\alpha_{i})\\},\qquad a=\min_{j\neq r}\\{a_{j}\\}.$

Then $R$ is quasihomogeneous if $o(\alpha_{r})+a\geqslant c_{R}$.

This provides a quick way to check quasihomogeneity of $R$. Of course, the
downside is that we need to have information on the valuation $c_{R}$ which,
in general can be hard to compute. But this result gives a quick numerical
characterization of quasihomogeneity and easily deployable in computational
packages such as Macaulay2.

For the second characterization, we use the invariant defined in [8]. We use
the description of the trace ideal,
$\text{tr}_{R}(\Omega_{R})=\sum\alpha(\Omega_{R})$ where
$\alpha\in\Omega_{R}^{*}=\operatorname{Hom}_{R}(\Omega_{R},R)$. Since the
valuation function $v(\cdot)$ can be extended to all fractional ideal in
$\overline{R}$, we get another characterization of the quasihomogeneity using
the valuation of the trace ideal $\text{tr}_{R}(\Omega_{R})$. Since $R$ is
quasihomogeneous, we have $v(\mathfrak{m})\geqslant
v(\text{tr}_{R}(\Omega_{R}))$, by definition. In fact since $R$ is non-
regular, this is an equality. However, the converse is unclear (see [8,
Conclusion 3]). We settle this completely in the following result.

######  Theorem B.

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular complete local one
dimensional domain with $\mathbb{k}$ algebraically closed of characteristic
$0$. Let $\Omega_{R}$ be the module of differentials and
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ with valuation function $v$.
Then $R$ is quasihomogeneous if and only if
$v(\text{tr}_{R}(\Omega_{R}))=v(\mathfrak{m}).$

### Acknowledgements

We thank Prof. Craig Huneke for discussions regarding both the main results in
the article. We deeply thank the referee for pointing out various improvements
(especially Theorem 3.1) to the article.

##  2\. Preliminaries

Throughout this article, $(R,\mathfrak{m},\mathbb{k})$ will denote a non-
regular complete one dimensional local domain which is a $\mathbb{k}$-algebra
with $\mathbb{k}$ algebraically closed of characteristic $0$. Hence
$R={P}/{I}$ where
$P=\mathbb{k}\llbracket{X_{1},\cdots,X_{n}}\rrbracket,n\geqslant 2$ and
$I\subseteq\mathfrak{m}_{P}^{2}=(X_{1},\ldots,X_{n})^{2}$, a prime ideal. We
denote $x_{i}$ to be the image of $X_{i}$.

Notice that in this situation, the integral closure $\overline{R}$ of $R$ in
its fraction field, $Q=\operatorname{Frac}(R)$, is a Discrete Valuation Ring
($DVR$) by [12, Theorem 4.3.4]. Thus, henceforth we fix a uniformizing
parameter $t$ and denote $\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$.
Using the inclusion $R\subseteq\overline{R}$, we can write every element of
$R$ as a power series in $t$. Thus,
$R=\llbracket{\alpha_{1}t^{a_{1}},\ldots,\alpha_{n}t^{a_{n}}}\rrbracket$
(i.e., $x_{i}=\alpha_{i}t^{a_{i}}$) where each $\alpha_{i}$ is a unit in
$\mathbb{k}\llbracket{t}\rrbracket$. We also arrange
$a_{1}<a_{2}<\dots<a_{n}$.

For each power series $p(t)=\sum_{i=0}^{\infty}c_{i}t^{i}$, we define the
order valuation $v(p(t)):=\min\\{i\mid c_{i}\neq 0\\}$. When $p(t)=0$, define
$v(0)=\infty$. Thus, $v(x_{i})=a_{i}$ for each $i$. We denote $v(R)$ to be the
valuation semigroup of $R$. Notice that this extends to a discrete valuation
on $Q$ by setting $v(p/q)=v(p)-v(q)$, where $p,q\in R$, $q\neq 0$.

For each fractional ideal $I$, i.e., a finitely generated $R$-submodule of
$Q$, we denote

$v(I):=\min\\{v(y)\mid y\in I\\}.$

Also, for each fractional ideal $I$, we denote $I^{-1}:=R:_{Q}I=\\{y\in Q\mid
yI\subseteq R\\}$.

One of the ideals which is important for our purposes is the conductor
$\mathfrak{C}_{R}$. It is defined to be
$\mathfrak{C}_{R}=R:_{Q}\overline{R}=\\{\alpha\in
Q\mid\alpha\overline{R}\subseteq R\\}$. One can show that it is the largest
common ideal of $R$ and $\overline{R}$. Since
$\displaystyle\overline{R}=k\llbracket{t}\rrbracket$, we have that
$\mathfrak{C}_{R}=(t^{i})_{i\geqslant c_{R}}\overline{R}$. Here $c_{R}$ is the
smallest integer such that $t^{c_{R}-1}\not\in R$, and $t^{c_{R}+i}\in R$ for
all $i\geqslant 0$. It is clear from this discussion that there cannot be any
element $r\in R$, such that $v(r)=c_{R}-1$. Since $\displaystyle\overline{R}$
is finitely generated over $R$ ([12, Theorem 4.3.4]), $\mathfrak{C}_{R}\neq
0$, and it is never equal to $R$ unless $R$ is regular.

###  2.1. Universally Finite Module of Differentials

######  Definition 2.1.

Let $(R,\mathfrak{m},\mathbb{k}),P,I$ be as above. Let $I=(f_{1},\dots,f_{m})$
where $f_{j}\in P=\mathbb{k}\llbracket{X_{1},\dots,X_{n}}\rrbracket,n\geqslant
2$. Then the universally finite module of differentials over $\mathbb{k}$,
denoted by $\Omega_{R}$, has a (minimal) presentation given as follows:

$R^{m}\xrightarrow{\left[\frac{\partial f_{j}}{\partial
x_{i}}\right]}R^{n}\to\Omega_{R}\to 0$

where $\left[\frac{\partial f_{j}}{\partial x_{i}}\right]$ is the Jacobian
matrix of $I$, with entries in $R$.

For further details regarding the module of differentials, we refer the reader
to the excellent book by E. Kunz [5].

###  2.2. Quasihomogeneous Rings

######  Definition 2.2.

([11, Satz 9.8], [1, Definition 3]) Let $(R,\mathfrak{m},\mathbb{k})$ be a
complete local one dimensional non-regular domain which is a
$\mathbb{k}$-algebra where $\mathbb{k}$ is algebraically closed of
characteristic $0$. Let $\Omega_{R}$ denote the module of differentials as in
Definition 2.1. Then $R$ is called quasihomogeneous if there exists an exact
sequence

$\Omega_{R}\to\mathfrak{m}\to 0.$

Notice that the hypotheses on $R$ in Definition 2.2 can be relaxed to quite an
extent and the surjection from $\Omega_{R}$ to $\mathfrak{m}$ is the essential
part (for instance, one may only require $R$ to be reduced with its
$\mathfrak{m}$-adic completion being a domain). However, for the purposes of
this article, we stick to the above definition.

######  Example 2.3.

Let
$R=\mathbb{C}\llbracket{X,Y,Z}\rrbracket/(XZ-Y^{2},X^{3}-Z^{2})\cong\mathbb{C}\llbracket{t^{4},t^{5},t^{6}}\rrbracket$
is quasihomogeneous. First notice that $\Omega_{R}=RdX\oplus RdY\oplus RdZ/U$
where $U$ is the submodule spanned by $zdX-2ydY+xdZ,3x^{2}dX-2zdZ$. The map

$dX\to 4x,dY\to 5y,dZ\to 6z$

defines an $R$-linear surjection to the maximal ideal $(x,y,z)$ of $R$.

######  Remark 2.4.

For rings in the above situation, the map $R$ to $R$ which is given by
$x\mapsto\text{deg}(x)x$ for any homogeneous element $x$ (known as the Euler
derivation map) lifts to a surjection from $\Omega_{R}$ to $\mathfrak{m}$
(known as the Euler homomorphism) (see the discussion following [2, Theorem
2.3] and also the proof of [2, Theorem 3.7]).

Rings of the form $\mathbb{k}\llbracket{t^{b_{1}},\dots,t^{b_{n}}}\rrbracket$
are all quasihomogeneous. To see this, first fix a presentation
$\mathbb{k}\llbracket{t^{b_{1}},\dots,t^{b_{n}}}\rrbracket\cong\mathbb{k}\llbracket{X_{1},\dots,X_{n}}\rrbracket/J$.
Now we can define the Euler homomorphism $\Omega_{R}\rightarrow R$ as
$dX_{i}\rightarrow b_{i}x_{i}$ which leads to the surjection
$\Omega_{R}\twoheadrightarrow\mathfrak{m}$.

##  3\. Quasihomogeneity via order valuation on units

In this section, we prove our first main result. we recall the following
notion of order valuation on the units which was utilised to solve partial
cases of a long standing conjecture, in a recent work, see [9, Notation 4.1,
Theorem 4.2].

###  3.1. Order Valuation of Units

The units in $\overline{R}$ are of the form
$\alpha=\sum_{i=0}^{\infty}c_{i}t^{i}$ with $c_{0}\neq 0,c_{i}\in\mathbb{k}$.
Let $\alpha_{j}=\sum_{i=0}^{\infty}c_{ji}t^{i}$ be such a unit. Then the order
valuation of $\alpha_{j}$ is given by

$\displaystyle o(\alpha_{j})=v(\alpha_{j}-c_{j0})=\min_{i\geqslant
1}\\{i~{}|~{}c_{ji}\neq 0\\}.$

######  Theorem 3.1.

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular, complete, local one
dimensional domain which is a $\mathbb{k}$-algebra with $\mathbb{k}$
algebraically closed of characteristic $0$. Let
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ with the conductor of $R$
given by $\mathfrak{C}_{R}=(t^{c_{R}})\overline{R}$. Writing
$R=\mathbb{k}\llbracket{\alpha_{1}t^{a_{1}},\ldots,\alpha_{n}t^{a_{n}}}\rrbracket$
with $a_{1}<a_{2}<\cdots<a_{n}$, set

$\displaystyle o(\alpha_{r})=\min_{1\leqslant i\leqslant
n}\\{o(\alpha_{i})\\},\qquad a=\min_{j\neq r}\\{a_{j}\\}.$

If $o(\alpha_{r})+a\geqslant c_{R}$, then $R$ is quasihomogeneous.

###### Proof.

As before, we write $x_{i}=\alpha_{i}t^{a_{i}}$. Now assume that $r\geqslant
2$. Then $a=a_{1}$. Choose any $1\leqslant j\leqslant n$. Then
$o(\alpha_{j})+a_{j}\geqslant o(\alpha_{r})+a\geqslant c_{R}$. Now

$x_{j}=\alpha_{j}t^{a_{j}}=\alpha_{j0}t^{a_{j}}+t^{o(\alpha_{j})+a_{j}}h$

where $h\in\overline{R}$. Since $o(\alpha_{j})+a_{j}\geqslant c_{R}$,
$t^{o(\alpha_{j})+a_{j}}\in\mathfrak{C}_{R}\subseteq R$. Thus,
$t^{a_{j}}=\frac{1}{\alpha_{j0}}(x_{j}-t^{o(\alpha_{j})+a_{j}}h)\in R$. Thus,
$R=\mathbb{k}\llbracket{t^{a_{1}},\ldots,t^{a_{n}}}\rrbracket$ and the
quasihomogeneity of $R$ follows from Remark 2.4.

Finally suppose that $r=1$. Then $a=a_{2}$. Thus, we get that
$o(\alpha_{j})+a_{j}\geqslant o(\alpha_{1})+a_{2}\geqslant c_{R}$ for all
$j\geqslant 2$. Also, we may assume that $\alpha_{1}=1+\sum_{k\geqslant
1}\alpha_{1k}t^{k}$ where $\alpha_{1k}\in\mathbb{k}$. Now let
$\beta=\alpha_{1}^{1/a_{1}}=1+\sum_{k\geqslant
1}\beta_{k}t^{o(\alpha_{1})+k-1},\beta_{k}\in\mathbb{k}$, a unit in
$\overline{R}$ (such a $\beta$ exists due to Hensel’s Lemma; see for example
[9, Section 2]). We change the parameter to

$\displaystyle s=\beta
t=t+\beta_{1}t^{o(\alpha_{1})+1}+\beta_{2}t^{o(\alpha_{1})+2}+\cdots$

We now identify $R$ inside $\mathbb{k}\llbracket{s}\rrbracket$. Notice that
$s^{c_{R}-1}\not\in R$ but $s^{c_{R}+i}\in R$ for all $i\geqslant 0$ using the
relation $s=\beta t$. In other words, the conductor valuation stays the same.
We denote the ring $R$ as $\mathbb{k}\llbracket{x_{1},\ldots,x_{n}}\rrbracket$
where each $x_{i}$ has a representation both in terms of $s$ and $t$,
depending on the choice of the parameter. Notice that both $s^{c_{R}+i}$ and
$t^{c_{R}+i}$ can be written purely in terms of $x_{1},\ldots,x_{n}$ and
consequently $t^{c_{R}+i}$ can be written purely in terms of $s$ (with
valuation equal to $c_{R}+i$).

Our goal is to write $t=\gamma s$ for some
$\gamma\in\mathbb{k}\llbracket{s}\rrbracket$ which establishes that
$\mathbb{k}\llbracket{t}\rrbracket=\mathbb{k}\llbracket{s}\rrbracket$. Notice
that

$\displaystyle s$ $\displaystyle=\beta
t=t+\beta_{1}t^{o(\alpha_{1})+1}+\beta_{2}t^{o(\alpha_{1})+2}+\cdots$
$\displaystyle\beta_{1}s^{o(\alpha_{1})+1}$
$\displaystyle=\beta_{1}t^{o(\alpha_{1})+1}+\beta_{1}((o(\alpha_{1})+1)\beta_{1})t^{2o(\alpha_{1})+1}+\cdots.$

Thus

(3.1) $\displaystyle s-\beta_{1}s^{o(\alpha_{1})+1}$
$\displaystyle=(t+\beta_{2}t^{o(\alpha_{1})+2}+\cdots)-(\beta_{1}((o(\alpha_{1})+1)\beta_{1})t^{2o(\alpha_{1})+1}+\cdots)$
$\displaystyle=\beta^{\prime}t\text{ for
}\beta^{\prime}\in\mathbb{k}\llbracket{t}\rrbracket.$

Notice that $o(\beta^{\prime})$ is at least $o(\alpha_{1})+1$ or
$2o(\alpha_{1})$. In either case, we see that $o(\beta^{\prime})\geqslant
o(\alpha_{1})+1$. We keep repeating this process of subtracting terms
involving higher powers of $t$ in the expression (3.1) to arrive at

$\displaystyle s-f(s)=t+\sum_{k\geqslant 0}\eta_{k}t^{c_{R}+k}\text{ where
}f(s)\in(s)\mathbb{k}\llbracket{s}\rrbracket,\eta_{k}\in\mathbb{k}.$

Now as explained in the paragraphs above,
$t^{c_{R}+k}=g_{k}(s)\in(s)\mathbb{k}\llbracket{s}\rrbracket$ (i.e., they can
be written purely in terms of $s$). Thus we arrive at

$\displaystyle s-f(s)-\sum_{k\geqslant 0}\eta_{k}g_{k}(s)=t.$

Thus $t=\gamma s,\gamma\in\mathbb{k}\llbracket{s}\rrbracket$. Now notice that
$o(\gamma)=o(\alpha_{1})=o(\beta)$.

Now $s^{a_{1}}=(\beta t)^{a_{1}}=\alpha_{1}t^{a_{1}}=x_{1}$. Also

$\displaystyle x_{i}=\alpha_{i}t^{a_{i}}$
$\displaystyle=\alpha_{i0}t^{a_{i}}+\alpha_{i1}t^{o(\alpha_{i})+a_{i}}+\alpha_{i2}t^{o(\alpha_{i})+a_{i}+1}+\cdots$
$\displaystyle=\alpha_{i0}(\gamma s)^{a_{i}}+\alpha_{i1}(\gamma
s)^{o(\alpha_{i})+a_{i}}+\alpha_{i2}(\gamma s)^{o(\alpha_{i})+a_{i}+1}+\cdots$
$\displaystyle=\alpha^{\prime}_{i0}s^{a_{i}}+\alpha^{\prime}_{i1}s^{o(\alpha_{1})+a_{i}}+\cdots$
$\displaystyle=\alpha_{i}^{\prime}s^{a_{i}}.$

Notice that for all $i\geqslant 2$, $o(\alpha_{1})+a_{i}\geqslant
o(\alpha_{1})+a_{2}\geqslant c_{R}$. Thus following the same method as in the
beginning of this proof, we can rewrite $x_{i}=s^{a_{i}}$. Since this can be
performed for all $x_{i}$, we have
$R=\mathbb{k}\llbracket{s^{a_{1}},\dots,s^{a_{n}}}\rrbracket$. Thus
quasihomogeneity of $R$ now follows from Remark 2.4. ∎

######  Example 3.2.

Let $R=\mathbb{k}\llbracket{t^{4}+t^{5},t^{7},t^{8},t^{9}}\rrbracket$. Then
the conductor $\mathfrak{C}_{R}=(t^{7})\overline{R}$. Here $o(\alpha_{1})=1$
and $a=7$. Since $o(\alpha_{1})+a\geqslant c=7$, $R$ is a quasihomogeneous
ring. Here $R=\mathbb{k}\llbracket{s^{4},s^{7},s^{8},s^{9}}\rrbracket$.

######  Example 3.3.

Let $R=\mathbb{k}\llbracket{t^{5},t^{6},t^{8}+t^{9}}\rrbracket$. Then [13,
Page 207,(1)] shows that $R$ is not quasihomogeneous. Notice that
$o(\alpha_{3})+a_{1}=1+5=6<10=c_{R}$.

##  4\. Quasihomogeneity via trace ideal of $\Omega_{R}$

We recall the following notions and results which will be crucial in
establishing our second main result.

######  Definition 4.1.

Let $R$ be a local domain with fraction field $Q$. Then the rank of any module
$M$ is defined to be $\operatorname{rank}(M):=\dim_{Q}(M\otimes_{R}Q)$.

######  Definition 4.2.

Let $M$ be a finitely generated $R$-module. Then the trace ideal of $M$,
denoted $\text{tr}_{R}(M)$, is the ideal $\sum\alpha(M)$ where $\alpha$ ranges
over $M^{*}:=\operatorname{Hom}_{R}(M,R)$. An ideal $I$ is said to be a trace
ideal if $I=\text{tr}_{R}(M)$ for some module $M$.

Clearly, $I\subseteq\text{tr}_{R}(I)$. For further details on trace ideals and
their properties, we refer the reader to resources such as [4, Proposition
2.4] and [7]. we shall need the following properties in this work.

######  Lemma 4.3.

Let $(R,\mathfrak{m},\mathbb{k})$ be a one dimensional non-regular complete
local domain which is a $\mathbb{k}$-algebra, where $\mathbb{k}$ is
algebraically closed of characteristic $0$. Let
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ with $v$ the order valuation.
Let $I$ be a fractional ideal of $R$, i.e., an $R$-submodule of $Q$. Then the
following statements hold.

  1. (1)

For any $R$-module $N$, we have that
$\text{tr}_{R}(\text{tr}_{R}(N))=\text{tr}_{R}(N)$.

  2. (2)

Suppose $J$ is another fractional ideal that is isomorphic to $I$. Then
$\text{tr}_{R}(I)=\text{tr}_{R}(J)$

  3. (3)

$\text{tr}_{R}(I)=II^{-1}$.

  4. (4)

$v(\text{tr}_{R}(I))=v(I)+v(I^{-1})$.

  5. (5)

Suppose $M$ is a finitely generated $R$-module of rank one. Suppose $f:M\to R$
is any non-zero map and let $I=f(M)$. Then
$\text{tr}_{R}(M)=\text{tr}_{R}(I)$.

###### Proof.

Suppose $\phi:I\to J$ be the isomorphism. Then by composing any map
$\alpha:J\to R$ with $\phi$ we get a map from $I\to R$. Thus,
$\text{tr}_{R}(J)\subseteq\text{tr}_{R}(I)$. Now a symmetric argument finishes
the proof of $(2)$. Statement $(1)$ also follows a similar reasoning.

Statement $(3)$ appears as [4, Proposition 2.4(2)] whereas statement $(5)$ is
[8, Proposition 3.5].

Statement $(4)$ is immediate from $(3)$. ∎

###  4.1. Fractional Ideal of Derivatives

Let $R=\mathbb{k}\llbracket{x_{1},\ldots,x_{n}}\rrbracket,n\geqslant 2$ where
each $x_{i}=\alpha_{i}t^{a_{i}}$ where $a_{1}<a_{2}<\ldots<a_{n}$ as in our
original setup. Let $\Omega_{R}$ be the module of differentials. From [3,
Section 2.3] and ignoring $dt$, we see that $\Omega_{R}$ surjects to the
$R$-span of $x_{i}^{\prime}(t):=\frac{dx_{i}}{dt}$. This is clearly a
fractional ideal and we call it the fractional ideal of derivatives.
Henceforth we shall use the following notation –

$\displaystyle\mathcal{D}:=Rx_{1}^{\prime}(t)+Rx_{2}^{\prime}(t)+\cdots+Rx_{n}^{\prime}(t).$

Notice that some power series of high enough valuation multiplies
$\mathcal{D}$ into $R$ and hence we get an ideal of $R$ to which $\Omega_{R}$
surjects. This ideal of course is isomorphic to $\mathcal{D}$ as the map is
simply by multiplication by a non-zero element of $Q$. For instance,
$t^{c_{R}}\mathcal{D}$ is such an ideal, where $c_{R}=v(\mathfrak{C}_{R})$.

###  4.2. The $\operatorname{h}(\cdot)$ Invariant

######  Definition 4.4.

[8, Definition 2.1] Let $R$ be a one dimensional local domain and $M$ be a
finitely generated $R$-module. Then

$\operatorname{h}(M)=\min\\{\ell(R/I)\mid\text{there exists a
homomorphism}~{}\phi:M\to R~{}\text{with~{}}\phi(M)=I\\}$

where $\ell(\cdot)$ denotes length.

Notice that $\operatorname{h}(\cdot)$ is a non-negative integer or takes the
value $\infty$.

######  Lemma 4.5.

[8, Theorem 5.2] Let $R,\Omega_{R},\mathcal{D}$ be as discussed above. Then

$\operatorname{h}(\Omega_{R})=\ell(\overline{R}/\mathcal{D})-\ell(\overline{R}/R)+v(\mathcal{D}^{-1})\leqslant
v(\mathcal{D}^{-1}).$

######  Theorem 4.6.

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular complete local one
dimensional domain with $\mathbb{k}$ algebraically closed of characteristic
$0$. Let $\Omega_{R}$ be the module of differentials and
$\overline{R}=\mathbb{k}\llbracket{t}\rrbracket$ with valuation function $v$.
Then $R$ is quasihomogeneous if and only if

$v(\text{tr}_{R}(\Omega_{R}))=v(\mathfrak{m}).$

###### Proof.

Throughout the proof, we assume that
$R=\mathbb{k}\llbracket{x_{1},\ldots,x_{n}}\rrbracket,n\geqslant 2$ with
$x_{i}=\alpha_{i}t^{a_{i}}$, $a_{1}<a_{2}<\cdots<a_{n}$.

Assume that $R$ is quasihomogeneous. So there exists a surjection from
$\Omega_{R}$ to $\mathfrak{m}$. Since $\operatorname{rank}(\Omega_{R})=1$, we
have that $\text{tr}_{R}(\Omega_{R})=\text{tr}_{R}(\mathfrak{m})$ by Lemma
4.3$(5)$. Now $\mathfrak{m}\subseteq\text{tr}_{R}(\mathfrak{m})$ and if the
latter is equal to $R$, then $\mathfrak{m}$ has a free direct summand by [7,
Proposition 2.8(iii)], contradicting the non-regularity assumption of $R$.
Thus $\text{tr}_{R}(\mathfrak{m})=\mathfrak{m}$ and hence
$v(\text{tr}_{R}(\Omega_{R}))=v(\mathfrak{m})$.

Conversely, suppose that
$v(\text{tr}_{R}(\Omega_{R}))=v(\mathfrak{m})=v(x_{1})$. In order to prove
that $R$ is quasihomogeneous, it is enough to show that
$\displaystyle\operatorname{h}(\Omega_{R})=1$: for suppose
$1=\displaystyle\operatorname{h}(\Omega_{R})=\ell(R/I)$ for some ideal $I$
which is a surjective image of $\Omega_{R}$. Now
$I\subseteq\mathfrak{m}\subseteq R$ and hence
$1=\ell(R/I)\geqslant\ell(R/\mathfrak{m})=1$ which implies that
$I=\mathfrak{m}$.

By Lemma 4.3, we get that
$\text{tr}_{R}(\Omega_{R})=\text{tr}_{R}(\mathcal{D})=\mathcal{D}\mathcal{D}^{-1}$
and hence $v(\text{tr}_{R}(\Omega_{R}))=v(\mathcal{D})+v(\mathcal{D}^{-1})$.
Since $\mathbb{k}$ is of characteristic $0$, we get that
$v(\mathcal{D})=v(x_{1}^{\prime}(t))=v(x_{1})-1=v(\mathfrak{m})-1$. Using this
and the hypothesis, we get that
$v(\mathcal{D}^{-1})=v(\text{tr}_{R}(\Omega_{R}))-v(\mathcal{D})=v(\mathfrak{m})-v(\mathfrak{m})+1=1$.

Thus, using Lemma 4.5, we get that

$\displaystyle\operatorname{h}(\Omega_{R})=\ell(\overline{R}/\mathcal{D})-\ell(\overline{R}/R)+1\leqslant
1.$

If $\displaystyle\operatorname{h}(\Omega_{R})=0$, then $R$ is regular (for
instance, see [8, Remark 4.2]), a contradiction. Thus
$\displaystyle\operatorname{h}(\Omega_{R})=1$, thereby finishing the proof. ∎

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