Invariance for ODE
Rough differential equations
Invariance theorem
Invariance for rough differential equations
Laure COUTIN1 and Nicolas MARIE2
–
1 Mathematical Institut of Toulouse
[email protected]
–
2 Paris 10 University and ESME Sudria
[email protected]
Invariance for ODE
Rough differential equations
Invariance theorem
K closed subset of Rd .
• Viability of a function : f ([0, T ]) ⊂ K.
• Invariance of a set by a differential equation :
DXt = f (Xt )dt, X0
Invariance for ODE
Rough differential equations
Invariance theorem
Example
Let
ẏt = yt (1 − yt )
Solutions of (1) :
t 7−→
]0, 1[ is invariant by (1).
1
.
1
−t
1− 1−
e
y0
(1)
Invariance for ODE
Rough differential equations
Invariance theorem
Tangent cone TK (x) at K in x :
dK (x + hv)
d
=0
TK (x) = v ∈ R | lim
h→0,h>0
h
Nagumo’s Theorem,
1942
K is invariant by
dxt = b(xt )dt
(2)
if and only if
b(x) ∈ TK (x) ∀ x ∈ K.
Invariance for ODE
Rough differential equations
Invariance theorem
Idea’s of proof
Aubin book’s.
• Necessary condition x solution of (2) starting from
ξ ∈ K, such that x([0, T ]) ⊂ K.
dK (ξ + hb(ξ))
kξ + hb(ξ) − x(h)k + o(h)
≤
→ 0.
h
h
• Sufficient condition Modified Euler scheme
xk + hb(xk ) ∼ xk+1 ∈ K
A priori estimation
Extraction of a converging subsequence to a solution of (2).
Invariance for ODE
Rough differential equations
Let
Z
t
Z
b(ys )ds +
yt = y0 +
0
with B (stochastic) process.
t
σ(ys )dBs
0
Invariance theorem
(3)
Invariance for ODE
Rough differential equations
SNC of K invariance by (3) ?
Invariance theorem
Invariance for ODE
Rough differential equations
• 1942. Nagumo (ODE).
• 1990. Aubin et DaPrato.
• K closed convex set.
• B MB.
• (3) in Itô sens.
• 2009. Ciotir et Rascanu.
• K closed set.
• B MBF Hurst index > 1/2.
• (3) Young meaning.
Invariance theorem
Invariance for ODE
Rough differential equations
Invariance theorem
Integration: Young 36
• Set of α Hölder continuous paths C α ([0, T ], Rn )
kwkα :=
kwt − ws k
.
α
s,t∈[0,T ] |t − s|
sup
• Integration : α + β > 1
∃! I bilinear continuous
C α ([0, T ], Rn ) × C β ([0, T ], L(Rn , Rd )) → C βR([0, T ], Rn )
.
(w, w̃)
7→
0 w̃u dwu
such that
Z t
w̃u dwu − w̃s (wt − ws ) ≤ Ckwkα kw̃kβ |t − s|α+β ,
s
Z
t
w̃u dwu = lim
s
kπk→0
X
ti ∈π
w̃ti [wti+1 − wti ].
Invariance for ODE
Rough differential equations
Invariance theorem
Young Existence and uniquness Theorem :
Lyons 94
B α Hölder, α > 1/2.
b and σ Cb2 ,
Z
yt = y0 +
t
Z
b(ys )ds +
0
t
σ(ys )dBs
0
has an unique solution such that
|yt − ys − b(y − s)(t − s) − σ(ys ).(Bt − Bs )| ≤ C|t − s|2α ,
(4)
Invariance for ODE
Rough differential equations
Invariance theorem
If B α Hölder with α ∈]1/3, 1/2]
Assumption 1
There exists a sequence (B n ) of Lipschitz functions such that
• (B n )n converges to B is C α ([0, T ], Rn )
•
R
t
( s Bn (u) ⊗ dBn (u), s, t ∈ [0, T ]2 ) converge to B (2) for
n
the seminorm 2α Hölder.
Fractional Brownian motion with Hurst index H > 1/3 fulfills
Assumption 1
Invariance for ODE
Rough differential equations
Invariance theorem
Theorem Lyons 98
If b and σ are Cb3 , B fulfills Assumption 1, (3) has a unique
solution y :
lim ky − y n k∞,T = 0
n→∞
with
ytn
Z
= y0 +
0
t
b(ysn )ds
Z
+
t
σ(ysn )dBsn
(5)
0
and Euler scheme approximation
(2) 3α
y
−
y
−
b(y
)(t
−
s)
−
σ(y
)(B
−
B
)
−
∇σ.σ(y
)B
y
s
s
s
t
s
s
s,t ≤ C|t − s|
Invariance for ODE
Rough differential equations
Invariance theorem !
Invariance theorem
Invariance for ODE
Rough differential equations
Invariance theorem
K closed convex of Rd and x ∈ K.
Tangent cone TK (x) and normal cone NK (x) to K in x :
NK (x) = TK (x)O = {s ∈ Rd , hs, vi ≤ 0 ∀v ∈ TK (x)}
Invariance for ODE
Rough differential equations
Assumption 2
Forall x ∈ ∂K et k ∈ J1, eK, b(x), ±σ.,k (x) ∈ TK (x).
Figure: f = b, f = σ.,k ou f = −σ.,k
Invariance theorem
Invariance for ODE
Rough differential equations
K half space of Rd
• K = {x : hv, x − ai 6 0}.
• ∂K = {x : hv, x − ai = 0}.
• Let x ∈ ∂K.
NK (x) = {λv ; λ > 0}.
Then
TK (x) = {δ : ∀s ∈ NK (x), hs, δi 6 0}
= {δ : hv, δi 6 0}.
• b and σ fulsfills Assumption 2 is equivalent to
hv, b(x)i 6 0 et hv, σ.,k (x)i = 0 ; ∀x ∈ ∂K.
Invariance theorem
Invariance for ODE
Rough differential equations
Invariance theorem
Assumption 3
B visits all direction of Rd .
From the law of iterated logarithm, Assumption 3 is satisfied by
fractional Brownian motion
Invariance for ODE
Rough differential equations
Invariance theorem
B α Hölder continuous with
-α > 1/2 or
- 1/3 < α ≤ 1/2 and fulfills Assumption 1
Theorem
• If b and σ fulfills Assumption 2, K is closed
then K is invariant by (3) ,
• If K is invariant by (3), convex, and B fulfills Assumption
3,
then b and σ fulfills Assumption 2.
Application : Logistic equation for MBF.
Invariance for ODE
Rough differential equations
Sufficient condition : idea’s of proof
Invariance theorem
Invariance for ODE
Rough differential equations
Invariance theorem
Step 1
Strong Assumption 2
Forall x ∈ Rd and k ∈ J1, eK, b(x), ±σ.,k (x) ∈ TK (pK (x)).
Under this assumption there exists C > 0 such that :
ϕn (t + h) − ϕn (t) 6 Ch2
with
ϕn (t) := d2K (ytn ).
Invariance for ODE
Rough differential equations
Invariance theorem
Step 2
1. Under strong assumption 2, from step 1 :
D↑ ϕn (t)(1) := lim inf
h→0+ ,u→1
ϕn (t + hu) − ϕn (t)
h
6 0.
2. Then ϕn is non increasing (see Aubin (1990) ):
dK (ytn ) = 0.
Invariance for ODE
Rough differential equations
Invariance theorem
Step 3
1. Under Assumption 2, K is invariant by
Z t
Z t
Ytn = y0 +
(b ◦ pK )(Ysn )ds +
(σ ◦ pK )(Ysn )dBsn
0
0
From step 2.
2. Since
b ◦ pK |K = b et σ ◦ pK |K = σ,
K is invariant by (5).
Invariance for ODE
Rough differential equations
Step 4
Under Assumption 2, K is invariant by (3) since
lim ky n − yk∞,T = 0.
n→∞
Invariance theorem
Invariance for ODE
Rough differential equations
Necessary condition : idea’s of proof
Invariance theorem
Invariance for ODE
Rough differential equations
Step 1
If the half plane
Dν := {x : x(ν) > 0}
is invariant by (3), then
b(ν) (x − x(ν) eν ) > 0 et σν,. (x − x(ν) eν ) = 0.
Tools :
• Euler scheme .
• Same idea’s as in the ODE’s case.
Invariance theorem
Invariance for ODE
Rough differential equations
Step 2
If the half space
Hs,a := {x : hs, x − ai 6 0}
is invariant by (3), then
hs, b(x)i 6 0 et hs, σ.,k (x)i = 0 ; ∀x ∈ ∂Hs,a .
Going back to step 1 by a change of variable.
Invariance theorem
Invariance for ODE
Rough differential equations
Invariance theorem
Step 3
• y0 ∈ ∂K such that for f = b, f = σ.,k or f = −σ.,k :
∃s ∈ NK (y0 ) : hs, f (y0 )i > 0.
• Hy0 ,s := {x : hs, x − y0 i 6 0}.
Invariance for ODE
Rough differential equations
Invariance theorem
Fractional Hodgkin-Huxley model for application in
biology
Invariance for ODE
Rough differential equations
Thanks for your attention !
Invariance theorem