General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Semiclassical analysis of a random walk on a
manifold
L. Michel
(joint work with G. Lebeau)
Laboratoire J.-A. Dieudonné
Université de Nice
April 4, 2008
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
General framework
Let (M, g ) be a smooth, compact, connected Riemannian manifold
of dimension d, equipped with its canonical volume form dg x. We
denote
dg (x, y ) the Riemannian distance on M × M.
for x ∈ M and h > 0, B(x, h) = {y , dg (x, y ) ≤ h}
R
|B(x, h)| = B(x,h) dg y
For any given h > 0, let Th be the operator acting on continuous
functions on M
Z
1
(Th f )(x) =
f (y )dg y
|B(x, h)| B(x,h)
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
The Markov kernel
We denote by Kh the kernel of Th , which is given by
Kh (x, y )dg y =
1{dg (x,y )≤h}
dg y
|B(x, h)|
for any x ∈ M, Kh (x, y )dg y is a probability measure on M
(hence, Kh is a Markov kernel)
Kh is associated to the following natural random walk (Xn ) on
M: if the walk is at x, then it moves to a point y ∈ B(x, h)
with a probability given by Kh (x, y )dg y . For f ∈ C 0 (M),
Th (f )(Xn ) = E (f (Xn+1 )|Xn ). Or equivalently, for A, B
measurable,
P(Xn+1 ∈ A and Xn ∈ B) = E (Th (1A )(Xn )1B (Xn )).
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Basic properties
We define dνh = |B(x,h)|
dg x where Zh is chosen so that dνh is a
h d Zh
probability measure. We have the following facts:
Th is self-adjoint on L2 (M, dνh ).
Th is compact
For all p ∈ [1, ∞], kTh kLp →Lp = 1.
Hence, the spectrum of Th is made of eigenvalues and {0} is the
only possible accumulation point. We denote
1 = µ0 (h) ≥ µ1 (h) ≥ µ2 (h) ≥ ... ≥ µk (h)... > 0
the positive eigenvalues, (ekh )k the associated normalized
eigenfunctions.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Reference operator
We use the following notations:
∆g is the (negative) Laplace-Beltrami operator on (M, g ).
0 = λ0 < λ1 ≤ λ2 ≤ ... ≤ λn ≤ ... denotes the spectrum of
the self adjoint operator −∆g on L2 (M, dg x).
for ξ ∈ Rd
Z
1
Gd (ξ) =
e iy ξ dy
cd |y |≤1
where cd = volume of the unit ball inRd .
the function Gd is radial and we let Γd be such that
Gd (ξ) = Γd (|ξ|2 ). Then, Γd is analytic and near s = 0 we have
Γd (s) = 1 −
s
+ O(s 2 )
2(d + 2)
We denote Γd,h = Γd (−h2 ∆g ).
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Localization an Weyl asymptotics
Resolvent
Theorem 1 (part 1)
Let h0 > 0 be small. There exist γ < 1 such that for any h ∈]0, h0 ]
one has Spec(Th ) ⊂ [−γ, 1] and 1 is a simple eigenvalue of Th .
For any given L > 0, there exists C such that for all h ∈]0, h0 ] and
all k ≤ L, one has
|
1 − µk (h)
λk
−
| ≤ Ch2
2
h
2(d + 2)
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Localization an Weyl asymptotics
Resolvent
Theorem 1 (part 2)
Let N(a, h) = card(Spec(Th ) ∩ [a, 1]). For any δ ∈]0, 1[, there exist
C > 0 s.t. for any h ∈]0, h0 ] and any τ ∈ [0, (1 − δ)h−2 ], we have
Z
d−1
2
−d
|N(1 − τ h , h) − (2πh)
dxdξ| ≤ C (1 + τ ) 2
Γd (|ξ|2x )∈[1−τ h2 ,1]
In particular, one has
N(1 − τ h2 , h) ≤ C (1 + τ )d/2
Denote ekh the eigenfunction of Th associated to µk (h) ∈ [δ, 1],
and set τk (h) = h−2 (1 − µk (h)), then
kekh kL∞ ≤ C (1 + τk (h))d/4 kekh kL2 .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Localization an Weyl asymptotics
Resolvent
Let |∆h | be the positive, bounded, self adjoint operator on
L2 (M, dνh ) defined by
1 − Th =
h2
|∆h |
2(d + 2)
Let F1 and F2 be the two closed subset of C,
F1 = {z, dist(z, spec(−∆g )) ≤ ε}
F2 = {z, Re(z) ≥ A, |Im(z)| ≤ εRe(z)}
with ε > 0 small and A > 0 large. Let F = F1 ∪ F2 and U = C \ F .
Theorem 2
There exists C , h0 > 0 such that for all h ∈]0, h0 ], and all z ∈ U
k(z − |∆h |)−1 − (z + ∆g )−1 kL2 →L2 ≤ Ch2
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Case of the geodesic random walk
The Metropolized operator
Convergence to stationary measure
As Th 1 = 1 and Th self-adjoint on L2 (M, dνh ) then dνh is a
stationary measure for Kh (x, y )dg y , ie: for all f ∈ C (M)
Z
Z
f (x)dνh (x) =
Th f (x)dνh (x)
M
M
it follows from general theory of Markov chains that for all
x ∈ M, Khn (x, y )dg y → dνh (y ), when n → ∞, where
Khn (x, y )dg y denotes the kernel of Thn .
Question: how fast does it converge?
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Case of the geodesic random walk
The Metropolized operator
Total variation estimate
The total variation distance between two probability measures µ, ν
is defined by
kµ − νkTV =
|µ(A) − ν(A)|
sup
A measurable
Theorem 3
Let h0 > 0 small. There exists C > 0 such that for all h ∈]0, h0 ]
the following holds true :
0
2
e −γ (h)nh ≤ 2 sup kKhn (x, y )dg y −dνh kTV ≤ Ce −γ(h)nh
2
for all n
x∈M
Here γ(h), γ 0 (h) are positive functions s.t. γ(h) ' γ 0 (h) '
when h → 0.
L. Michel (joint work with G. Lebeau)
λ1
2(d+2)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Case of the geodesic random walk
The Metropolized operator
The Metropolized operator
If one is interested in sampling the probability measure
dµM = dg x/Vol(M), we can use a kernel modified according to
the Metropolis strategy. Remark that dµM = ρh (x)dνh with
h d Zh
ρh (x) = Vol(M)|B(x,h)|
. We define
Mh (x, dy ) = mh (x)δy =x + Kh (x, y )dg y
where the functions mh and Kh are defined by
ρ (y ) 1
|B(x, h)| d (x,y )≤h
h
,1 = g
min
,1
Kh (x, y ) := Kh (x, y )min
ρh (x)
|B(x, h)|
|B(y , h)|
Z
mh (x) = 1 −
Kh (x, y )dg y
M
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Case of the geodesic random walk
The Metropolized operator
Then, Mh (x, dy ) is still a Markov kernel and the operator
Z
Mh (f )(x) =
f (y )Mh (x, dy )
M
is self-adjoint on the space L2 (M, dg x), and therefore dµM is
invariant for Mh .
Theorem 4
Let h0 > 0 small. There exists C > 0 such that for all h ∈]0, h0 ]
the following holds true :
0
2
e −γ (h)nh ≤ 2 sup kMhn (x, dy )−dµM kTV ≤ Ce −γ(h)nh
2
for all n
x∈M
Here γ(h), γ 0 (h) are two positive functions such that
λ1
γ(h) ' γ 0 (h) ' 2(d+2)
when h → 0.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Structure of the operator
Let x = (x1 , . . . , xd ) be a local system of coordinates and gi,j (x)
the metric g . In these coordinates, the geodesic ball of radius r
centred at x is given by
X
B(x, r ) = {x + u,
ki,j (x, u)ui uj ≤ r 2 }
where (ki,j (x, u)) is a symmetric matrix smooth w.r.t. (x, u) such
that ki,j (x, 0) = gi,j (x). In these local coordinates, we have
Z
p
1
Th f (x) =
f (x + u) det(g (x + u))du
|B(x, h)| t uk(x,u)u≤h2
Z
hd
f (x + hm(x, hv )v )ρ(x, hv )dv
=
|B(x, h)| |v |≤1
where m(x, w ) is a symmetric and positive matrix s.t.
m(x, 0) = g −1/2 (x) and ρ(x, 0) = 1.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Computation of the symbol
We define the symbol of Th by σ(Th )(x, ξ, h) = e −ixξ/h Th (e ixξ/h ).
Then,
Z
hd
t
σ(Th )(x, ξ, h) =
e i ξ.m(x,hv )v ρ(x, hv )dv
|B(x, h)| |v |≤1
In particular, since m(x, 0) = g −1/2 (x) and ρ(x, 0) = 1, one has
σ(Th )(x, ξ, 0) = Γd (|ξ|2x )
Remark
∀α, β, ∃Cα,β independent of h such that
|∂xα ∂ξβ σ(Th )(x, ξ, h)| ≤ Cα,β (1 + |ξ|)|α|
Hence, σ(Th )(x, ξ, h) is not in a good symbol class.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Class of operator
Definitions
a(x, ξ, h) ∈ S −∞ if |∂xα ∂ξβ a(x, ξ, h)| ≤ Ck (1 + |ξ|)−k , ∀k ∈ N.
Scl−∞ = {a ∈ S −∞ , a has an expansion in powers of h}.
A family of operators (Rh )h∈]0,1] is smoothing if for all s, t, N
we have
kRh kH s →H t ≤ Cs,t,N hN , ∀h ∈]0, 1]
Ah ∈ Ecl−∞ if Ah = a(x, hDx , h) + Rh with a ∈ Scl−∞ and Rh
smoothing.
A family of operators (Ch )h∈]0,1] on D0 (M), belongs to Eecl0 if
Ch is bounded uniformly in h on L2 (M) and for any Φ0 ∈ C0∞ :
Φ0 (−h2 ∆g )Ch
and Ch Φ0 (−h2 ∆g )
L. Michel (joint work with G. Lebeau)
belongs to
Ecl−∞
Semiclassical analysis of a random walk on a manifold
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Lemma 1
Let h0 small. For h ∈]0, h0 ], the operator Th belongs to the class
Eecl0 .
Proof. Let φ0 ∈ C0∞ .
Observe that φ0 (−h2 ∆g ) ∈ Ecl−∞ . As Th is bounded on
C k (M) for all k, it suffices to show that for all a ∈ Scl−∞ ,
Th a(x, hDx ) ∈ Ecl−∞ .
In local coordinates, we have Th a(x, hDx ) = b(x, hDx ) with
hd
b(x, ξ, h) =
|B(x, h)|
Z
ei
t ξ.m(x,hv )v
|v |≤1
a(x + hm(x, hv )v , ξ, h)ρ(x, hv )dv
thanks to the cut off function a, it is clear that b ∈ Scl−∞ .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Lemma 2
Let Φ0 ∈ C0∞ ([0, ∞[), then (Th − Γd,h )Φ0 (−h2 ∆g ) = h2 Ah with
Ah ∈ Ecl−∞ . Its principal symbol, σ0 (Ah ), satisfies near ξ = 0
S(x)
|ξ|2x (Γ00d (0) − Γ0d (0)2 )
σ0 (Ah )(x, ξ) =
3
Γ00 (0)
+ d
Ric(x)(ξ, ξ) Φ0 (|ξ|2x ) + O(ξ 3 )
3
where Ric(x) and S(x) are the RicciP
tensor and the scalar
curvature at x. Moreover, denoting
hk ak (x, ξ) the symbol of
Ah , we have ak (x, 0) = 0 for all k ≥ 0.
In particular for f ∈ H 2 (M) we have k(Th − Γd,h )f kL2 ≤ Ch4 kf kH 2 .
It is one of the main ingredient in the proof of Theorem 2.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Proof. Let m0 ∈ M be fixed. We use the geodesic coordinates x,
centred at m0 . Thanks to the formula giving the principal symbol
of a pseudo conjugate by a change of variable, it suffices to
compute σ(Th ) in x = 0. In the geodesic coordinates we have
p
det(g )(y ) = 1 − 61 Ric(y , y ) + O(y 3 ),
m(0, v ) = Id, ρ(0, v ) = 1
Consequently, |B(0, h)| = hd cd (1 +
Γ0d (0)
2
3 S(0)h )
+ O(h3 ) and
Z
p
hd
σ(Th )(0, ξ, h) =
e iξ.v det(g )(hv )dv
|B(0, h)| |v |≤1
Γ0 (0)
1X
∂ 2 Gd
= Γd (|ξ|2 ) + h2 − Γd (|ξ|2 ) d
S+
Ricj,k
(ξ) + O(h3 )
3
6
∂ξj ∂ξk
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Control of high frequencies
Lemma 3
Let χ ∈ C0∞ (R) be equal to 1 near 0. There exists h0 > 0, C > 0
such that
−h2 ∆g
C
kTh (1 − χ)(
)kL2 →L2 ≤ √
s
s
for all h ∈]0, h0 ] and all s ≥ 1 .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Proof. Thanks to the support properties of χ we have
k(
−h2
−h2
∆g )−1/2 (1 − χ)(
∆g )kL2 →L2 = O(1)
s
s
Hence, it suffices to prove that
kTh (−h2 ∆g )1/2 kL2 →L2 = O(1)
On the other hand, in local coordinates we have
Z
−1
−d
Th f (x) = Ch
e ih (x−y )ξ b(x, ξ, h)f (y )dy
with
Z
b(x, ξ, h) =
e ih
−1 t ξm(x,hv )v
ρ(x, hv )dv ,
|v |≤1
m(x, 0) = g (x)−1/2 and ρ(x, 0) = 1.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Using the stationary phase method, we can compute the symbol
b(x, ξ, h):
b(x, ξ, h) = e iΦ+ (x,ξ,h) τ+ (x, ξ, h) + e iΦ− (x,ξ,h) τ− (x, ξ, h) + n(x, ξ, h)
where n ∈ S −∞ , τ± are two symbols of degree −(d + 1)/2 and the
phase φ± satisfy
φ± (x, ξ, h) = |ξ|x + O(h)
Therefore, Th is locally the sum of two quantized canonical
transformation with symbol of degree −(d + 1)/2 and phases close
to (x − y )ξ + h|ξ|x . In particular , as (d + 1)/2 ≥ 1 we gain one
h-derivative.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Estimate on eigenfunctions(I)
Let δ ∈]0, 1[ and for h > 0, e h ∈ L2 and zh ∈ [δ, 1] be such that
ke h kL2 = 1 and (Th − zh )e h = 0.
Lemma 4
There exists h0 > 0, and for all j ∈ N there exists Cj > 0, such
that, the following inequality holds true
sup k(−h2 ∆g + 1)j e h kL2 ≤ Cj
h∈]0,h0 ]
Proof. We localize as before. As e h = z1h Th e h and Th is the sum
of two quantized canonical transformation of degree
−(1 + d)/2 ≤ −1, we gain one h-derivative, and we can iterate
this argument.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Estimate on eigenfunctions(II)
Lemma 5
Let s1 be such that |Γd (s)| ≤ δ/2 for s ≥ s1 − 1 and let
χ1 ∈ C0∞ (R+ ) be equal to 1 on [0, s1 ]. Then,
(1 − χ1 (−h2 ∆g ))e h = OC ∞ (h∞ )
Proof. Recall that zh ∈ [δ, 1]. We write 1 − χ1 = χ2 + χ3 with χ3
supported in [s2 , ∞[, s2 >> 1. For j = 2, 3 we have
δ
χj (−h2 ∆g )(zh − Th )χj (−h2 ∆g ) ≥ χj (−h2 ∆g )
3
(apply Lemma 2 in the case j=2 and Lemma 3 in the case j=3).
Hence, kχj (−h2 ∆g )e h kL2 = O(h∞ ). To gain regularity, use
interpolation and the preceding Lemma.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Estimate on eigenfunctions(III)
Lemma 6
For all j ∈ N, there exists Cj such that for all h ∈]0, h0 ], we have
ke h kH j (M) ≤ Cj (1 + h−2 (1 − zh ))j/2
Proof. Let ẽ h = χ1 (−h2 ∆g )e h . From Lemma 5 and Lemma 2 we
have
O(h∞ ) = (Th − zh )ẽ h = ((Γd − 1)(−h2 ∆g ) + O(h2 ) + (1 − zh ))ẽ h
Moreover, there exists a smooth function Fd non vanishing on
[0, s1 + 2] s.t. (Γd − 1)(s) = sFd (s). Hence,
−∆g Fd (−h2 ∆g )ẽ h = (O(1) +
1 − zh h
)ẽ
h2
This shows that kẽ h kH j ≤ C (1 + h−2 (1 − zh ))j/2 kẽ h kL2 .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Localization of the spectrum
Recall that (−∆g − λk )ek = 0, (Th − µk (h))ekh = 0 and
1 − Th =
h2
|∆h |
2(d + 2)
Let k ∈ N be fixed and χ ∈ C0∞ (R+ ). Then
for h > 0 small enough, ek = χ(−h2 ∆g )ek
Lemma 2=⇒ (Th − Γd (−h2 ∆g ))ek = O(h4 ).
As Γd (−h2 ∆g )ek = (1 + h2 Γ0d (0)λk )ek + O(h4 )ek and Th is
selfadjoint on L2 (M, dνh ) it follows that for h > 0 small enough
card Spec(|∆h |) ∩ [λk − C0 h2 , λk + C0 h2 ] ≥ mk .
where mk is the multiplicity of λk .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Localization of the spectrum
On the other hand, if e h ∈ L2 (M) satisfies |∆h |e h = τ h e h with τ h
bounded, we have
e h = χ1 (−h2 ∆g )e h + OC ∞ (h∞ ) (thanks to Lemma 5 ).
hence, Lemma 2 =⇒ (Th − Γd (−h2 ∆g ))e h = O(h4 ) and
dist(τ h , Spec(−∆g )) = O(h2 )
Hence, it remains to show that for h > 0 small we have
card Spec(|∆h |) ∩ [λk − C0 h2 , λk + C0 h2 ] ≤ mk .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Multiplicity of the eigenvalues
Let p ≥ mk and e1h , . . . , eph satisfy (|∆h | − τl (h))elh = 0 with
τl (h) ∈ [λk − C0 h2 , λk + C0 h2 ], (elh ) orthonormal for the scalar
product h., .iL2 (M,dνh ) .
By Lemma 6, we have
kelh kH j ≤ Cj (1 + τl (h))j/2 , ∀j
and we can suppose that elh converges in H 2 when h → 0.
Denoting fl its limit we get −∆g fl = λk fl for all l = 1, . . . , p and
the functions fl are orthogonal for the scalar product h., .iL2 (M,dg x) .
Consequently, p ≤ mk .
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
Proof of Weyl type estimate
Use Lemma 2 and Lemma 3 to approximate the number of
eigenvalues of Th in an interval by those of Γd (−h2 ∆g ).
Use wellknow Weyl estimates on pseudodifferential operators
on a manifold.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
From Theorem 1 to Theorem 3
We can suppose that n ≥ h−2 . Recall that the total variation
between two probability µ, ν on M is given by
kµ − νkTV := sup |µ(A) − ν(A)| =
A⊂M
1
sup |µ(f ) − ν(f )|
2 kf kL∞ ≤1
Consequently,
1
sup kThn (x, dy ) − dνh kTV = kThn − Πh0 kL∞ →L∞
2
x∈M
where Πh0 is the orthogonal projector in L2 (M, dνh ) on the space of
constant functions.
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
From Theorem 1 to Theorem 3
Recall that ekh denotes the normalized eigenfunction of Th
associated to the eigenvalue µk (h) and let Πhk be the orthogonal
projector on span(ekh ). Then,
(Thn − Πh0 )e1h = µ1 (h)n e1h = (1 − h2
λ1
+ O(h4 ))n e1h
2(d + 2)
and the left inequality in Theorem 3 is straightforward.
To prove the right inequality, let δ ∈]0, 1[ and denote
Th − Π0 (h) = Th,1 + Th,2
with Th,1 =
P
1−δ≤µk ≤µ1
µk (h)Πk (h).
L. Michel (joint work with G. Lebeau)
Semiclassical analysis of a random walk on a manifold
General framework
Analysis of the spectrum
Convergence to stationary measure
Sketches of proof
Symbolic calculus of Th
The spectral theory of Th
Proof of Theorem 1
Proof of Theorems 3 and 4
From Theorem 1 to Theorem 3
Denote τk (h) = h−2 (1 − µk (h)), thanks to the eigenfunction
estimate, we have
X
n
kTh,1
kL∞ →L∞ ≤
(1 − h2 τk )n (1 + τk )α
τ1 ≤τk ≤h−2 (1−δ)
Thanks to Weyl law, we get for nh2 ≥ 1,
Z +∞
2
2
n
e −nh x (1 + x)β dx ≤ Ce −nh τ1
kTh,1 kL∞ →L∞ ≤ C
τ1
Finally, we observe that
n−1
n
kTh,2
kL2 ,L∞ ≤ kTh,2
kL2 ,L2 kTh,2 kL∞ ,L∞
≤ C (1 − δ)n << e −nh
L. Michel (joint work with G. Lebeau)
2 γ(h)
Semiclassical analysis of a random walk on a manifold