Probability Seminar, University of Warwick
An Eyring–Kramers formula
for one-dimensional parabolic SPDEs
with space-time white noise
Nils Berglund
MAPMO, Université d’Orléans
Warwick, 9 December 2015
Joint work with Barbara Gentz (Bielefeld)
Nils Berglund
nils.berglund@univ-orleans.fr
http://www.univ-orleans.fr/mapmo/membres/berglund/
Deterministic Allen–Cahn PDE
[Chafee & Infante 74, Allen & Cahn 75]
∂t u(x, t) = ∂xx u(x, t) + f (u(x, t))
.
x ∈ [0, L]
.
u(x, t) ∈ R
.
Either periodic or zero-flux Neumann boundary conditions
.
In this talk: f (u) = u − u 3 (results more general)
Energy function:
Z Lh
i
1 0 2 1
1
u (x) − u(x)2 + u(x)4 dx
V [u] =
2
2
4
0
→ min
Stationary solutions: u000 (x) = −u0 (x) + u0 (x)3
critical points of V
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
1/17
Deterministic Allen–Cahn PDE
[Chafee & Infante 74, Allen & Cahn 75]
∂t u(x, t) = ∂xx u(x, t) + f (u(x, t))
.
x ∈ [0, L]
.
u(x, t) ∈ R
.
Either periodic or zero-flux Neumann boundary conditions
.
In this talk: f (u) = u − u 3 (results more general)
Energy function:
Z Lh
i
1 0 2 1
1
u (x) − u(x)2 + u(x)4 dx
V [u] =
2
2
4
0
→ min
Stationary solutions: u000 (x) = −u0 (x) + u0 (x)3
critical points of V
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
1/17
Stationary solutions
u000 (x) = −f (u0 (x)) = −u0 (x) + u0 (x)3
.
H = 12 (u 0 )2 + 12 u 2 − 14 u 4
u0
u± (x) ≡ ±1
.
u0 (x) ≡ 0
.
Nonconstant solutions satisfying b.c.
H=
u
−1
1
(expressible in terms of Jacobi elliptic fcts)
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
2/17
1
4
Stationary solutions
H = 12 (u 0 )2 + 12 u 2 − 14 u 4
u000 (x) = −f (u0 (x)) = −u0 (x) + u0 (x)3
.
u0
u± (x) ≡ ±1
H=
u
.
u0 (x) ≡ 0
.
Nonconstant solutions satisfying b.c.
−1
1
(expressible in terms of Jacobi elliptic fcts)
.
Neumann b.c: k nonconstant solutions when L > kπ
0
.
π
2π
3π
4π
L
Periodic b.c: k families when L > 2kπ
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
2/17
1
4
Stability of stationary solutions
u000 (x) = −u0 (x) + u0 (x)3
Variational eq around u0 : ∂t vt (x) = vt00 (x) + [1 − 3u0 (x)2 ]vt (x)
Sturm–Liouville spectrum of RHS determines stability of u0
.
u± ≡ ±1: always stable (global minima of V )
.
u0 ≡ 0: always unstable, eigenvalues λk =
βkπ 2
L
−1
(Neumann b.c.: β = 1, periodic b.c.: β = 2)
0
Neumann b.c.:
Number of positive
eigenvalues
1
1
2
π
0
(= unstable directions)
2
3
3
2π
1
4
3π
2
4π
L
3
0
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
3/17
Coarsening dynamics
[Carr & Pego 89, Chen 04]
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
(Link to simulation)
9 December 2015
4/17
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos kπx
= e−(kπ/L) t cos kπx
L
L
√ Z t ∆(t−s)
√
∆t
. u̇t = ∆ut + 2ε Ẅtx
⇒ ut = e u0 + 2ε
e
Ẇx (ds)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
0
wt (x) =
XZ
k
.
u̇t = ∆ut +
⇒
⇒
0
t
−(kπ/L)2 (t−s)
e
(k)
dWs cos kπx
L
s
=:wt (x)
α
∈ H ∩ C ∀s, α <
1
2
√
2ε Ẅtx + f (ut )
Z t
√
∆t
e∆(t−s) f (us ) ds =: Ft [u]
ut = e u0 + 2ε wt +
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
5/17
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos kπx
= e−(kπ/L) t cos kπx
L
L
√ Z t ∆(t−s)
√
∆t
. u̇t = ∆ut + 2ε Ẅtx
⇒ ut = e u0 + 2ε
e
Ẇx (ds)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
0
wt (x) =
XZ
k
.
u̇t = ∆ut +
⇒
⇒
0
t
−(kπ/L)2 (t−s)
e
(k)
dWs cos kπx
L
s
=:wt (x)
α
∈ H ∩ C ∀s, α <
1
2
√
2ε Ẅtx + f (ut )
Z t
√
∆t
e∆(t−s) f (us ) ds =: Ft [u]
ut = e u0 + 2ε wt +
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
5/17
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos kπx
= e−(kπ/L) t cos kπx
L
L
√ Z t ∆(t−s)
√
∆t
. u̇t = ∆ut + 2ε Ẅtx
⇒ ut = e u0 + 2ε
e
Ẇx (ds)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
0
wt (x) =
XZ
k
.
u̇t = ∆ut +
⇒
⇒
0
t
−(kπ/L)2 (t−s)
e
(k)
dWs cos kπx
L
∈
Hs
∩
=:wt (x)
C α ∀s, α
<
1
2
√
2ε Ẅtx + f (ut )
Z t
√
∆t
e∆(t−s) f (us ) ds =: Ft [u]
ut = e u0 + 2ε wt +
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
5/17
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
Ẅtx : space–time white noise (formal derivative of Brownian sheet)
Construction of mild solution via Duhamel formula:
. u̇t = ∆ut
⇒
ut = e∆t u0
2
where e∆t cos kπx
= e−(kπ/L) t cos kπx
L
L
√ Z t ∆(t−s)
√
∆t
. u̇t = ∆ut + 2ε Ẅtx
⇒ ut = e u0 + 2ε
e
Ẇx (ds)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
0
wt (x) =
XZ
k
.
u̇t = ∆ut +
⇒
⇒
0
t
−(kπ/L)2 (t−s)
e
(k)
dWs cos kπx
L
∈
Hs
∩
=:wt (x)
C α ∀s, α
<
1
2
√
2ε Ẅtx + f (ut )
Z t
√
∆t
e∆(t−s) f (us ) ds =: Ft [u]
ut = e u0 + 2ε wt +
0
Existence and a.s. uniqueness [Faris & Jona-Lasinio 1982]
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
5/17
Stochastic partial differential equations
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
(∆ ≡ ∂xx , f (u) = u − u 3 )
∞
1 X
Fourier variables: ut (x) = √
zk (t) ei πkx/L
L k=−∞
X
√
1
(k)
⇒
dzk = −λk zk dt −
zk1 zk2 zk3 dt + 2ε dWt
L
k1 +k2 +k3 =k
(k)
dWt :
Itō SDE,
independent Wiener processes
λk = −1 + (πk/L)2
Energy functional:
∞
1
1 X
λk |zk |2 +
V [u] =
2
4L
k=−∞
⇒
dzt = −∇V (zt ) dt +
X
zk1 zk2 zk3 zk4
k1 +k2 +k3 +k4 =0
√
2ε dWt
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
6/17
Coarsening dynamics with noise
(Link to simulation)
Question: transition time from u− ≡ −1 to u+ ≡ +1?
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
7/17
Coarsening dynamics with noise
(Link to simulation)
Question: transition time from u− ≡ −1 to u+ ≡ +1?
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
7/17
Metastability and Eyring–Kramers law
dxt = −∇V (xt ) dt +
Mont
Puget
√
2ε dWt
Col de Sugiton
V : Rd → R confining potential
z
τyx = inf{t > 0 : xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ' e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 · · · 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 · · · 6 λd
E[τyx ] = 2π
q
λ2 ...λd
|λ1 |ν1 ...νd
e[V (z)−V (x)]/ε 1 + Oε (1)
Arrhenius’ law: proved by Freidlin, Wentzell (1979) using large deviations
Eyring–Kramers law: Bovier, Eckhoff, Gayrard, Klein (2004) using potential theory,
Helffer, Klein, Nier (2004) using Witten Laplacian, . . .
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
8/17
Metastability and Eyring–Kramers law
dxt = −∇V (xt ) dt +
Mont
Puget
√
2ε dWt
Col de Sugiton
V : Rd → R confining potential
z
τyx = inf{t > 0 : xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ' e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 · · · 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 · · · 6 λd
E[τyx ] = 2π
q
λ2 ...λd
|λ1 |ν1 ...νd
e[V (z)−V (x)]/ε 1 + Oε (1)
Arrhenius’ law: proved by Freidlin, Wentzell (1979) using large deviations
Eyring–Kramers law: Bovier, Eckhoff, Gayrard, Klein (2004) using potential theory,
Helffer, Klein, Nier (2004) using Witten Laplacian, . . .
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
8/17
Metastability and Eyring–Kramers law
dxt = −∇V (xt ) dt +
Mont
Puget
√
2ε dWt
Col de Sugiton
V : Rd → R confining potential
z
τyx = inf{t > 0 : xt ∈ Bε (y )}
first-hitting time of small ball Bε (y ),
when starting in x
Luminy
Calanque de Sugiton
x
y
Arrhenius’ law (1889): E[τyx ] ' e[V (z)−V (x)]/ε
Eyring–Kramers law (1935, 1940):
Eigenvalues of Hessian of V at minimum x: 0 < ν1 6 ν2 6 · · · 6 νd
Eigenvalues of Hessian of V at saddle z: λ1 < 0 < λ2 6 · · · 6 λd
E[τyx ] = 2π
q
λ2 ...λd
|λ1 |ν1 ...νd
e[V (z)−V (x)]/ε 1 + Oε (1)
Arrhenius’ law: proved by Freidlin, Wentzell (1979) using large deviations
Eyring–Kramers law: Bovier, Eckhoff, Gayrard, Klein (2004) using potential theory,
Helffer, Klein, Nier (2004) using Witten Laplacian, . . .
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
8/17
Potential theory and Eyring–Kramers law
Consider first Brownian motion Wtx = x + Wt
Fact 1: wA (x) = E[τAx ] satisfies
1
− 2 ∆wA (x) = 1
x ∈ Ac
wA (x) = 0
x ∈A
y
A
function
GAc (x, y ) Green’s
Z
GAc (x, y ) dy
⇒ wA (x) =
Ac
Fact 2: hA,B (x) = P[τAx
 1
 − 2 ∆hA,B (x) = 0
hA,B (x) = 1

hA,B (x) = 0
< τBx ] satisfies
x ∈ (A ∪ B)c
x ∈A
x ∈B
ρA,B : “surfaceZcharge density” on ∂A
GB c (x, y ) ρA,B (dy )
⇒ hA,B (x) =
∂A
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
9/17
Potential theory and Eyring–Kramers law
Consider first Brownian motion Wtx = x + Wt
Fact 1: wA (x) = E[τAx ] satisfies
1
− 2 ∆wA (x) = 1
x ∈ Ac
wA (x) = 0
x ∈A
y
A
+ + ++
+
+
+ A
+
+
+
+ +
+
−
−
−
−
−
function
GAc (x, y ) Green’s
Z
GAc (x, y ) dy
⇒ wA (x) =
Ac
Fact 2: hA,B (x) = P[τAx
 1
 − 2 ∆hA,B (x) = 0
hA,B (x) = 1

hA,B (x) = 0
< τBx ] satisfies
x ∈ (A ∪ B)c
x ∈A
x ∈B
ρA,B : “surfaceZcharge density” on ∂A
GB c (x, y ) ρA,B (dy )
⇒ hA,B (x) =
∂A
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
−
− −−
−
−
−
−
−
−
− − −−
B
1V
9 December 2015
9/17
Potential theory and Eyring–Kramers law
Z
ρA,B (dy )
capA (B)
ρA,B (dy ) ⇒ νA,B (dy ) =
Capacity: capA (B) =
∂A
Variational representation: Dirichlet form
Z
capA (B) =
k∇hA,B (x)k2 dx = inf
Z
h∈HA,B
(A∪B)c
prob measure
k∇h(x)k2 dx
(A∪B)c
(HA,B : set of sufficiently smooth functions satisfying b.c.)
Key observation: let C = Bε (x), then (using G (y , z) = G (z, y ))
Z Z
Z
hC ,A (y ) dy =
GAc (y , z)ρC ,A (dz) dy
ZAc
ZAc ∂C
hC ,A (y ) dy =
wA (z)ρC ,A (dz) = capC (A)E νC ,A [wA ]
Ac
∂C
Harnack inequalities ⇒
E x [τA ]
'
E νC ,A [wA ]
=
Z
1
capC (A)
hC ,A (y ) dy
Ac
General case: replace 12 ∆ by ε∆ − ∇V (x) · ∇ and dx by e−V (x)/ε dx
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
10/17
Potential theory and Eyring–Kramers law
Z
ρA,B (dy )
capA (B)
ρA,B (dy ) ⇒ νA,B (dy ) =
Capacity: capA (B) =
∂A
Variational representation: Dirichlet form
Z
capA (B) =
k∇hA,B (x)k2 dx = inf
Z
h∈HA,B
(A∪B)c
prob measure
k∇h(x)k2 dx
(A∪B)c
(HA,B : set of sufficiently smooth functions satisfying b.c.)
Key observation: let C = Bε (x), then (using G (y , z) = G (z, y ))
Z Z
Z
hC ,A (y ) dy =
GAc (y , z)ρC ,A (dz) dy
ZAc
ZAc ∂C
hC ,A (y ) dy =
wA (z)ρC ,A (dz) = capC (A)E νC ,A [wA ]
Ac
∂C
Harnack inequalities ⇒
E x [τA ]
'
E νC ,A [wA ]
=
Z
1
capC (A)
hC ,A (y ) dy
Ac
General case: replace 12 ∆ by ε∆ − ∇V (x) · ∇ and dx by e−V (x)/ε dx
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
10/17
Potential theory and Eyring–Kramers law
Z
ρA,B (dy )
capA (B)
ρA,B (dy ) ⇒ νA,B (dy ) =
Capacity: capA (B) =
∂A
Variational representation: Dirichlet form
Z
capA (B) =
k∇hA,B (x)k2 dx = inf
Z
h∈HA,B
(A∪B)c
prob measure
k∇h(x)k2 dx
(A∪B)c
(HA,B : set of sufficiently smooth functions satisfying b.c.)
Key observation: let C = Bε (x), then (using G (y , z) = G (z, y ))
Z Z
Z
hC ,A (y ) dy =
GAc (y , z)ρC ,A (dz) dy
ZAc
ZAc ∂C
hC ,A (y ) dy =
wA (z)ρC ,A (dz) = capC (A)E νC ,A [wA ]
Ac
∂C
Harnack inequalities ⇒
E x [τA ]
'
E νC ,A [wA ]
=
Z
1
capC (A)
hC ,A (y ) dy
Ac
General case: replace 12 ∆ by ε∆ − ∇V (x) · ∇ and dx by e−V (x)/ε dx
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
10/17
Eyring–Kramers law for SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0 : kut − u+ kL∞ } < ρ
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
(
u0 (x) ≡ 0
if L 6 βπ
uts (x) =
with ev λk =( βkπ
)2 −1
L
u1 (x) β-kink stationary sol. if L > βπ
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ' e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
q
Q
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] = 2π |λ01|ν0 ∞
k=1 νk e
.
Rigorous proof?
.
What happens when L → π as then λ1 → 0?
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
11/17
Eyring–Kramers law for SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0 : kut − u+ kL∞ } < ρ
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
(
u0 (x) ≡ 0
if L 6 βπ
uts (x) =
with ev λk =( βkπ
)2 −1
L
u1 (x) β-kink stationary sol. if L > βπ
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ' e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
q
Q
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] ' 2π |λ01|ν0 ∞
k=1 νk e
.
Rigorous proof?
.
What happens when L → π as then λ1 → 0?
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
11/17
Eyring–Kramers law for SPDEs: heuristics
√
u̇t (x) = ∆ut (x) + f (ut (x)) + 2ε Ẅtx
Initial condition: uin near u− ≡ −1
(∆ ≡ ∂xx , f (u) = u − u 3 )
with eigenvalues νk
Target: u+ ≡ 1, τ+ = inf{t > 0 : kut − u+ kL∞ } < ρ
Transition state: (β = 1 for Neumann b.c., β = 2 for periodic b.c.)
(
u0 (x) ≡ 0
if L 6 βπ
uts (x) =
with ev λk =( βkπ
)2 −1
L
u1 (x) β-kink stationary sol. if L > βπ
[Faris & Jona-Lasinio 82]: large-deviation principle
⇒ Arrhenius law: E uin [τ+ ] ' e(V [uts ]−V [u− ])/ε
[Maier & Stein 01]: formal computation; for Neumann b.c.
q
Q
λk (V [uts ]−V [u− ])/ε
⇒ E uin [τ+ ] ' 2π |λ01|ν0 ∞
k=1 νk e
.
Rigorous proof?
.
What happens when L → βπ as then λ1 → 0?
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
11/17
Eyring–Kramers law for SPDEs: main result
Theorem 1: Neumann b.c. [B & Gentz, Elec J Proba 2013]
.
If L < π − c with c > 0, then
v
u
∞
u 1 Y
λk (V [uts ]−V [u− ])/ε uin
e
E [τ+ ] = 2π t
1 + O(ε1/2 |log ε|3/2 )
|λ0 |ν0
νk
k=1
1
2
.
If L > π + c, then same formula with extra factor
(since 2 saddles)
.
If π − c 6 L 6 π, then
v
√
u
∞
u λ1 + C ε Y
λk e(V [uts ]−V [u− ])/ε uin
√
E [τ+ ] = 2π t
1 + R(ε)
|λ0 |ν0 ν1
νk Ψ+ (λ1 / C ε)
k=2
where Ψ+ explicit, involves Bessel function K1/4 , limα→∞ Ψ+ (α) = 1
.
If π 6 L 6 π + c, then same formula, with another function Ψ− ,
involving Bessel function I±1/4 , limα→∞ Ψ+ (α) = 2
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
12/17
Eyring–Kramers law for SPDEs: main result
Theorem 1: Neumann b.c. [B & Gentz, Elec J Proba 2013]
.
If L < π − c with c > 0, then
v
u
∞
u 1 Y
λk (V [uts ]−V [u− ])/ε uin
e
E [τ+ ] = 2π t
1 + O(ε1/2 |log ε|3/2 )
|λ0 |ν0
νk
k=1
1
2
.
If L > π + c, then same formula with extra factor
(since 2 saddles)
.
If π − c 6 L 6 π, then
v
p
u
∞
u λ1 + 3ε/2L Y
λk e(V [uts ]−V [u− ])/ε uin
p
E [τ+ ] = 2π t
1 + R(ε)
|λ0 |ν0 ν1
νk Ψ+ (λ1 / 3ε/2L)
k=2
where Ψ+ explicit, involves Bessel function K1/4 , limα→∞ Ψ+ (α) = 1
.
If π 6 L 6 π + c, then same formula, with another function Ψ− ,
involving Bessel functions I±1/4 , limα→∞ Ψ− (α) = 2
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
12/17
Sketch of proof: Spectral Galerkin approximation
(d)
ut (x) =
√1
L
Pd
i πkx/L
k=−d zk (t) e
⇒ dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 21 ) there exists an a.s. finite r.v. Z : Ω → R+ s.t. ∀ω ∈ Ω
(d)
sup kut (ω) − ut (ω)kL∞ < Z (ω)d −γ
∀d ∈ N
06t6T
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
13/17
Sketch of proof: Spectral Galerkin approximation
(d)
ut (x) =
√1
L
Pd
i πkx/L
k=−d zk (t) e
⇒ dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 21 ) there exists an a.s. finite r.v. Z : Ω → R+ s.t. ∀ω ∈ Ω
(d)
sup kut (ω) − ut (ω)kL∞ < Z (ω)d −γ
∀d ∈ N
06t6T
Proposition (using potential theory)
∃ε0 > 0 : ∀ε < ε0 ∃d0 (ε) < ∞ : ∀d > d0 ∃νd proba measure on ∂Br (u− )
Z
(d)
E v0 [τ+ ]νd (dv0 ) = C (d, ε) eH(d)/ε 1 + R(ε)
∂Br (u− )
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
13/17
Sketch of proof: Spectral Galerkin approximation
(d)
ut (x) =
√1
L
Pd
i πkx/L
k=−d zk (t) e
⇒ dzt = −∇V (zt ) dt +
√
2ε dWt
Theorem [Blömker & Jentzen 13]
For all γ ∈ (0, 21 ) there exists an a.s. finite r.v. Z : Ω → R+ s.t. ∀ω ∈ Ω
(d)
sup kut (ω) − ut (ω)kL∞ < Z (ω)d −γ
∀d ∈ N
06t6T
Proposition (using potential theory)
∃ε0 > 0 : ∀ε < ε0 ∃d0 (ε) < ∞ : ∀d > d0 ∃νd proba measure on ∂Br (u− )
Z
(d)
E v0 [τ+ ]νd (dv0 ) = C (d, ε) eH(d)/ε 1 + R(ε)
∂Br (u− )
Proposition (using large deviations and lots of other stuff)
H0 := V [uts ] − V [u− ]. ∀η > 0 ∃ε0 , T1 , H1 : ∀ε < ε0 ∃d0 < ∞ s.t. ∀d > d0
sup
v0 ∈Br (u− )
E v0 [τ+2 ] 6 T12 e2(H0 +η)/ε , sup
sup
d>d0 v0 ∈Br (u− )
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
(d)
E v0 [(τ+ )2 ] 6 T12 e2H1 /ε
9 December 2015
13/17
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
(d)
(d)
ΩK ,d =
sup kvt − vt kL∞ 6 δ, τB− 6 KTKr
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
(d)
(d)
(d)
τB+
(d)
E v0 [τB− ]
KTKr
6 τB 6
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
(d)
(d)
⇒ E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ]+
Integrate against νq
d and use Cauchy–Schwarz to bound error terms:
v
0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,d ), take d → ∞ and finally K large
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
14/17
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
(d)
(d)
ΩK ,d =
sup kvt − vt kL∞ 6 δ, τB− 6 KTKr
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
(d)
(d)
(d)
τB+
(d)
E v0 [τB− ]
KTKr
6 τB 6
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
(d)
(d)
⇒ E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ] + E v0 [τB 1{ΩcK ,d } ]
Integrate against νq
d and use Cauchy–Schwarz to bound error terms:
v
0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,d ), take d → ∞ and finally K large
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
14/17
Main step of the proof
Set TKr = C (∞, ε) eH0 /ε
Let B = Bρ (u+ ) and define nested sets B− ⊂ B ⊂ B+ at L∞ -distance δ
(d)
(d)
ΩK ,d =
sup kvt − vt kL∞ 6 δ, τB− 6 KTKr
t∈[0,KTKr ]
(d)
P(ΩcK ,d ) 6 P{Z > δd γ } +
On ΩK ,d one has
(d)
(d)
(d)
(d)
(d)
τB+
(d)
E v0 [τB− ]
KTKr
6 τB 6
Cauchy–
Schwarz
.
⇒
lim sup P(ΩcK ,d ) =
d→∞
M(ε)
K
(d)
τB−
(d)
(d)
⇒ E v0 [τB+ 1{ΩK ,d } ] 6 E v0 [τB 1{ΩK ,d } ] 6 E v0 [τB− 1{ΩK ,d } ]
(d)
(d)
(d)
(d)
⇒ E v0 [τB+ ] − E v0 [τB+ 1{ΩcK ,d } ] 6 E v0 [τB ] 6 E v0 [τB− ] + E v0 [τB 1{ΩcK ,d } ]
Integrate against νq
d and use Cauchy–Schwarz to bound error terms:
v
0
E [τB 1{ΩcK ,d } ] 6 E v0 [τB2 ]P(ΩcK ,d ), take d → ∞ and finally K large
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
14/17
Eyring–Kramers law for SPDEs: main result
Theorem 2: periodic b.c. [B & Gentz, Elec J Proba 2013]
.
If L < 2π − c, then
∞
Y
2π
λk (V [uts ]−V [u− ])/ε E uin [τ+ ] = p
e
1 + O(ε1/2 |log ε|3/2 )
|λ0 |ν0 k=1 νk
.
If 2π − c 6 L 6 2π, then
E
uin
2π
[τ+ ] = p
|λ0 |ν0
λ1 +
∞
3ε/L Y λk e(V [uts ]−V [u− ])/ε p
1 + R(ε)
ν1
νk Θ+ (λ1 / 3ε/L)
k=2
p
where Θ+ explicit, involves error function, limα→∞ Θ+ (α) = 1
.
Similar expression for 2π 6 L 6 2π + c with λ1 = 0
.
If L > 2π + c, then
E
uin
√
∞ p
2πελ1 Y λk λ−k e(V [uts ]−V [u− ])/ε 1 + R(ε)
[τ+ ] = p
0
ν1
νk
Lkuts kL2
|λ0 |ν0
k=2
2π
0 k
Lkuts
L2 = “length of the saddle”
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
15/17
Eyring–Kramers law for SPDEs: main result
Theorem 2: periodic b.c. [B & Gentz, Elec J Proba 2013]
.
If L < 2π − c, then
∞
Y
2π
λk (V [uts ]−V [u− ])/ε E uin [τ+ ] = p
e
1 + O(ε1/2 |log ε|3/2 )
|λ0 |ν0 k=1 νk
.
If 2π − c 6 L 6 2π, then
E
uin
2π
[τ+ ] = p
|λ0 |ν0
λ1 +
∞
3ε/L Y λk e(V [uts ]−V [u− ])/ε p
1 + R(ε)
ν1
νk Θ+ (λ1 / 3ε/L)
k=2
p
where Θ+ explicit, involves error function, limα→∞ Θ+ (α) = 1
.
Similar expression for 2π 6 L 6 2π + c with λ1 = 0
.
If L > 2π + c, then
E
uin
√
∞ p
2πελ1 Y λk λ−k e(V [uts ]−V [u− ])/ε 1 + R(ε)
[τ+ ] = p
0
ν1
νk
Lkuts kL2
|λ0 |ν0
k=2
2π
0 k
Lkuts
L2 = “length of the saddle”
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
15/17
Outlook
.
Potentials with more than two wells: essentially understood
.
Potentials invariant under symmetry group: done for SDEs
.
Allen–Cahn in dimension d > 2: only possible since Martin Hairer’s
theory of regularity structures
[Link]
.
Open problem: non-gradient systems
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
9 December 2015
16/17
References
.
N. B. and Barbara Gentz, Sharp estimates for metastable lifetimes in
parabolic SPDEs: Kramers’ law and beyond, Electronic J. Probability
18, (24):1–58 (2013)
.
, Anomalous behavior of the Kramers rate at bifurcations in classical
field theories, J. Phys. A: Math. Theor. 42, 052001 (2009)
.
, The Eyring–Kramers law for potentials with nonquadratic saddles,
Markov Processes Relat. Fields 16, 549–598 (2010)
.
N. B., Kramers’ law: Validity, derivations and generalisations, Markov
Processes Relat. Fields 19, 459–490 (2013)
.
N. B. and Sébastien Dutercq, The Eyring–Kramers law for Markovian jump
processes with symmetries, J. Theoretical Probability (2015)
.
, Interface dynamics of a metastable mass-conserving spatially
extended diffusion, J. Statist. Phys., to appear
Thanks for your attention
An Eyring–Kramers formula for parabolic SPDEs with space-time white noise
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