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Vol. 3 (1998) Paper no. 4, pages 1–36.
Journal URL
http://www.math.washington.edu/˜ejpecp/
Paper URL
http://www.math.washington.edu/˜ejpecp/EjpVol3/paper4.abs.html
GEOMETRIC EVOLUTION UNDER
ISOTROPIC STOCHASTIC FLOW
M. Cranston and Y. LeJan
University of Rochester
Rochester, New York, USA
cran@math.rochester.edu
Université de Paris, Sud
Orsay 91405, France
Yves.LeJan@math.u-psud.fr
Abstract. Consider an embedded hypersurface M in R3 . For Φt a stochastic flow
of differomorphisms on R3 and x ∈ M, set xt = Φt (x) and Mt = Φt (M). In this
paper we will assume Φt is an isotropic (to be defined below) measure preserving
flow and give an explicit descripton by SDE’s of the evolution of the Gauss and mean
curvatures, of Mt at xt . If λ1 (t) and λ2 (t) are the principal curvatures of Mt at xt
then the vector of mean curvature and Gauss curvature, (λ1 (t) + λ2 (t), λ1 (t)λ2 (t)),
is a recurrent diffusion. Neither curvature by itself is a diffusion. In a separate
addendum we treat the case of M an embedded codimension one submanifold of
Rn . In this case, there are n−1 principal curvatures λ1 (t), . . . , λn−1 (t). If Pk , k = 1,
n − 1 are the elementary symmetric polynomials in λ1 , . . . , λn−1 , then the vector
(P1 (λ1 (t), . . . , λn−1 (t)), . . . , Pn−1 (λ1 (t), . . . , λn−1 (t)) is a diffusion and we compute
the generator explicitly. Again no projection of this diffusion onto lower dimensions
is a diffusion. Our geometric study of isotropic stochastic flows is a natural offshoot
of earlier works by Baxendale and Harris (1986), LeJan (1985, 1991) and Harris
(1981).
1991 Mathematics Subject Classification. 60H10, 60J60.
Key words and phrases. Stochastic flows, Lyapunov exponents, principal curvatures.
Research supported by NSA and Army Office of Research through MSI at Cornell. Submitted
to EJP on February 19, 1996. Final version accepted on February 12, 1998.
Typeset by AMS-TEX
1
2
M. CRANSTON AND Y. LEJAN
§1 Isotropic Flows and Geometric Setting.
We begin with a nonnegative measure F (dρ) on [0, +∞) with finite moments of
all orders. The pth moment of F will be denoted by µp . To F we can associate a
covariance
Z ∞Z
(1.1)
Cij (z) =
eiρhz,ti (δji − ti tj )σn−1 (dt)F (dρ),
S n−1
0
where σn−1 is normalized Lebesgue measure on S n−1 . By a result of Kolmogorov,
to Cij is associated a smooth (in x) vector-field valued Brownian motion Ut (x) such
that
(1.2)
EUti (x)Usj (y) = (t ∧ s)C ij (x − y),
1 ≤ i, j ≤ n.
A Brownian flow on Rn is constructed by solving the equation
Z
(1.3)
t
Φt (x) = x +
∂Us (Φs (x))
0
(∂ denotes Stratonovich differential.) Existence and uniqueness of such flows was
proved in Baxendale (1984), LeJan and Watanabe (1984). Then if Ta is translation
by a ∈ Rn , Ta Φt T−a and Φt are identical in law so Φ is stationary. If R is unitary,
RΦt R−1 and Φt are identical in law. When these two properties hold we say Φ is
isotropic. These properties both follow from the nature of Cij so our Φ is isotropic.
The field Ut (x) is smooth so we differentiate it writing
Wji =
(1.4)
i
Bjk
=
(1.5)
∂ i
U
∂xj
∂2
U i.
∂xj ∂xk
Then we have as direct consequence of (1.1) and (1.2) (using h, i to denote both
Euclidean inner product as well as quadratic variation for martingales, d will denote
the Itô differential.)
(1.6)
hdWji (t, y), dW`k (t, y)i =
(1.7)
µ2
[(n + 1)δki δ`j − δji δ`k − δ`i δjk ]dt
n(n + 2)
i
hdBjk
(t, y), dWqp (t, y)i = 0
and for vectors u, v ∈ Rn ,
(1.8)
hhdB(u, u), vi, hdB(u, u), vii =
3µ4
[(n + 3)||u||4||v||2 − 4hu, vi2 ||u||2 ]dt
n(n + 2)(n + 4)
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
3
By isotropy, (1.6), (1.7) and (1.8) remain valid under any unitary change of coordinates. We define,
ik
Cj`
dt = hdWji , dW`k i =
µ2
[(n + 1)δki δ`j − δji δ`k − δ`i δjk ]dt
n(n + 2)
where both dWji , dW`k are evaluated at (t, y), and
ij
Ck`pq
dt = hdB i (ek , e` ), dB j (ep , eq )i
again both dB terms are evaluated at (t, y).
A quick calculation shows
Pn
i 2
E
= 0 so div U = 0 and therefore Φ is a measure-preserving flow.
i=1 dWi
(n+3)µ4
nn
nn
nn
Also, it’s rather easy to check that C1111
= 3C1122
and C1122
= n(n+2)(n+4)
.
ik
dt for isotropic flows is
The most general form of hdWji , (t, y), dW`k (t, y)i = Cj`
ik
i,k
i k
i k
Cj` = aδ δj,` + bδj δ` + cδ`δj where a + c, a − c, a + c + db are nonnegative (see
LeJan (1985) and the references therein). We retrict our attention to the present
µ2
µ2
µ2
covariance structure; a = (n+1) n(n+2)
, b = − n(n+2)
, c = − n(n+2)
, as computations
are massively simplified by the fact that certain Ito correction terms vanish. The
works of Baxendale and Harris (1986) and LeJan (1985) focused on first order
properties of the general isotropic flows in Euclidean space. That is, on properties
of the derivative flow DΦt (x). Perhaps the most fundamental information about
the derivative flow is its Lyapunov spectrum. To describe the Lyapunov spectrum,
first extend DΦt to p-forms α = v1 ∧ · · · ∧ vp (we’re in Euclidean space so we can
identify forms and vectors) by defining DΦt (x)α = DΦt (x)v1 ∧ · · · ∧ DΦt (x)vp .
Then if v1 , . . . , vp are linearly independent, a.s.
1
log kDΦt (x)αk = γ1 + · · · + γp
t→∞ t
lim
is the sum of the first p Lyapunov exponents. In the case of the isotropic, measure2
preserving flow on Rn , γ1 + · · · + γp = np(n−p)µ
(see LeJan 1985). The problem
2(n+2)
dealt with in this work involves second order behavior of the flow. The evolving
curvature of a submanifold moving under the flow would depend on the properties
of the second order flow,
(x, u, v) → (Φt (x), DΦt (x)u, D2 Φt (x)(DΦt (x)u, DΦt (x)u) + DΦt (x)v) .
In particular, our results rely on the properties given by (1.6), (1.7) and (1.8).
Initially, we will consider M an embedded hypersurface in Rn , later specializing
to the case n = 3. Take Πt : Txt Rn → Txt Mt to be the orthogonal projection
and ∇ to be the canonical connection on Euclidean space. Then St , the second
fundamental form of Mt at xt , is defined for u, v ∈ Txt Mt by St (u, v) = (I −Πt )∇u v
where v is extended in an arbitrary but smooth manner in a neighborhood of xt .
Now St induces a linear map from the exterior algebra Λk (Txt Mt ) → Λk (Txt Mt )
4
M. CRANSTON AND Y. LEJAN
for k = 1, 2, . . . , n − 1. This arises as follows. First observe that since we are
in Euclidean space we can identify vectors and forms. Next take νt ∈ Tx⊥t Mt to
be a unit vector (there are two choices and we may select the process νt to be
continuous in t.) Now in the case k = 1, u → hSt (u, ·), νt i is a map from Λ1 (Txt Mt )
to Λ1 (Txt Mt ). When k > 1 we need to introduce the index set
*
Ik = {m ∈ {1, . . . , n − 1}k : m1 < m2 < · · · < mk }
and for σ ∈ Sk , the set of permutations on k letters, define (−1)σ to be the sign of
σ. Then for u1, . . . , uk , v1 , . . . , vk ∈ Txt Mt , set
S (k)(u1 ∧ · · · ∧ uk , v1 ∧ · · · ∧ vk ) =
X
(−1)σ
k
Y
hSt (uσ(j) , vj ), νt i
j=1
σ∈Sk
This gives rise to the linear map
u1 ∧ · · · ∧ uk → S (k)(u1 ∧ · · · ∧ uk , ·)
from Λk (Txt Mt ) → Λk (Txt Mt ).
The object of our study will be the trace of S (k), T rS (k) , for k = 1, 2, . . . , n − 1.
This is obtained via contraction on indices (see Bishop and Goldberg (1980)). Take
{u1 , . . . , un−1} to be any basis for Txt Mt . For k ∈ {1, . . . , n − 1}, ~` ∈ Ik , set
*
| ` | = `1 + · · · + `k ,
α* = u`1 ∧ u`2 ∧ · · · ∧ u`k
`
*
*
α ` = (−1)| ` |+k u1 ∧ · · · ∧ û`1 ∧ · · · ∧ û`k ∧ · · · ∧ un−1
( ˆ indicates the indicated vector is omitted from the wedge product)
α = u1 ∧ · · · ∧ un−1 .
Define an inner product
* i = dethu` , umj i.
hα* , αm
i
`
Then we may express T rS (k) as
*
(1.9)
*
`
m
* )hα , α
T rS (k) = S (k) (α* , αm
i.||α||−2 ,
`
* *
where the sum (Einstein’s convention) is over ` , m ∈ Ik . Now this is invariant under
change of basis so we may select {u1 , . . . , un−1 } to be the unit principal directions
of curvature (see Spivak (1975).) Namely, the eigenvectors of S (1) : Λ1 (Txt Mt ) →
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
5
Λ1 (Txt Mt ). Let λ1 , . . . , λn−1 be the corresponding eigenvalues, otherwise known
as the principal curvatures. Now the ui are orthonormal so
*
* *
*
hα ` , αm i||α||−2 = δ ` ,m
and as hSt (uj , v), νt i = λj huj , vi
X
σ∈Sk
(−1)σ
k
Y
hSt (u`σ(j) , u`j ), νt i = λ`1 . . . λ`k ≡ λ*
Thus
(1.10)
`
j=1
T rS (k) =
X
*
λ* ≡ Pk (λ1 , . . . , λn−1 ).
`
` ∈Ik
Pk is the elementary symmetric polynomial of degree k.
The process ξt = (xt , Πt , νt , P1 (λ1 , . . . , λn−1 ), · · · Pn−1 (λ1 , . . . , λn−1 )) will, in
our case be a diffusion. By the translation invariance part of isotropy, it is clear
that xt is not needed, i.e., (Πt , νt , P1 (λ1 , . . . , λn−1 ), · · · , Pn−1 (λ1 , . . . , λn−1 )) has
the Markov property. By the invariance of the law of the flow under unitary actions,
it follows that Πt and νt are also superfluous. Thus, for isotropic flows the vector
(P1 (λ1 , . . . , λn−1 ), . . . , Pn−1 (λ1 , . . . , λn−1 )) is a diffusion. The point of the present
article is to describe this diffusion explicitly in the measure-preserving isotropic
case. We can derive consequences about the behavior of this diffusion from the
explicit description. These properties suggest what sort of behavior one might have
with more general flows. What we expect for more general flows is that the process
of curvatures for a k-dimensional submanifold evolving under a general flow will be
recurrent (though not necessarily a diffusion) if γk > 0. That is, generically the k
dimensional tangent space Txt Mt will ‘see’ only stretching directions and this will
tend to smooth out the curvatures. This condition may even be too strong.
§2 Preliminary Results.
Recall the correlations of dW given by (1.6). The flow Φt : Rn → Rn is a C ∞ diffeomorphism whose first derivative satisfies the Stratonovich stochastic differential
equation, where x ∈ M,
(2.1)
dDΦt (x) = ∂W (xt )DΦt (x)
Given u ∈ Tx M, DΦt (x)u = u(t) ∈ Txt Mt satisfies
(2.2)
du = ∂W u
where we have suppressed the variable (xt ) in ∂W . In fact, due to the choice of C
in (1.1)
(2.3)
du = dW u,
6
M. CRANSTON AND Y. LEJAN
that is, the Itô corrections vanish. Furthermore, if we select a basis {u1 , . . . , un−1}
for Tx M, then {u1(t), . . . , un−1 (t)}, where uj (t) = DΦt (x)uj , forms a basis for
Txt Mt . We will write uj for uj (t) for simplicity of notation.
We now proceed to derive a Stratonovich equation for the second fundamental
form St of Mt at xt . Given vectors u, v ∈ Tx M move them via the flow, u(t) =
DΦt (x)u, v(t) = DΦt (x)v. Extend v to a smooth vector field V in a neighborhood
−1
of x. Set Vt (y) = DΦt (Φ−1
t (y))V (Φt (y)) so that Vt is a vector field near xt such
that Vt (xt ) = v(t). Then if Zt ≡ ∇u(t)Vt (xt ) we claim
(2.4)
dZ(u, v) = ∂W Z(u, v) + ∂B(u, v).
For proof of (2.4), take γ to be a curve on M with γ(0) = x, γ 0 (0) = u and define
γt = Φt (γ) so that γt0 (0) = u(t) = DΦt u. Then,
∇ut Vt = lim s−1 (Vt (γt (s)) − Vt (γt (0))
s→0
= lim s−1 (DΦt (γ(s))V (γ(s)) − DΦt (x)v(x))
s→0
= lim s−1 [(DΦt (γ(s)) − DΦt (x))V (γ(s))
s→0
+ DΦt (x)(V (γ(s)) − v(x))]
= D2 Φt (x)(u, v) + DΦt (x)∇u v
Since
Z
t
Φt (x) = x +
∂U(Φs (x))
Z
0
t
DΦt (x) = I +
∂W (Φs (x))DΦs (x)
0
Z t
2
D Φt (x)(·, ·) =
∂B(Φs (x))(DΦs (x)·, DΦs (x)·)
0
Z t
∂W (Φs (x))D2 Φs (x)(·, ·)
+
0
(2.4) follows. A word about notation: ∂Wt Z is an abbreviation for ∂Wji (Φt (x))(Ztj )
i
and ∂B(u, v) is short for (∂Bjk
(Φt (x))uj (t)vk (t).
Next we need an equation for Πt : Txt Rn → Txt Mt . For any ξ ∈ Txt Mt , let
{u1 , . . . , un−1} be the moving frame mentioned above, set α = u1 ∧ · · · ∧ un−1 and
defining
(2.5)
αi = (−1)i+1 u1 ∧ · · · ∧ ûi ∧ · · · ∧ un−1
we have
(2.6)
Πt ξ =
n−1
X
hξ ∧ αi , αi||α||−2 ui .
i=1
The next two lemmas are elementary but crucial.
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
7
Lemma 2.1. If ξt is a Txt Mt -valued continuous semimartingale, then
(I − Πt ) · ∂W ξt = ∂Πt ξt .
Proof. Suppose Πt ξt = ξt is as in the statement of the lemma with ξt = xit ui . Then
(I − Πt )∂W ξt − ∂Πt ξt = ∂W ξt − Πt ∂W ξt
h∂αt , αt i
ξt − ∂W ξt
||αt ||2
hξt ∧ αit , ∂αt i
hξt ∧ ∂αit , αt i
−
u
−
ui
i
||αt ||2
||αt ||2
h∂αt , αt i
= −Πt ∂W ξt + 2
ξt
||αt ||2
hαt , ∂αt i
i hui ∧ ∂αi , αt i
− xit
u
−
x
ui
i
t
||αt ||2
||αt ||2
+2
Since
dαt = −ui ∧ ∂αi + ∂W ui ∧ αi
the last term becomes
hui ∧ ∂αi , αi
ui
||αt ||2
h∂αt , αt i
h∂W ui ∧ αit , αt i i
=−
ξt +
xt ui
||αt ||2
||αt ||2
h∂αt , αt i
=−
ξt + Πt ∂W ξt
||αt ||2
− xit
From this we get the conclusion of the lemma.
Lemma 2.2. If ηt ∈ Txt Mt⊥ is a continuous semimartingale, then
∂Πt ηt = Πt ∂ W̃ ηt
where W̃ is the transpose of W .
Proof. Recalling that
h∂αt , αt i
Πt v + ∂W Πt v
||αt ||2
hv ∧ αit , ∂αt i
hv ∧ ∂αit , αt i
+
ui +
ui
||αt ||2
||αt ||2
∂Πt (v) = −2
8
M. CRANSTON AND Y. LEJAN
we see that since Πt ηt = 0,
∂Π(ηt ) =
hηt ∧ αit , ∂αt i
ui .
||αt ||2
But
∂αt = ∂W uk ∧ αkt
so
∂Πt (ηt ) =
hηt ∧ αit , ∂W uk ∧ αkt i
ui
||αt ||2
=
h∂ W̃ ηt , uk (t)ihαkt (t), αkt (t)iui (t)
||α(t)||2
=
h∂ W̃ ηt ∧ αit , αt i i
ut
||αt ||2
= Πt ∂ W̃ ηt
and the lemma is proved. Using S = (I − Π)Z together with Lemmas 2.1 and 2.2 we obtain the following.
Proposition 2.3. For u, v ∈ Tx M, u(t) = DΦt (x)u, v(t) = DΦt (x)v,
(2.7)
dS(u, v) = (I − Π)dB(u, v) + ∂W S(u, v) − Π(∂W + ∂ W̃ )S(u, v)
where W̃ denotes the transpose of W .
Proof. We shall abbreviate by suppressing the dependence on u and v. So,
dS = (I − Π)∂Z − ∂ΠZ
= (I − Π)dB + (I − Π)∂W Z − ∂ΠZ, by (2.4),
= (I − Π)dB + (I − Π)∂W S
+ (I − Π)∂W R − ∂ΠR − ∂ΠS
, ΠZ ≡ R
= (I − Π)dB + (I − Π)∂W S − ∂ΠS,
by Lemma 2.1
= (I − Π)dB + ∂W S − Π(∂W + ∂ W̃ )S, by Lemma 2.2.
The equation (2.7) can be simplified. Set
dP = (I − Π)dW
(2.8)
dQ = ΠdW̃
and define λ and µ so that
(I − Π)∂W = dP + dλ
(2.9)
Π∂ W̃ = dQ + dµ.
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
Lemma 2.4.
dΠ = dP Π + dλΠ + dQ(I − Π) + dµ(I − Π).
(2.10)
1
dS(u, v) = (I − Π)dB(u, v) + (dP − dQ)S(u, v) + (dP − dQ)2 S(u, v)
2
(2.11)
+ d(λ − µ)S(u, v)
Proof. Let ζt be an Rn -valued semimartingale. Then
dΠ = (I − Π)∂W Π + Π∂ W̃ (I − Π)
(2.12)
holds since
∂Πt ζt = ∂Πt (Πt ζt ) + ∂Πt ((I − Πt )ζt )
= (I − Πt )∂Wt Πt ζt + Πt ∂ W̃t (I − Πt )ζt ,
by Lemmas 2.1 and 2.2. Thus, we have the following quadratic variations,
1
(−(I − Π)dW ΠdW − ΠdW̃ (I − Π)dW )
2
1
= (−dP ΠdW − dQ(I − Π)dW ) ,
2
1
dµ = ((I − Π)dW ΠdW̃ + ΠdW̃ (I − Π)dW̃ )
2
1
= (dP ΠdW̃ + dQ(I − Π)dW̃ ).
2
dλ =
Consequently, performing Itô corrections on (2.12), we get
dΠ = dP Π + dλΠ + dQ(I − Π) + dµ(I − Π)
which proves (2.10). In order to derive (2.11), rewrite (2.7) as
dS(u, v) = (I − Π)dB(u, v) + (I − Π)∂W S(u, v) − Π∂ W̃ S(u, v).
Then (2.11) follows from the definitions of dP, dQ, dλ and dµ in (2.9).
Lemma 2.5. If ζt is a Txt Mt⊥ -valued semi-martingale, then
(2.12)
dλζ =
(n − 1)µ2
ζdt
2n(n + 2)
9
10
M. CRANSTON AND Y. LEJAN
(2.13)
dµζ =
(n − 1)(n + 1)µ2
ζdt
2n(n + 2)
(n − 1)nµ2
1
(dP − dQ)2 ζ = −
ζdt
2
2n(n + 2)
(2.14)
Proof. Select an orthonormal basis {e1 , . . . , en } for Txt Mt such that en ∈ Txt Mt⊥ .
Then,
2dλζ = −dΠdW ζ − dP ΠdW ζ − dQ(I − Π)dW ζ
= −(I − Π)dW ΠdW ζ − ΠdW̃ (I − Π)dW ζ
n−1
X
= −(
in
Cni
)ζdt, since hdWin , dWnn i = 0, 1 ≤ i ≤ n − 1,
i=1
(n − 1)µ2
ζdt, by (1.6)
=
n(n + 2)
2dµζ = dP ΠdW̃ ζ + dQ(I − Π)dW̃ ζ
n−1
X
=(
ii
Cnn
)ζdt
i=1
=
(n − 1)(n + 1)µ2
ζdt, by (1.6).
n(n + 2)
Thus,
1
1
(dP − dQ)2 ζ = (dP − dQ)((I − Π)dW ζ − ΠdW̃ ζ)
2
2
n−1
X
1
n
= (dP − dQ)(dWn ζ −
dW̃in ei hζ, en i)
2
i=1
X
1
(dWnn dWnn ζ −
dWni dW̃in ζ)
2
i=1
n−1
=
n−1
X
1 nn
ii
= (Cnn ζ − (
Cnn
)ζ)dt
2
i=1
µ2
=
((n − 1) − (n − 1)(n + 1))ζdt , by (1.6),
2n(n + 2)
(n − 1)µ2
=−
ζdt.
2(n + 2)
Recall that we have selected an initial basis for Tx M with un ∈ Tx M ⊥ and
ui (t) = DΦt (x)ui gives a basis for Txt Mt when i runs through {1, . . . , n − 1}.
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
11
(I−Π)un
Theorem 2.6. If ν(t) denotes the unit vector ν = ||(I−Π)u
perpendicular to Mt
n||
at xt , then
(n2 − 1)µ2
dν = −dQν −
νdt.
2n(n + 2)
Proof. Set vn = (I − Π)un . Then by Lemmas 2.2 and 2.3,
dvn = −Π∂ W̃ vn + (I − Π)∂W vn
1
= (dP − dQ)vn + (dP − dQ)2 vn + (dλ − dµ)vn
2
(n − 1)µ2
vn dt, by Lemma 2.5
= (dP − dQ)vn −
(n + 2)
Then, applying
d||vn ||2 = 2hvn , dvn i + h(dP − dQ)vn i
2(n − 1)µ2
||vn ||2 dt + h(dP − dQ)vn i
(n + 2)
2(n − 1)µ2
(n − 1)µ2
||vn ||2 dt +
||vn ||2 dt
= 2||vn ||2 dWnn −
(n + 2)
n
(n − 1)(n − 2)µ2
||vn ||2 dt
= 2||vn ||2 dWnn −
n(n + 2)
= 2hvn , (dP − dQ)vn i −
and so
d||vn||−1 = −||vn ||−1 dWnn +
Consequently, as ν =
(n − 1)(n + 1)µ2
||vn ||−1 dt.
2n(n + 2)
vn
||vn || ,
(n − 1)µ2
(n − 1)(n + 1)µ2
νdt − (I − Π)dW ν +
νdt
(n + 2)
2n(n + 2)
− h(dP − dQ)ν, dWnn i
dν = (dP − dQ)ν −
(n − 1)2 µ2
(n − 1)µ2
νdt −
νdt
2n(n + 2)
n(n + 2)
(n + 1)(n − 1)µ2
= dQν −
νdt , as desired.
2n(n + 2)
= −dQν −
We can now derive the equation for hS(u, v), νi.
12
M. CRANSTON AND Y. LEJAN
Theorem 2.7. Suppose u(t) = DΦt u, v(t) = DΦt v and ν is as in Theorem 2.6.
Then
dhS(u, v), νi = hdB(u, v), νi + hS(u, v), νihdW ν, νi −
(n − 1)2 µ2
hS(u, v), νidt.
2n(n + 2)
Moreover, if ui (t) = DΦt ui , then
33
hdhS(ui , uj ), νi, dhS(uk , u` ), νii = C1122
[hui , uj ihuk , u` i
+ hui , uk ihuj , u` i + hui , u`ihuj , uk i]dt
+
µ2 (n − 1)
hS(ui , uj ), νihS(uk , u` i, νidt.
n(n + 2)
Proof. From Lemmas 2.4 and 2.5 we have
dS(u, v) = (I − Π)dB(u, v) + (dP − dQ)S(u, v) −
(n − 1)µ2
S(u, v)dt.
(n + 2)
Also
−h(dP − dQ)S(u, v), dQνi = hdQS(u, v), dQνi
= hS(u, v), νi
n−1
X
ii
Cnn
dt
i=1
(n − 1)µ2
hS(u, v), νidt.
n(n + 2)
2
=
So,
dhS(u, v), νi = hdS(u, v), νi + hS(u, v), dνi + hdS(u, v), dνi
= hdB(u, v), νi + hS(u, v), νidWnn −
(n − 1)µ2
hS(u, v), νidt
(n + 2)
(n2 − 1)µ2
hS(u, v), νidt − h(dP − dQ)S(u, v), dQνi
2n(n + 2)
(n − 1)(n − 1)2 µ2
= hdB(u, v), νi + hS(u, v), νidWnn −
hS(u, v), νidt
2n(n + 2)
−
nn
For the quadratic variation term, since B and W are independent and C111
=
nn
3C1122,
hdS(ui , uj ), νihdS(uk , u` ), νi = hhdB(ui , uj ), νihdB(uk , u` ), νii
nn
+ hS(ui , uj ), νihS(uk , u`), νiCnn
dt
nn
= C1122
[hui , uj ihuk , u`i + hui , uk ihuj , u` i
+ hui , u` ihuj , uk i]dt
+
(n − 1)µ2
hS(ui , uj ), νihS(uk , u` ), νidt.
n(n + 2)
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
13
We now split the exposition. The case of a hypersurface in R3 will be developed
below. The case of a hypersurface in Rn , n > 3, will be treated in an addendum.
The latter involves lengthy computations together with combinatorial arguments,
whereas the former is more readable.
Theorem 2.8. Let u, v ∈ Tx M with u and v linearly independent and define
u(t) = DΦt u, v(t) = DΦt v, (however in our computations we shall suppress the t
dependence). The process (T r S (1), T r S (2)) satisfies the equations
d T r S (1) = [hdB(u, u), νikvk2 + hdB(v, v), νikuk2 − 2hdB(u, v), νihu, vi]kαk−2
+ T r S (1) (dW33 − 2(dW11 + dW22 ))
+ 2[S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)]kαk−2
4µ2
+
T r S (1) dt
15
d T r S (2) = [S(v, v)hdB(u, u), νi + S(u, u)hdB(v, v), νi − 2S(u, v)hdB(u, v), νi]kαk−2
+ 2 T r S (2) (dW33 − (dW11 + dW22 ))
4µ2
T r S (2) dt .
+
15
16µ4
µ2
(1) 2
(2)
+
[14(T r S ) − 24 T r S ] dt
hd T r S i =
35
15
2µ4
32µ2
(2)
(1) 2
(2)
(2) 2
hd T r S i =
(3(T r S ) − 4 T r S ) +
(T r S ) dt
35
15
8µ4
16µ2
(1)
(2)
(1)
(1)
(2)
hd T r S , dT r S i =
Tr S +
Tr S Tr S
dt .
35
15
(1)
Proof. Recall we have selected an orthomal basis {e1 , e2 , e3 } for Txt Mt such that
ν = e3 . Then with α = u ∧ v, from LeJan (1986) we have
(2.15)
dkαk−2
2µ2
= −2(dW11 + dW22 ) −
dt .
−2
kαk
15
Using (1.6), it follows that
(2.16)
dhu, vi = hdW u, vi + hu, dW vi +
10µ2
hu, vidt .
15
Now by (1.9), the mean curvature is
(2.17)
T r S (1) = [S(u, u)kvk2 + S(v, v)kuk2 − 2S(u, v)hu, vi]kαk−2
14
M. CRANSTON AND Y. LEJAN
so by Itô’s formula,
d T r S (1) = [kvk2 dS(u, u) + kuk2 dS(v, v) − 2hu, vidS(u, v)
+ S(u, u)dkvk2 + S(v, v)dkuk2 − 2S(u, v)dhu, vi
+ hdS(u, u), dkvk2 i + hdS(v, v), dkuk2 i − 2hdS(u, v), dh(u, v)i]kαk−2
+ T r S (1)
dkαk−2
+ hd(T r S (1) kαk2 ), dkαk−2 i
kαk−2
= [hdB(u, u), νikvk2 + hdB(v, v), νikuk2 − 2hdB(u, v), νihu, vi]kαk−2
2µ2
T r S (1) dt
+ T r S (1) dW33 −
15
+ [S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)]kαk−2
10µ2
2µ2
+
T r S (1) dt −
T r S (1) dt
15
15
2µ2
− 2 T r S (1) (dW11 + dW22 ) −
T r S (1) dt ,
15
using Theorem 2.7, (2.15), (2.16) and the fact that
hd(T r S (1) kαk2 ), dkαk−2 i = −2kαk−2 [hkαk2 T r S (1) dW33 , dW11 + dW22 i
+ 2hS(u, u)hdW v, vi + S(v, v)hdW u, ui
− S(u, v)(hdW u, vi + hu, dW vi), dW11 + dW22 i]
h −2µ
i
2µ2
2
= −2
+
T r S (1) dt
15
15
=0.
Thus,
(2.18)
d T r S (1) = [hdB(u, u), νikvk2 + hdB(v, v), νikuk2 − 2hdB(u, v), νi hu, vi]kαk−2
+ T r S (1)(dW33 − 2(dW11 + dW22 ))
+ 2[S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi
+ hu, dW vi)]kαk−2
4µ2
+
T r S (1)dt .
15
Consequently, the quadratic variation of T r S (1) is given by
(2.19)
hd T r S (1) i = [ hdB(u, u), νikvk2 + hdB(v, v), νikuk2 − 2hdB(u, v), νi hu, vi ]kαk−4
+ (T r S (1) )2 hdW33 − 2(dW11 + dW22 )i
+ 4[S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)]kαk−4
+ 4T r S (1) hdW33 − 2(dW11 + dW22 ), S(u, u)hdW v, vi + S(v, v)hdW u, ui
− S(u, v)(hdW u, vi + hu, dW vi)ikαk−2
Setting
qn =
(n + 3)µ4
nn
= C1122
n(n + 2) (n + 4)
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
and if ui , uj , uk , u` are all orthogonal to ν, then
(2.20)
hdB(ui , uj ), νi hdB(uk , u` ), νi = qn [hui , uj i huk , u` i + hui , uk i huj , u`i
+ hui , u` i huj , uk i]dt .
Then
(2.21)
h hdB(u, u), νikvk2 + hdB(v, v), νikuk2 − 2hdB(u, v), νi hu, viikαk−4
= q3 [3kuk4kvk4 + 3kvk4 kuk4 + 4(2hu, vi2 + kuk2 kvk2 )hu, vi2
+ 2(kuk2 kvk2 + 2hu, vi2 )kuk2 kvk2 − 4(3kuk2 hu, vi)kvk2 hu, vi
− 4(3kvk2 hu, vi)kuk2 hu, vk]kαk−4 dt
= q3 [8kuk4 − 16kuk2kvk2 hu, vi2 + 8hu, vi4 ]kαk−4 dt
= 8q3 dt .
Next, observe
hdW33 − 2(dW11 + dW22 )i = hdW33 i − 4hW33 , dW11 + dW22 i + 4hdW11 + dW22 i
18µ2
=
dt
15
and
hhdW v, vii =
hhdW v, vi hdW u, uii =
hhdW u, vi, hu, dW vii =
hhdW u, vi, hdW u, vii =
hhdW u, vi, hdW v, vii =
hhdW v, ui, hdW u, uii =
hdW v, ui hv, vi =
2µ2
kvk4 dt
15
µ2
(3hu, vi2 − kuk2 kvk2 )dt
15
µ2
(3hu, vi2 − kvk2 )dt
15
µ2
(4kuk2 kvk2 − 2hu, vi2 )dt
15
2µ2
hu, vikvk2 dt
15
2µ2
hu, vikuk2 dt
15
2µ2
hu, vikvk2 dt .
15
15
16
M. CRANSTON AND Y. LEJAN
Thus,
[S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)]
µ2
=
[2S(u, u)2 kvk4 + 2S(v, v)2 kuk4
15
+ 2S(u, v)2 (3kuk2 kvk2 + hu, vi2 ) + 2S(u, u)S(v, v)(3hu, vi2 − kuk2 kvk2 )
− 2S(u, u)S(u, v)(1hu, vikuk2 + 2hu, vikuk2 )kvk2
− 2S(v, v)S(u, v)(2hu, vikuk2 + 2hu, vikuk2 )kuk2 ]dt
µ2
=
[2{S(u, u)2 kvk4 + S(v, v)2 kuk4 + 2S(u, u)S(v, v)kuk2 kvk2
15
− 4S(u, u)S(u, v)kuk2 kvk2 hu, vi
− 4S(v, v)S(u, v)kuk2 kvk2 hu, vi
4S(u, v)2 hu, vi2 }
− 6(S(u, u)S(v, v) − S(u, v)2 )kαk2 ]dt
µ2
=
[2(T r S (1) )2 − 6T r S (2) ]kαk4 dt
15
It’s easy to verify that
µ2
kvk2 dt
15
µ2
hdW33 , hdW u, vii = hdW33 , hu, dW vii = − hu, vidt
15
µ2
1
2
hdW1 + dW2 , hdW u, vii =
hu, vidt
15
hdW33 , hdW v, vii = −
so
hdW33 , S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)i
µ2
= − [S(u, u)kvk2 + S(v, v)kuk2 − 2S(u, v)hu, vi]dt
15
µ2
= − T r S (1)kαk2 dt ,
15
1
hdW1 + dW22 , S(u, u)hdW v, vi + S(v, v)hdW u, ui − S(u, v)(hdW u, vi + hu, dW vi)i
µ2
[S(u, u)kvk2 + S(v, v)kuk2 − 2S(u, v)hu, vi]dt
=
15
µ2
=
T r S (1)kαk2 dt .
15
Therefore
4 T r S (1)kαk−2 hdW33 − 2(dW11 + dW22 ), S(u, u)hdW v, vi + S(v, v)hdW u, ui
(2.23)
− S(u, v)(hdW u, vi + hu, dW vi)i
µ2
2µ2
(1) 2
= 4(T r S ) −
−
dt
15
15
12µ2
=−
(T r S (1))2 dt .
15
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
17
Combining (2.20), (2.21) and (2.22) we arrive at
(2.24)
hd T rS (1) i =
µ2
16µ4
dt +
[14(T r S (1))2 − 24 T r S (2)]dt .
35
15
Turning now to the Gauss curvature, by (1.9) we have,
(2.25)
T r S (2) = [S(u, u)S(v, v) − S(u, v)2 ]kαk−2 .
So, by Itô’s formula,
d T r S (2) = (S(v, v)dS(u, u) + S(u, u)dS(v, v) − 2S(u, v)dS(u, v)
+ hdS(u, u), dS(v, v)i − hdS(u, v), dS(u, v)i]kαk−2
+ [S(u, u)S(v, v) − S(u, v)2 ]dkαk−2
+ hS(v, v)dS(u, u) + S(u, u)dS(v, v) − 2S(u, v)dS(u, v), dkαk−2 i
= [S(v, v)hdB(u, u), νi + S(u, u)hdB(v, v), νi
− 2S(u, v)hdB(u, v), νi]kαk−2
4µ2
+ 2T r S (2)dW33 −
T r S (2)dt
15
+ [hhdB(u, u), νi, hdB(v, v), νii − hdB(u, v), νi, hdB(u, v), νii]kαk−2 dt
2µ2
+
T r S (2)dt
15
2µ2
T r S (2)dt
− 2 T r S (2)(dW11 + dW22 ) −
15
8µ2
+
T r S (2)dt .
15
Thus,
dT r S (2) = [S(v, v)hdB(u, u), νi + S(u, u)hdB(v, v), νi
(2.26)
− 2S(u, v)hdB(u, v), νi]kαk−2
4µ2
T r S (2)dt ,
15
since hhdB(u, u), νi, hdB(v, v), νii = h(dB(u, v), νi, hdB(u, v), νii .
+ 2T r S (2)(dW33 − (dW11 + dW22 )) +
18
M. CRANSTON AND Y. LEJAN
Using (2.20), (2.26) and our remarks above on W correlations, we get
(2.27)
hd T r S (2)i = hS(v, v)hdB(u, u), νi + S(u, u)hdB(v, v), νi
− 2S(u, v)hdB(u, v), νiikαk−4 + 4(T r S (2))2 hdW 3 − (dW11 + dW22 )i
= q3 [3S(v, v)2 kuk4 + 3S(u, u)2kvk4 + 4S(u, v)2 (kuk2 kvk2 + 2hu, vi2 )
+ 2S(u, u)S(v, v)(kuk2kvk2 + 2hu, vi2 )
− 12S(v, v)S(u, v)kuk2hu, vi − 12S(u, u)S(u, v)kvk2hu, vi]kαk−2 dt
32µ2
+
(T r S (2))2 dt
15
32µ2
= 3q3 (T r S (1))2 dt − 4q3 T r S (2)dt +
(T r S (2))2 dt
15
32µ2
2µ4
(3(T r S (1))2 − 4 T r S (2))dt +
(T r S (2))2 dt .
=
35
15
Finally we need
(2.28)
hdT r S (1) , dT r S (2) i = 2T r S (1) T r S (2) hdW33 − 2(dW11 + dW22 ), dW33
− (dW11 + dW22 )i + 4T r S (2) hS(u, u)hdW v, v)
+ S(v, v)hW u, ui − S(u, v)(hdW u, vi
+ hu, dW vi), dW33 − (dW11 + dW22 )ikαk−2
+ kαk−4 hhdB(u, u), νikvk2 + hdB(v, v), νikuk2
− 2hdB(u, v), νihu, vi , S(v, v)hdB(u, u), νi
+ S(u, u)hdB(v, v), νi − 2S(u, v)hdB(u, v), νii
24µ2
8µ2
=
T r S (1) T r S (2) dt −
T r S (1) T r S (2) dt + 4q3 T r S (1) dt
15
15
16µ2
8µ4
=
T r S (1) T r S (2) dt +
T r S (1) dt .
15
35
This completes the proof. Theorem 2.9. The process (T r St(1), T r St(2)) is a recurrent diffusion with generator
1 µ2
16µ4 ∂ 2 f
2
Lf(κ, m) =
(14κ − 24m) +
(κ, m)
2 15
35
∂κ2
2
16µ2
8µ4
∂ f
κm +
κ
(κ, m)
+
15
35
∂κ ∂m
2
1 32µ2 2 2µ4
∂ f
2
+
m +
(3κ − 4m)
(κ, m)
2
15
35
∂m2
4µ2 ∂f
4µ2 ∂f
+
κ (κ, m) +
m
(κ, m) .
15 ∂κ
15 ∂m
This diffusion never enters the region κ2 ≤ 4m at strictly positive times a.s. .
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
19
Remarks.
(1) In n = 3, the three Lyapunov exponents are (see LeJan 1986) γ1 = 3, γ2 = 0,
γ3 = −3. Thus there is always a stretching direction, a neutral direction and
a compressing direction. A curve will always “see” the stretching direction.
This roughly explains the positive recurrence of the curvature of a curve
proved in LeJan 1986. In the case of a surface, the tangent plane will “see”
the stretching and the neutral direction. This also explains recurrence. The
authors plan a complete account of the situation in higher dimensions in a
forthcoming paper.
(2) Since X = T r S (1) = λ1 +λ2 and Y = T r S (2) = λ1 λ2 , it follows that (X, Y )
stays in the region κ2 ≥ 4m. The boundary of this region corresponds to
λ1 = λ2 . That is κ2 = 4m when the surface in question has an umbilic (i.e.
a point where λ1 = λ2 ). Thus, x may be an umbilic for M but xt , t > 0,
will never be an umbilic for Mt .
One easily checks that the matrix a appearing in the generator for this
diffusion (2nd order derivative coefficients) degenerates on the curve κ2 −
4m = 0. Moreover, the two eigenvectors of this matrix are, respectively,
perpendicular to and tangent to the curve κ2 − 4m = 0. The eigenvector
perpendicular to this curve has eigenvalue zero. Notice the drift vector
4µ2
4µ2
2
15 κ , 15 m points away from the region κ − 4m < 0. Thus, when the
point x on the initial manifold M is an umbilic point, the diffusion (Xt , Yt )
has diffusion component only tangential the curve κ2 = 4m and drifts away
from this curve never to return.
(3) Neither T r S (1) nor T r S (2) is a diffusion by itself.
(4) In the case n = 3 one can assert that (Xt , Yt ) does not spend a lot of time in
the region λ = {(κ, m) : κ > M1 , m > M2 } with M1 and M2 large positive
constants. Recall the following results of LeJan (1986): draw a curve γ on
the initial manifold M with γ(0) = x. Set γt = Φt (γ). Then the curvature
τt of γt at γt (0) is a positive recurrent diffusion. Thus, the average amount
of time that τt spends above a large positive constant is small. Since it
is impossible to draw a curve on Mt with zero curvature (or even small
curvature) when (Xt , Yt ) ∈ Λ, (Xt , Yt ) can not stay for long in Λ. More
precise statements can be made by referring to the article of LeJan where
the exact form of the invariant measure for τt is given.
Proof (of Theorem 2.9). The claim
follows immediately
√ that (Xt , Yt ) is a diffusion
µ2
2
from Theorem 2.8. Define h = κ − 4m. Then with a = 15 , b = µ354 , in (m, h)coordinates
∂2f
∂2f
+
14aκh
∂κ2
∂κ∂h
2
∂ f
+ (3aκ2 + 4ah2 + 4b) 2
∂h
∂f
∂f
+ 4aκ
+ h−1 (3aκ2 + ah2 + 4b)
.
∂κ
∂h
Lf(κ, h) = (4aκ2 + 3ah2 + 8b)
20
M. CRANSTON AND Y. LEJAN
We prove the claim regarding lack of umbilics first as it eases a technical point
which might arise in the proof of the recurrence. For this, it suffices to show that
Ht does not hit zero a.s. where (Xt , Ht ) is diffusion given in (κ, h) coordinates.
Rt
First define ηt = 3aXt2 + 4aHt2 + 4b and then τt = 0 dx
ηs . Then
Z
τt
Hτt = h + bt +
Z
Hs−1 (3aXs2 + aHs2 + 4b)ds
0
t
= h + bt +
0
Hτ−1
(3aXτ2s + aHτ2s + 4b)ητ−1
ds
s
s
with bt a one-dimensional Brownian motion. Fix now an ∈ (0, 1/6) and suppose
0 < h < 1/6 as well. Set
ρt
= h + bt +
q
If σ = inf{t > 0 : Hτt > 2
2
( 12 −)b
(1+4)a },
Z t
1
ds
+
.
2
0 ρs
then since for h ≤
q
( 12 −)b
(1+4)a ,
one has
2
3aκ +ah +4b
( 12 + ) < 3aκ
2 +4ah2 +4b for all κ. By an elementary comparison theorem (see IkedaWatanabe, 1989), it follows that ρt ≤ Hτt for t < σ . Since ρt can not hit zero (it
is a Bessel process above the critical dimension), Ht can’t hit zero either.
For recurrence, we shall show (Xt , Ht ) is recurrent which will suffice.
f(κ, h) = `n(κ6 + h6 ). Then
6κ5
6h5
,
f
=
h
κ6 + h6
κ6 + h6
30κ4 h6 − 6κ10
36κ5 h5
=
,
f
=
−
;
κh
(κ6 + h6 )2
(κ6 + h6 )2
fκ =
fκκ
fhh =
30κ6 h4 − 6h10
(κ6 + h6 )2
and
Lf = 6(κ6 + h6 )−2 [(4aκ2 + 3ah2 + 8b)(5κ4 h6 − κ10 ) − 84aκ6 h6
+ (3aκ2 + 4ah2 + 4b)(5κ6 h4 − h10) + 4aκ12 + 4aκ6 h6
+ h4 (3aκ2 + ah2 + 4b)(κ6 + h6 )]
= 6(κ6 + h6 )−2 [−a(3κ10 h2 − 18κ8 h4 + 39κ6 h6 − 15κ4h8 + 3h12 )
− b(8κ10 − 24κ6 h4 − 40κ4 h6 )]
κ = uh
= −6(κ6 + h6 )−2 h10 [3ah2 (u10 − 6u8 + 13u6 − 5u4 + 1)
+ 8bu4 (u6 − 3u2 − 5)]
The polynomial P (u) = u10 − 6u8 + 13u6 − 5u4 + 1 > 0 for all u.
Set
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
21
Also, there is a u0 > 0 such that Q(u) = u6 − 3u2 − 5 ≥ 0 whenever |u| ≥ u0.
Thus,
Lf = −6(κ6 + h6 )−2 h10 [3ah2 P (u) + 8bu4 Q(u)]
has
{Lf ≤ 0} ⊇ {(κ, h) : |κ| ≥ u0 h} .
Also, denote M = supu
Q(u)
P (u) ,
one sees
(
{Lf ≤ 0} ⊇
r
(κ, h) : h ≥
86M
3a
)
Consequently, off of the compact triangle
(
T = {(κ, h) : |κ| ≤ u0 h} ∩
(κ, h) : h ≤
r
86M
3a
)
we have Lf ≤ 0. Now, define
C(r) = {(κ, h) : h > 0, κ6 + h6 < r}
and select r0 large enough so that C(r0 ) ⊃ T . Put for R > r0
`n(κ6 + h6 ) − `nR
.
`nr0 − `nR
Then on C(R) \ C(r0 ), LuR ≥ 0 and uR ∂C(r0 )∩{h>0} = 1, uR ∂C(R)∩{h>0} = 0.
Thus, if τr = inf{t > 0 : (Xt , Ht ) ∈ ∂C(r)} (recall, Ht never hits 0), then as uR is
L-subharmonic and has the same boundary values as P(κ,w) (τr0 < TR )
uR (κ, h) =
uR (κ, h) ≤ P(κ,h) (τr0 < τR ) .
But limR↑∞ uR (κ, h) = 1 which implies the recurrence of (Xt , Ht ) and therefore the
recurrence of (Xt , Yt ). 22
M. CRANSTON AND Y. LEJAN
References
P. Baxendale, (1984), Brownian motions in the diffeomorphism Group. I,
Compositio Math. 53, 19–50.
P. Baxendale and T.Harris, (1986), Isotropic stochastic flows, Annals of
Prob., Vol 14, 1155–1179.
R. Bishop and S. Goldberg, (1980), Tensor Analysis on Manifolds, Dover,
Toronto.
Hardy, Littlewood, Polya, (1973), Inequalities, Cambridge University Press,
Cambridge.
T. Harris, (1981), Brownian motions on the homeomorphisms of the plane,
Annals of Prob. Vol. 9, 232–254.
N. Ikeda, S. Watanabe, (1989), Stochastic Differential Equations and Diffusion Processes, North-Holland, New York.
Y. LeJan, (1985), On isotropic Brownian motions, Zeit. für Wahr., 70,
609–620.
Y. LeJan, (1991), Asymototic properties of isotropic Brownian flows, in
Spatial Stochastic Processes, ed. K. Alexander, J. Watkins, Birkhäuser,
Boston, 219–232.
Y. LeJan, S. Watanabe (1984), Stochastic flows of diffeomorphisms, Proc.
of the Taniguchi Symposium, 307–332, North Holland, New York.
M. Spivak, (1979), A comprehensive introduction to differential geometry,
Publish or Perish, Berkeley.
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
23
Addendum. Higher dimensional hypersurfaces.
~ ~ ∈ Ik . Define
Recall α~` = u`1 ∧ · · · ∧ u`k , αm
~ = um1 ∧ · · · ∧ umk , for `, m
α~` = (−1)p+1 u`1 ∧ · · · ∧ û`p ∧ · · · ∧ u`k ,
i+1
αm
um1 ∧ · · · ∧ ûmi ∧ · · · ∧ umk .
~ i = (−1)
p
(p)
(i)
Then α~`p = (−1)p+1 u`1 ∧ · · · ∧ up`
(i)
(i)
(i)
and um , and set S (0)(·, ·) ≡ 1.
Theorem A.1.
dS
(k)
k
X
(α~`, αm
~)=
S (k−1)(α~`p , αm
~ i )hdB(u`p , umi ), νi
i,p=1
n
+ kS (k)(α~`, αm
~ )dWn −
k(n − k)(n − 1)µ2 (k)
S (α~`, αm
~ )dt ,
2n(n + 2)
or, in a more synthetic form
(k)
dS(α)
= S (k−1) ∧ dB ν (α) + kS (k)(α)hdhW ν, νi −
−k(n − k)(n − 1) (k)
S (α) ,
2n(n + 2)
with hB ν u, vi = hB(u, v), νi.
Proof. By Theorem 2.7 and Itô’s formula,
dS (k) (α~`, αm
~) =
X
(−1)σ
k
k Y
X
[
hS(u`σ(j) , umj ), νi]hdB(u`σ(i) , umi ), νi
i=1 j=1
j6=i
σ∈Sk
n
+ kS (k) (α~`, αm
~ )dWn −
+
1
2
X
X
(−1)σ
σ∈Sk
1≤i6=p≤k
[
k
Y
k(n − 1)(n − 1)µ2 (k)
S (α~`, αm
~ )dt
2n(n + 2)
hS(u`σ(j) , umj ), νi][dB n (u`σ(i) , umi ), dB n (u`σ(p) , ump )i
j=1
j6={i,p}
+hdB n (u`σ(i) , ump ),dB n (u`σ(p) , umi )i + hdB n (u`σ(i) , u`σ(p) ), dB n (umi , ump )i]
+
k(k − 1)(n − 1)µ2 (k)
S (α~`, αm
~ )dt.
2n(n + 2)
The first term may be written
k
X
X
(−1)σ [
=
[
X
p+i
(−1)
k
Y
hS(u`σ(j) , umj ), νi]hdB(u`p , umi ), νi
j=1
j6=i
i,p=1 σ∈Sk
σ(i)=p
k
X
σ
(−1) [
k
Y
(p)
hS(u`
σ(j)
(i)
, umj ), νi]]hdB(u`p , umi ), νi
j=1
j6=i
i,p=1 σ=Sk−1
=
k
X
S (k−1) (α~` , αm
~ i )hdB(u`p , umi ), νi.
p
i,p=1
(p)
p+1
αm
um1 ∧ · · · ∧ umk−1 defines u`
~ i = (−1)
k−1
24
M. CRANSTON AND Y. LEJAN
The third term combines with the fifth term to give
−
k(n − k)(n − 1)µ2 (k)
S (α~`, αm
~ )dt.
2n(n + 2)
Finally, in the fourth term, notice that given σ ∈ Sk and i, p ∈ {1, . . . , k} there is
exactly one other η ∈ Sk such that both {σ(i), σ(p)} = {η(i), η(p)} and σ(j) = η(j)
for j ∈
/ {i, p}. Furthermore, (−1)σ = −(−1)η , and
[
k
Y
hS(u`σ(j) , umj ), νi][hB n (u`σ(i) , umi ), dB n (u`σ(p) , ump )i
j=1
j ∈{i,p}
/
+hdB n (u`σ(i) , ump ), dB n (u`σ(p) , umi )i + hdB n (u`σ(i) , u`σ(p) ), dB n (umi , ump )i]
=[
k
Y
hS(u`η(j) , umj ), νi][hdB n (u`η(i) , umi ), dB n (u`η(p) , ump )i
j=1
j ∈{i,p}
/
+hdB n (u`η(i) , ump ), dB n (u`η(p) , umi )i
+ hdB n (u`η(i) , u`η(p) ), dB n (umi , ump )i]
Thus exchanging the sums in the fourth term reveals the cancellation of terms from
these σ − η pairs and so the whole term vanishes. This finishes the proof. From LeJan (1985) we recall the notation
τ`j β = ej ∧ i(e` )β
where i(e` )β = Σ(−1)k+1 hβk , e` iβ1 ∧ · · · ∧ β̂k ∧ · · · ∧ βm . The situation for general
n is given by the following Theorem which we prove in the Addendum.
Theorem A.2. For 1 ≤ k ≤ n − 1,
dT rS
(k)
k
X
=[
~
m
~
−2
`
S (k−1) (α~` , αm
~ i )hdB(u`p , umi ), νi]hα , α i||α||
p
i,p=1
+ T rS
(k)
[kdWnn
−2
n−1
X
dWii ]
i=1
~
j
m
~
`
+ S (k) (α~`, αm
~ )(hτi α , α i
−
~
~
+ hα` , τij αm
i)dWji ||α||−2
(n + 1)k(n − k)µ2
T rS (k) dt , or
2n(n + 2)
d T rS (k) = dhS (k)α, αi kαk2
hdW α, αi
= hS
∧ dB (α) + hdW ν, νi − 2
S (k)(α)
kαk2
(n + 1)k(n − k)µ2 (k)
+ S (k) ∧ dW (α), αi kαk2 −
hS (α), αi kαk2 .
2n(n + 2)
(k−1)
ν
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
Proof. For ~` ∈ Ik ,
~
~
+
(k + 1)(n − 1 − k)µ2 ~` m
hα , α ~ idt
n
n−1
X
d||α||−2
(n − 2)(n − 1)µ2
=
−2
dWii −
dt
−2
||α||
n(n + 2)
i=1
(A.2)
(A.3)
~
~
~
~
dhα` , αm
i = (hτij α` , αm
i + hα` , τij αm
i)dWji
(A.1)
d||α||−2 = −2||α||−2dWii + At ,
Since
At has bounded variation
~
`
m
~
−2
Tr S (k) = S (k)(α~`, αm
~ )hα , α i||α||
by Itô’s formula,
~
`
m
~
−2
dT rS (k) = (dS (k)(α~`, αm
~ ))hα , α i||α||
~
`
m
~
−2
+ S (k)(α~`, αm
~ )(dhα , α i)||α||
+ T rS (k)
d||α||−2
||α||−2
~
`
m
~
−2
+ hdS (k) (α~`, αm
~ ), dhα , α ii||α||
~
−2
~
+ hdS (k) (α~`, αm
ihα` , αm
i
~ ), d||α||
~
`
m
~
−2
+ S (k)(α~`, αm
i
~ )hdhα , α i, d||α||
Proceeding in order through the terms, we have using Theorem A.1
(A.4)
~
`
m
~
−2
dS (k)(α~`, αm
~ )hα , α i||α||
=[
k
X
~
`
m
~
−2
S (k−1) (α~`p , αm
~ i )hdB(u`p , umi ), νi]hα , α i||α||
i,p=1
+ kT rS (k) dWnn −
k(2n − k)(n − 1)µ2
T rS (k) dt.
2n(n + 2)
From (A.1) follows,
(A.5)
~
`
m
~
−2
S (k)(α~`, αm
~ )dhα , α i||α||
~
~
j `
j m
`
m
~
~
i
−2
= S (k)(α~`, αm
~ )(hτi α , α i + hα , τi α i)dWj ||α||
+
(k + 1)(n − k − 1)µ2
T rS (k) dt.
n
25
26
M. CRANSTON AND Y. LEJAN
By (A.2)
(A.6)
d||α||−2
T rS (k)
||α||−2
= −2T rS
(k)
n−1
X
dWii −
i=1
(n − 2)(n − 1)µ2
T rS (k) dt.
n(n + 2)
Combining Theorem A.1 and (A.1),
~
m
~
−2
`
hdS (k) (α~`, αm
~ ), dhα , α ii||α||
~
~
j `
j m
m
~
~
i
n
`
= kS (k) (α~`, αm
~ )(hτi α , α i + hα , τi α i)hdWj , dWn i
kµ2
~
j ~
j m
m
~
~
i j
i n
i j
`
`
S (k) (α~`, αm
~ )(hτi α , α i + hα , τi α i)[(n + 1)δn δn − δj δn − δn δn ]dt
n(n + 2)
kµ2
~
i ~
m
~
i m
~
`
`
=−
S (k) (α~`, αm
~ )(hτi α , α i + hα , τi α i)dt
n(n + 2)
2k(n − k − 1)µ2
T rS (k) dt.
=−
n(n + 2)
=
By Theorem A.1 and (A.3)
~
−2
~
hdS (k) (α~`, αm
ihα` , αm
i
~ ), d||α||
(A.7)
`
n
i
m
~
−2
= −2kS (k) (α~`, αm
~ )hdWn , dWi ihα , α i||α||
~
=
2k(n − 1)µ2
T rS (k) dt.
n(n + 2)
From (A.1) and (A.3) we get
(A.8)
~
m
~
−2
`
S (k) (α~`, αm
i
~ )hdhα , α i, d||α||
~
~
j `
j m
p
m
~
~
i
−2
`
= −2S (k) (α~`, αm
~ )(hτi α , α i + hα , τi α i)hdWp , dWj i||α||
2µ2
~
i ~
m
~
i m
~
−2
`
`
=
S (k) (α~`, αm
dt
~ )(hτi α , α i + hα , τi α i)||α||
n(n + 2)
4(n − k − 1)µ2
=−
T rS (k) dt
n(n + 2)
Summing (A.4)–(A.8) shows
dTr S
(k)
=[
k
X
~
i
(−1)i+p+1 S (k−1) (α~p , αm
)hdB(u`p , umi ), νi]hα` , αm
i||α||−2
~
`
~
m
`
i,p=1
+ T rS (k) [kdWnn − 2dWii ]
~
~
m
~
m
~
i
−2
`
`
+ S (k) (α~`, αm
~ )(hτi α , α i + hα , τi α i)dWj · kαk
j
+
k(n − k)(n + 1)µ2
T rS (k) dt
2n(n + 2)
and the proof is complete. j
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
27
Theorem A.3. For n − 1 ≥ k ≥ ` ≥ 1,
hdT rS (k) , dT rS (`) i =
µ2
[(k` + 2(k + `) + 4)(n − 1)
n(n + 2)
− 2(k + 2)(n − ` − 1) − 2(` + 2)(n − k − 1) − 4(n − ` − 1)(n − k − 1)]T rS (k) T rS (`) dt
X
(n−k−1)∧`
k
+ (1 − δn−1
)
4µ2
(n + 2)
d(n, (k, `); j)T rS (k+j) T rS (`−j) dt
j=0
X
(n−k)∧(`−1)
nn
+ C1122
a(n, (k, `); j)T rS (k+j−1) T rS (`−j−1) dt
j=0
where
d(n, (k, `); 0) = (n − 1 − k)
k − ` + 2j X
k − ` + 2j d(n, (k, `); j) = (n − 1 − k − j)
−
d(n, (k, `); r)
,j ≥ 1
j
j −r
j−1
r=0
b(n, (k, `); 0) = (n − k)(n − ` + 2)
k − ` + 2j k − ` + 2j − 2
b(n, (k, `); j) = 3(n − k − j)
+ (k − ` + 2j)(k − ` + 2j − 1)
j
j−1
k − ` + 2j − 1
k − ` + 2j − 1
+ (k − ` + 2j)(n − k − j)[
+
]
j
j −1
k − ` + 2j + (n − k − j)(n − k − j − 1)
,j ≥ 1
j
a(n, (k, `); 0) = b(n, (k, `); 0)
a(n, (k, `); j) = b(n, (k, `); j) −
j−1
X
k − ` + 2j j−m
a(n, (k, `); m)
m=0
nn
C1122
(n2 − 1) =
(n + 3)µ4
.
n(n + 2)(n + 4)
Proof. We compute the quadratic variation terms in hdT rS (k) , dT rS (`)i, k ≥ `,
which do not arise from dB. These are
hT rS (k) [kdWnn − 2
n−1
X
~
~
m
~
m
~
−2
`
`
dWii ] + S (k) (α~`, αm
dWjh ,
~ )(hτh α , α i + hα , τh α i)||α||
j
j
i=1
X
n−1
T rS (`) [`dWnn − 2
m
dWm
] + S (`) (α~r , α~s )(hτq α~r , α~s i + hα~r , τq α~s i)||α||−2 dWp i
p
p
q
m=1
unless k = n − 1 or both ` = k = n − 1 in which case, the term S (k)(α~`, αm
~)
~
j ~
`
` j m
m
~
~
−2
i
(hτi α , α i + hα , τi α i)||α|| dWj = 0 or both this and the corresponding S (`)
term vanishes. Thus we get
T rS (k) T rS (`) hkdWnn − 2
n−1
X
i=1
X
n−1
dWii , `dWnn − 2
j=1
dWjj i
28
M. CRANSTON AND Y. LEJAN
`
+T rS (k) S (`) (α~r , α~s )(hτqp α~r , α~s i + hα~r , τqp α~s i)||α||−2 (1 − δn−1
)
hkdWnn − 2
n−1
X
dWii , dWpq i
i=1
+T rS
(`)
S
(k)
h`dWnn
~
~
m
~
m
~
−2
k
`
`
(α~`, αm
(1 − δn−1
)
~ )(hτh α , α i + hα , τh α i)||α||
−2
j
n−1
X
j
dWii , dWjh i
i=1
+S
(k)
~
~
(`)
~
~
(α~`, αm
(α~r , α~s )(hτhj α` , αm
i + hα` , τhj , αm
i)(hτqp α~r , α~s i
~ )S
`
+ hα~r , τqp α~s i)(1 − δn−1
)
k
· (1 − δn−1
)||α||−4 hdWjh , dWpq i
=
µ2
T rS (k) T rS (`) [k · `(n − 1) + 2(k + `)(n − 1) + 4(n − 1)]dt
n(n + 2)
X ip
µ2
np
`
+
T rS (k) S (`) (α~r , α~s )(hτpp α~r , α~s i + hα~r , τpp α~s i)||α||−2 (1 − δn−1
)[kCnp
−2
Cip ]dt
n(n + 2)
n−1
i=1
X ij
µ2
~
j ~
j m
nj
m
~
~
−2
k
`
`
+
T rS (`) S (k) (α~`, αm
(1 − δn−1
)[`Cnj
−2
Cij ]dt
~ )(hτj α , α i + hα , τj α i)||α||
n(n + 2)
n−1
i=1
µ2
~
~
(k)
~
~
S (k) (α~`, αm
(α~r , α~s )(hτhj α` , αm
i + hα` , τhj αm
i)(hτqp α~r , α~s i
+
~ )S
n(n + 2)
k
+ hα~r , τqp α~s i)(1 − δn−1
)||α||−4 [(n + 1)δqh δpj − δjh δpq − δph δjq ]dt
We now proceed singly through the last three terms. Using the invariance of our
expressions under a change of basis we may take the ui to be the unit principal
curvature directions. Then
(A.9)
~
`
S (k) (α~`, αm
~ ) = λ~
~
` δm
S (`) (α~r , α~s ) = λ~r δ~s~r
hτqp α~r , α~s iδ~s~r = δqp δ~s~r if and only if p ∈ ~r , = 0
~
~
`
hτhj α` , αm iδm
~
=
j ~
`
δh
δm
~
if and only if j ∈ ~
`,= 0
otherwise,
otherwise.
Thus,
(A.10)
X ip
µ2
np
`
T rS (k) S (`) (α~r , α~s )(hτpp α~r , α~s i + hα~r , τpp α~s i)(1 − δn−1
)||α||−2 [kCnp
−2
Cip ]dt
n(n + 2)
n−1
i=1
µ2
p
`
n p
=
T rS (k) λ~r hτp α~r , α~r i(1 − δn−1
)[k(nδpn − δn
δp ) − 2]dt
n(n + 2)
2(k + 2)(n − ` − 1)µ2
=−
T rS (k) T rS (`) dt.
n(n + 2)
Similarly,
(A.11)
X ij
µ2
~
j ~
j m
nj
m
~
~
−2
k
`
`
T rS (`) S (k) (α~`, αm
(1 − δn−1
)(`Cnj
−2
Cij )dt
~ )(hτj α , α i + hα , τj α i)||α||
n(n + 2)
n−1
i=1
−2(` + 2)(n − k − 1)µ2
=
T rS (k) T rS (`) dt.
n(n + 2)
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
29
Finally, the last term is
(A.12)
4µ2
~ ~
`
λ~λ~r hτhj α` , α` ihτqp α~r , α~r i(1 − δn−1
)[(n + 1)δqh δpj − δjh δpq − δph δjq ]dt
n(n + 2) `
4µ2
~ ~
~ ~
k
=(1 − δn−1
)
λ~`λ~r [(n + 1)hτhj α` , α` ihτhj α~r , α~r i − hτjj α` , α` ihτpp α~r , α~r i
n(n + 2)
k
(1 − δn−1
)
~
~
−hτhj α` , α` ihτjh α~r , α~r i]dt
4µ2
~ ~
~ ~
k
λ~λ~r [nhτjj α` , α` ihτjj α~r , α~r i − hτjj α` , α` ihτpp α~r , α~r i]dt
)
=(1 − δn−1
n(n + 2) `
=(1
k
− δn−1
)
4µ2
λ~λ~r [nδj6∈~`δj6∈~r − δj6∈~`δp6∈~r ]dt
n(n + 2) `
(δj6∈~` =
1, when j 6∈ ~
`,
)
0, otherwise
4nµ2
4(n − k − 1)(n − ` − 1µ2 )
k
=(1 − δn−1
)(
λ~λ~r · δj6∈~`δj6∈~r dt −
T rS (k) T rS (`) dt).
n(n + 2) `
n(n + 2)
However, by summing on j we see that
(A.13)
λ~`λ~r δj6∈~`δj6∈~r =
n−`−1
X
mλ~`λ~r
m=0
rc |=m
|~
`c ∩~
which is a symmetric homogeneous polynomial in λ1 , . . . , λn−1 of degree k + ` so
we seek an expansion for (A.13) of the form
X
(n−k−1)∧`
(A.14)
d(n, (k, `); j)T rS (k+j) T rS (`−j) .
j=0
The principal term in T rS (k+j) T rS (`−j) (which doesn’t appear in
T rS (k+m) T rS (`−m) for m > j) is
In
λ21 . . . λ2`−j λ`−j+1 . . . λk+j .
Pn−`−1
r
m=0 mλ~
` λ~
|~
`∩~
r |=m
this term appears (n − 1 − k − j)
In T rS (k+r) T rS (`−r) for r < j this term appears
P
expansion for n−`−1
mλ~`λ~r
m=0
k−`+2j
j
k−`+2j
j−r
times.
times. Thus in the
rc |=m
|~
`c ∩~
d(n, (k, `); 0) = (n − 1 − k)
k − ` + 2j X
k − ` + 2j −
d(n, (k, `); r)
,j ≥ 1.
j
j−r
j−1
d(n, (k, `); j) = (n − 1 − k − j)
r=0
Putting (A.10), (A.11) and (A.12) together results in
(A.15)
µ2
T rS (k) T rS (`) [(k` + 2(k + `) + 4)(n − 1) − 2(k + 2)(n − ` − 1)
n(n + 2)
− 2(` + 2)(n − k − 1) − 4(n − ` − 1)(n − k − 1)]
[
X
(n−k−1)∧`
+ (1
k
− δn−1
)
4µ2
n(n + 2)
j=0
d(n, (k, `); j)T rS (k+j) T rS (`−j) ]dt.
30
M. CRANSTON AND Y. LEJAN
The quadratic variation term in hdT rS (k) , dT rS (`)i, k ≥ `, arising from dB is,
Q(n, (k, `))dt
=h
k
X
~
m
~
`
S (k−1) (α~`p , αm
~ i )hα , α ihdB(u`p , umi ), νi,
i,p=1
X̀
S (`−1) (α~rq , α~sj )hα~r , α~s ihdB(urq , usj ), νii||α||−4
j,q=1
nn
= C1122
k
X
X̀
(`−1)
S (k−1) (α~`p , αm
(α~rq , α~sj )[hu`p , umi ihurq , usj i
~ i )S
i,p=1 j,q=1
~
~
+ hu`p , urq ihumi , usj i + hu`p , usj ihurq , umi i]hα` , αm
ihα~r , α~s i||α||−4 dt
and taking advantage of the invariance of this expression under a change of basis,
we take ui to be the ith unit principal direction of curvature. Then
~
~
m
~
`
` i
S (k−1) (α~`p , αm
~ i )hα , α i = λ~
~ δp
`p δm
S `−1 (α~rq , α~sj )hα~r , α~s i = λ~rq δ~s~r δqj
~
r
` ~
[hu`p , umi ihurq , usj i + hu`p , urq ihumi , usj i + hu`p , usj ihurq , umi i]δm
s
~ δ~
`
r
`
`
r
~
r
` ~
= (δmpi δsjq + δrpq δsmj i + δspj δmqi )δm
s.
~ δ~
Thus,
Q(n, (k, `)) =
nn
C1122
k X̀
X
`
λ~` λ~rq (1 + 2δrpq )dt
p
p=1 q=1
nn
= C1122
(3
k
X
X̀
λ~` λ~rq +
k
X
X̀
p
p=1 q=1
`p =rq
λ~` λ~rq ).
p
p=1 q=1
`p 6=rq
Since Q(n, (k, `)) is a symmetric homogeneous polynomial of degree k + ` − 2 in
λ1 , . . . , λn−1 , it may be written
X
(n−k)∧(`−1)
Q(n, (k, `)) =
nn
C1122
a(n, (k, `); j)T rS (k+j−1) T rS (`−j−1) .
j=1
We now employ some elementary counting arguments to compute the coefficients
a(n, (k, `); j).
In T rS (k+j−1) T rS (`−j−1) the term
(A.16)
λ21 . . . λ2`−j−1 λ`−j . . . λk+j−1
appears
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
31
but no such term (with ` − j − 1 squares) appears in T rS (k+m−1)T rS (`−m−1) for
m > j. However, if r < j,
λ21 . . . λ2`−j−1 λ`−j . . . λk+j−1
(A.17)
k−`+2j
j−r
appears
times in T rS (k+r−1) T rS (`−r−1) .
We now count the number of appearances of (A.16) in Q(n, (k, `)).
X̀
3
λ~`p λ~rq we must fill in the empty spaces in λ~r and λ~`
From
k
X
p=1 q=1
`p =rq
λ~r = λ1 . . . λ`−j−1 − · · · −
(A.18)
λ~` = λ1 . . . λ`−j−1 − · · · −` · · · −k
with the terms λ`−j , . . . , λk+j−1 , λx , λx , where x = `p = rq . Notice λt appears
before λu only if t < u. One λx must go in λ~r the other in λ~`. If the order is not
to be violated, this forces x > k − 1 + j and λx must end each of the terms λ~r and
λ~`. There are (n − 1) − (k − 1 + j) = n − k − j such choices for x. This leaves
k − 1 + j − (` − j − 1)
j
k − ` + 2j
=
j
remaining possibilities for filling in the empty spaces in (A.18) in λ~r and λ~`. Note
that if we interpret 00 = 1 this is still correct for k = ` and j = 0. Thus
(A.19)
3
k X̀
X
λ~`p λ~rq
p=1 q=1
`p =rq
k − ` + 2j
has 3(n − k − j)
j
terms of the form λ21 . . . λ2`−j−1 λ`−j . . . λk+j−1 .
In
k X̀
X
λ~`p λ~rq we must fill in the blanks of
p=1 q=1
`p 6=rq
λ~r = λ1 . . . λ`−j−1 − · · · −
λ~` = λ1 . . . λ`−j−1 − · · · −` · · · −k
with λ`−j , . . . , λk+j−1 , λx , λy with x 6= y, λx going into λ~r , λy going into λ~`.
If ` − j ≤ x, y ≤ k + j − 1, then λx and λy appear on the list λ`−j , . . . , λk+j−1 .
Since identical terms can’t appear twice in λ~r or λ~` the twin of λx must go in λ~`,
the twin of λy must go in λ~r . There are thus k−`+2j−2
ways to fill in the spaces
j−1
once λx and λy are given with ` − j ≤ x, y ≤ k + j − 1. There are (k + j − 1 − (` −
32
M. CRANSTON AND Y. LEJAN
j − 1))(k + j − 1 − (` − j − 1) − 1) choices for the ordered pair (x, y). Notice that
when j = 0 there are no such terms. Thus
(A.20)
` − j ≤ x, y ≤ k + j − 1 accounts for
k − ` + 2j − 2
(k − ` + 2j)(k − ` + 2j − 1)
j−1
terms of the form λ21 . . . λ2`−j−1 λ`−j . . . λk+j−1 in Q(n, (k, `)) (interpreting
0.)
z
−1
=
If ` − j ≤ x ≤ k + j − 1 < y ≤ n − 1, then λy must
terminate thesequence λ~`
k − ` + 2j − 1
and the twin of λx must go into λ~`. There are then
ways to fill
j
in the spaces in (A.18) once such a pair (x, y) is given. We interpret this expression
as 0 when k = ` and j = 0. There are (k − ` + 2j)(n − k − j) such pairs so we have
(A.21)
corresponding to ` − j ≤ x ≤ k +j − 1 < x ≤ n −
1
k − ` + 2j − 1
there are (k − ` + 2j)(n − k − j)
j
terms of the form λ21 · · · λ2`−j−1 λ`−j · · · λk+j−1 in
Q(n, (k, `)) .
If ` − j ≤ y ≤ k + j − 1 < x ≤ n − 1, then
the twin of λy must go in λ~r . When
k − ` + 2j − 1
j = 0, this is impossible. There are thus
ways to fill the spaces
j −1
in (A.18) once (x, y) is given and (k − ` + 2j)(n − k − j) such pairs so
(A.22)
corresponding to ` − j ≤ y ≤ k +j − 1 < x ≤ n− 1,
k − ` + 2j − 1
there are (k − ` + 2j)(n − k − j)
j−1
2
2
terms of the form λ1 · · · λ`−j−1 λ`−j · · · λk+j−1 in
z
Q(n, (k, `)), for j ≥ 0, (interpreting −1
= 0.)
k − ` + 2j
Finally, when k + j − 1 < x, y ≤ n − 1, there are
ways to fill in the
j
spaces in (A.18) once (x, y) is given and (n − k − j)(n − k − j − 1) such pairs (x, y).
Thus
(A.23)
corresponding to k + j −1 < x, y ≤ n− 1, there are
k − ` + 2j
(n−k−j)(n−k−j −1)
terms of the
j
form λ21 · · · λ2`−j−1 λ`−j · · · λk+j−1 in Q(n, (k, `)),
interpreting 00 = 1.
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
33
Totalling (A.18)–(A.23), we see
(A.24)
Q(n, k) has
b(n, (k, `); j) = 3(n − k − j)
k − ` + 2j
j
k − ` + 2j − 2
+ (k − ` + 2j)(k − ` + 2j − 1)
j−1
k − ` + 2j − 1
+ (k − ` + 2j)(n − k − j)
j
k − ` + 2j − 1
+ (k − ` + 2j)(n − k − j)
j−1
k − ` + 2j
+ (n − k − j)(n − k − j − 1)
j
z
= 0, −1
terms with exactly ` − j − 1 squares. Notice that we interpret −1
0 = 0,
0
= 1. Thus, b(n, (k, `); 0) = (n − k)(n − k − 2). Upon decomposing Q(n, (k, `))
0
into sums of expressions with exactly ` − j − 1 squares summed over j, we are lead
to, using (A.16) and (A.23),
X
(n−k)∧(`−1)
Q(n, k) =
nn
C1122
a(n, (k, `); j) Tr S (k+j−1) Tr S (`−j−1)
j=0
where
a(n, (k, `); j) = b(n, (k, `); j) −
j−1 X
k − ` + 2j
m=0
j−m
a(n, (k, `); m) ,
a(n, (k, `); 0) = b(n, (k, `); 0) = (n − k)(n − ` + 2) .
From Theorems A.2 and A.3, we get immediately
Theorem A.4. Suppose Φt is the isotropic, measure preserving flow on Rn given
by (1.3). Let M be a smooth hypersurface in Rn with x ∈ M. If (λ1 (t), . . . , λn−1 (t))
are the principal curvatures of Φt (M) at Φt (x), then
Tr S (k) =
X
λ~`
~
`∈Ik
and (Tr S (1), Tr S (2), . . . , Tr S (n−1)) is an (n − 1)-dimensional diffusion. The
34
M. CRANSTON AND Y. LEJAN
generator is given by
X
Lf(x1 , . . . , xn−1 ) =
1≤`≤k≤n−1
1 1
k
+ (1 − δ` )
2 2
µ2
[((k` + 2(k + `) + 4)(n − 1)
n(n + 2)
− 2(k + 2)(n − ` − 1) − 2(` + 2)(n − k − 1)
− 4(n − ` − 1)(n − k − 1))xk x`
+ 1−
k
δn−1
X
(n−k−1)∧`
4n
d(n, (k, `); j)xk+j x`−j ]
j=0
X
(n−k)∧(`−1)
nn
+C1122
j=0
+
n−1
X
k=1


∂2f
a(n, (k, `); j)xk+j−1 x`−j−1
 ∂xk ∂x`
(n + 1)k(n − k))µ2
∂f
xk
2n(n + 2)
∂xk
where, for k ≥ `
d(n, (k, `); 0) = (n − 1 − k)
d(n, (k, `); j) = (n − 1 − k − j)
k − ` + 2j
j
b(n, (k, `); j) = 3(n − k − j)
k − ` + 2j
j
−
d(n, (k, `); r)
k − ` + 2j − 1
j
k − ` + 2j
j
+
a(n, (k, `); j) = b(n, (k, `); j) −
r=0
nn
C1122
=
(n + 3)µ4
,
n(n + 2)(n + 4)
and x−1 = x0 = 1.
a(n, (k, `); r)
k − ` + 2j
j −r
,j > 0
,j ≥ 1
k − ` + 2j − 2
j −1
k − ` + 2j − 1
j −1
; j ≥ 0 , with
a(n, (k, `); 0) = b(n, (k, `); 0) = (n − k)(n − ` + 2),
j−1
X
k − ` + 2j
j−r
r=0
+ (n − k − j)(n − k − j − 1)
j−1
X
+ (k − ` + 2j)(k − ` + 2j − 1)
+ (k − ` + 2j)(n − k − j)
z −1 0
=
,
= 1,
−1
0
0
GEOMETRIC EVOLUTION UNDER ISOTROPIC STOCHASTIC FLOW
35
Remarks.
(1) Straightforward computations show that for n = 3
k = ` = 1;
d(3, (1, 1); 0) = 1
d(3, (1, 1); 1) = −2
a(3, (1, 1); 0) = 8
k = 2, ` = 1;
a(3, (2, 1); 0) = 4
k = 2, ` = 2;
a(3, (2, 2); 0) = 3
a(3, (2, 2); 1) = −4
Thus, when M is a hypersurface in R3 , the pair ( Tr S (1), Tr S (2)) is a
diffusion with generator
16µ4 ∂ 2 f
1 µ2
2
Lf(κ, m) =
(14κ − 24m) +
(κ, m)
2 15
35
∂κ2
16µ2
8µ4
∂2f
+
κm +
κ
(κ, m)
15
35
∂κ∂m
2
1 32µ2 2 2µ4
∂ f
2
+
m +
(3κ − 4m)
(κ, m)
2
15
35
∂m2
4µ2
∂f
4µ2 ∂f
κ (κ, m) +
mm
(κ, m)
+
15 ∂κ
15
∂m
which agrees with Theorem 2.9.
44
4
(2) For n = 4, a = µ242 , b = C1122
= 7µ
192 ,
1
∂2f
(a(19x2 − 32y) + 15b) 2
2
∂x
1
∂2 f
+ (a(44y2 − 32xz) + b(8x2 − 4y)) 2
2
∂y
2
1
∂ f
+ (a75z 2 + b(3y2 − 4xz)) 2
2
∂z
∂2f
∂2f
+ (a(22xy − 48z) + b10x)
+ (a25xz + b5y)
∂x∂y
∂x∂z
∂2f
+ (a50yz + b(4xy − 6z))
∂y∂z
15 ∂f
∂f
15 ∂f
+a x
+ a10y
+a z
.
2 ∂x
∂y
2 ∂z
Lf(x, y, z) =
(3) For any n, no proper subset of {Tr S (1), . . . , Tr S (n−1)} can be a diffusion.
36
M. CRANSTON AND Y. LEJAN
(4) The authors intend to resolve such issues as transience or recurrence for
higher dimensional cases.
(5) Since x = λ1 + λ2 + λ3 , y = λ1 λ2 + λ1 λ3 + λ2 λ3 , z = λ1 λ2 λ3 , Newton’s
inequalities, Hardy-Littlewood- Polya (1973), imply x2 ≥ 3y, y2 ≥ 3xz with
equality if and only if λ1 = λ2 = λ3 . As in the case n = 3, the diffusion
matrix degenerates on λ1 = λ2 = λ3 and one expects that the process will
not reach this set at positive times.