Stabilization by Noise of a C -valued Coupled System Fanny Shum

Stabilization by Noise of a C2 -valued Coupled System1
Fanny Shum
Joint work with
J.P. Chen, L. Ford, D. Kielty, R. Majumdar, H. McCain, D. O’Connell
University of Connecticut
November 19, 2015
14th Northeast Probability Seminar
New York University
1
Funded by the NSF under DMS award #1262929, Summer 2015 REU, Dept of Mathematics, University of Connecticut
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
1 / 13
Introduction
What is stabilization by noise?
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
2 / 13
Introduction
What is stabilization by noise?
I
When an explosive deterministic system can be stabilized by the
addition of noise or randomness.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
2 / 13
Example of perturbation
Im z
100
50
0
Re z
-50
-100
-100
-50
0
50
100
dz = z 2 dt
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
3 / 13
Example of perturbation
Im z
100
50
0
Re z
-50
-100
-100
-50
0
50
100
dz = z 2 dt
Fan Ny Shum (UConn)
dz = z 2 dt + σ dB
Stochastic Stabilization in C2
Northeast Probability Seminar
3 / 13
Recent Results
David Herzog and Jonathan Mattingly showed the stability of the
complex-valued SDE
dzt = (an+1 ztn+1 + an ztn + · · · + a0 )dt + σdBt
z0 ∈ C,
where n ≥ 1, an+1 6= 0, Bt = Bt1 + iBt2 is a complex Brownian motion, and
σ is some real constant; that is,
1
the process zt is nonexplosive
2
and there exists a unique invariant measure.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
4 / 13
Lyapunov Method
Let L be the infinitesimal generator of our process zt .
If there exists a function ϕ ∈ C 2 (C : [0, ∞)) such that
1
ϕ(z) → ∞ as |z| → ∞
2
and Lϕ(z) → −∞ as fast as possible as |z| → ∞,
then the process zt is nonexplosive and there exists an invariant measure.
Note: ϕ is a Lyapunov function.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
5 / 13
Next Natural Step
Now, we look at an analogous multivariable system of ODEs.

 żt = −νzt + αzt wt
ẇt = −νwt + βzt wt
where ν ∈ R+ , α, β ∈ R.

z0 , w0 ∈ C,
We shall assume α > 0 and β > 0 without loss of generality.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
6 / 13
Next Natural Step
Now, we look at an analogous multivariable system of ODEs.

 żt = −νzt + αzt wt
ẇt = −νwt + βzt wt
where ν ∈ R+ , α, β ∈ R.

z0 , w0 ∈ C,
We shall assume α > 0 and β > 0 without loss of generality.
Re w
3
2
1
0
Re z
-1
-2
-3
-3
Fan Ny Shum (UConn)
-2
-1
0
1
2
Stochastic Stabilization in C2
3
Northeast Probability Seminar
6 / 13
Coordinate Transformation

 żt = −νzt + αzt wt
ẇt = −νwt + βzt wt

z0 , w0 ∈ C,
Fan Ny Shum (UConn)
where ν ∈ R+ , α, β ∈ R.
Stochastic Stabilization in C2
Northeast Probability Seminar
7 / 13
Coordinate Transformation

 żt = −νzt + αzt wt
ẇt = −νwt + βzt wt

z0 , w0 ∈ C,
where ν ∈ R+ , α, β ∈ R.
We can decouple the system with the following coordinate transformation:

1
α
 z̃ = 2 (z + β w )

Fan Ny Shum (UConn)
w̃ = 12 (z − αβ w ).
Stochastic Stabilization in C2
Northeast Probability Seminar
7 / 13
Post-Coordinate Transformation

 d z̃t = (−ν z̃t + β z̃t2 − β w̃t2 ) dt

d w̃t = −ν w̃t dt,
with initial condition (z̃0 , w̃0 ). Observe that w̃t evolve autonomously with
solution
w̃ (t) = w̃0 e −νt .
This yields the following quasi 1-dimensional system:
d z̃t
= (−ν z̃t + β z̃t2 − β w̃02 e −2νt ) dt
∼ (−ν z̃t + β z̃t2 ) dt
as t → ∞.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
8 / 13
Phase Portrait Post-Coordinate Transformation
Figure: Phase portrait for large t (looks like ż = z 2 − z)
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
9 / 13
Ergodic theorem for the C2 -valued system
Theorem
The system of SDEs
dzt = (−νzt + αzt wt ) dt + σ dBt
dwt = (−νwt + βzt wt ) dt + αβ σ dBt
(1)
with initial condition (z0 , w0 ) ∈ C2 , where ν ∈ R+ , α, β ∈ R, σ ∈ R \ {0},
(1)
(2)
and Bt = Bt + iBt is a C-valued standard Brownian motion. Then the
system (??) is nonexplosive and has a unique invariant measure.
Proof. We used Girsanov transformation.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
10 / 13
Open Problem
Initial simulations suggest the stability of
dzt = (−νzt + αzt wt ) dt + i dBt
dwt = (−νwt + βzt wt ) dt.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
11 / 13
Open Problem
Initial simulations suggest the stability of
dzt = (−νzt + αzt wt ) dt + i dBt
dwt = (−νwt + βzt wt ) dt.
Under the coordinate transformation, it becomes

 d z̃t = (−ν z̃t + β z̃t2 − β w̃t2 ) dt + 2i dBt

Fan Ny Shum (UConn)
d w̃t = −ν w̃t dt + 2i dBt .
Stochastic Stabilization in C2
Northeast Probability Seminar
11 / 13
Open Problem
Initial simulations suggest the stability of
dzt = (−νzt + αzt wt ) dt + i dBt
dwt = (−νwt + βzt wt ) dt.
Under the coordinate transformation, it becomes

 d z̃t = (−ν z̃t + β z̃t2 − β w̃t2 ) dt + 2i dBt

d w̃t = −ν w̃t dt + 2i dBt .
Note: w̃t is the Ornstein-Uhlenbeck process.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
11 / 13
Current Problem
We will consider the homogeneous polynomial

 dzt = (zt2 + αzt wt ) dt + σ1 dBt1
dwt = (wt2 + βzt wt ) dt + σ2 dBt2

z0 , w0 ∈ C,
where α, β, σ1 , σ2 ∈ R and Bt1 , Bt2 are complex Brownian motions.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
12 / 13
Thank you!
Also thanks to Masha Gordina and David Herzog.
Fan Ny Shum (UConn)
Stochastic Stabilization in C2
Northeast Probability Seminar
13 / 13