Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Noise Suppresses or Expresses Exponential
Growth
Xuerong Mao
University of Strathclyde
Glasgow, G1 1XH
Joint work with Deng, Hu, Liu, Luo, Pang, Song
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
ODEs versus SDEs
Can a given ordinary differential equation (ODE)
ẏ (t) = f (y (t), t)
and its corresponding stochastic perturbed equation
dx(t) = f (x(t), t)dt + g(x(t), t)dB(t)
have significant differences?
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Hasminskii’s pioneering work on stability
The linear scalar ODE
ẏ (t) = y (t)
is unstable, while the SDE
dx(t) = x(t)dt + 2x(t)dB(t)
is almost surely exponentially stable.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Stochastic stabilization
Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any
linear system ẋ(t) = Ax(t) with trace(A) < 0 can be
stabilized by one real noise source.
Scheutzow (1993): Stochastic stabilization for two special
nonlinear systems.
Mao (1994): The general theory on stochastic stabilization
for nonlinear SDEs.
Mao (1996): Design a stochastic control that can
self-stabilize the underlying system.
Caraballo, Liu and Mao (2001): Stochastic stabilization for
partial differential equations (PDEs).
Appleby and Mao (2004): Stochastic stabilization for
functional differential equations.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Noise suppresses explosion established by Mao et al.
The solution to
dx(t)
= x(t)[1 + x(t)],
dt
t ≥ 0, x(0) = x0 > 0
explodes to infinity at the finite time
T = log
1 + x 0
x0
.
However, the SDE
dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]
will never explode with probability one as long as σ 6= 0.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
200
(a)
150
100
50
0
0
2
4
6
8
10
80
(b)
60
40
20
0
0
2
4
Xuerong Mao
6
Exponential Growth
8
10
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Note on the graphs:
In graph (a) the solid curve shows a stochastic trajectory
generated by the Euler scheme for time step ∆t = 10−7 and
σ = 0.25. The corresponding deterministic trajectory is shown
by the dot-dashed curve. In Graph (b) σ = 1.0.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Impact of the result
The result that noise suppresses explosion established by Mao
et al. in 2002 has made a significant impact on stochastic
population dynamics. Based on the data base of Google
Scholar, the paper has so far been cited by 69 papers.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Question:
What else significant effects can noise make?
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Consider the scalar ODE
ẏ (t) = a + by (t) on t ≥ 0
with initial value y (0) = y0 > 0, where a, b > 0. This equation
has its explicit solution
a bt a
y (t) = y0 +
e − .
b
b
Hence
1
log(y (t)) = b,
t→∞ t
lim
that is, the solution tends to infinity exponentially.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Consider the scalar SDE
dx(t) = [a + bx(t)]dt + σx(t)dB(t) on t ≥ 0 ,
with initial value x(0) = x0 > 0, where σ > 0 and B(t) is a
scalar Brownian motion. If σ 2 > 2b then the solution obeys
lim sup
t→∞
σ2
log(x(t))
≤ 2
log t
σ − 2b
a.s.
This shows that the noise suppresses the exponential growth.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Consider an n-dimensional SDE
dx(t) = f (x(t), t)dt + g(x(t), t)dB(t)
(2.1)
on t ≥ 0 with the initial value x(0) = x0 ∈ Rn , where B(t) is an
m-dimensional Brownian motion while
f : Rn × R+ → Rn
and
Xuerong Mao
g : Rn × R+ → Rn×m .
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Assumption
Assume that both coefficients f and g are locally Lipschitz
continuous. Assume also that there are nonnegative constants
α, β, η and γ such that
hx, f (x, t)i ≤ α + β|x|2 ,
|g(x, t)|2 ≤ η + γ|x|2
for all (x, t) ∈ Rn × R+ .
Xuerong Mao
Exponential Growth
(2.2)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
It is known that under this Assumption, the SDE has a unique
global solution x(t) on t ∈ R+ and the solution obeys
lim sup
t→∞
1
log(|x(t)|) ≤ β + 12 γ
t
a.s.
That is, the solution will grow at most exponentially with
probability one.
The following theorem shows that if the noise is sufficiently
large, it will suppress this potentially exponential growth and
make the solution grow at most polynomially.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Theorem
Let Assumption 1 hold. Assume that there are moreover two
positive constants δ and ρ such that
|x T g(x, t)|2 ≥ δ|x|4 − ρ
(2.3)
for all (x, t) ∈ Rn × R+ . If
δ > β + 21 γ,
(2.4)
then the solution of the SDE obeys
lim sup
t→∞
log(|x(t)|)
δ
≤
log t
2δ − 2β − γ
Xuerong Mao
Exponential Growth
a.s.
(2.5)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Key steps of the proof
Choose any such that
0<θ<
2δ − 2β − γ
2δ
and show
lim sup E[(1 + |x(t)|2 )θ ] ≤ C < ∞.
t→∞
Show
lim sup E
k →∞
sup (1 + |x(u)|2 )θ ≤ C.
k ≤u≤k +1
Show the assertion.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Key steps of the proof
Choose any such that
0<θ<
2δ − 2β − γ
2δ
and show
lim sup E[(1 + |x(t)|2 )θ ] ≤ C < ∞.
t→∞
Show
lim sup E
k →∞
sup (1 + |x(u)|2 )θ ≤ C.
k ≤u≤k +1
Show the assertion.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Key steps of the proof
Choose any such that
0<θ<
2δ − 2β − γ
2δ
and show
lim sup E[(1 + |x(t)|2 )θ ] ≤ C < ∞.
t→∞
Show
lim sup E
k →∞
sup (1 + |x(u)|2 )θ ≤ C.
k ≤u≤k +1
Show the assertion.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Consider a nonlinear ODE
ẏ (t) = f (y (t), t).
(2.6)
Here f : Rn × R+ → Rn is locally Lipschitz continuous and
obeys
hx, f (x, t)i ≤ α + β|x|2 ,
∀(x, t) ∈ Rn × R+ ,
(2.7)
for some positive constants α and β. Clearly, the solution of this
equation may grow exponentially.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Question: Can we design a linear stochastic feedback control
of the form
m
X
Ai x(t)dBi (t)
i=1
(i.e. choose square matrices Ai ∈ Rn×n ) so that the
stochastically controlled system
dx(t) = f (x(t), t)dt +
m
X
Ai x(t)dBi (t)
i=1
will grow at most polynomially with probability one?
Xuerong Mao
Exponential Growth
(2.8)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Corollary
Let (2.7) hold. Assume that there are two positive constants γ
and δ such that
m
X
|Ai x|2 ≤ γ|x|2 ,
|x T Ai x|2 ≥ δ|x|4
(2.9)
i=1
i=1
for all x ∈ Rn , and
m
X
δ > β + 21 γ.
(2.10)
Then, for any initial value x(0) = x0 ∈ Rn , the solution of the
stochastically controlled system (2.8) obeys
lim sup
t→∞
δ
log(|x(t)|)
≤
log t
2δ − 2β − γ
Xuerong Mao
Exponential Growth
a.s.
(2.11)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
We may wonder if the following more general stochastic
feedback control
m
X
(ai + Ai x(t))dBi (t)
i=1
could be better? Here ai ∈ Rn , 1 ≤ i ≤ n. The answer is not. In
fact, we can show in the same way as the above corollary was
proved that under the same conditions of Corollary 3, the
solution of the following SDE
m
X
dx(t) = f (x(t), t)dt +
(ai + Ai x(t))dBi (t)
i=1
still obeys (2.11).
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Question:
Are there matrices Ai that satisfy conditions (2.9) and (2.10)?
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Case 1: Let Ai = σi I for 1 ≤ i ≤ m, where I is the n × n identity
matrix and σi ’s are non-negative real numbers which represent
the intensity of the noise. In this case, the stochastically
controlled system becomes
dx(t) = f (x(t), t)dt +
m
X
σi x(t)dBi (t).
i=1
Corollary 3 shows that its solution obeys
Pm 2
σ
log(|x(t)|)
≤ Pm i=12 i
lim sup
log
t
t→∞
i=1 σi − 2β
a.s.
Hence, the solution will grow
Pm at 2most polynomially with
probability one provided i=1 σi > 2β.
Xuerong Mao
Exponential Growth
(2.12)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Case 2: For each i, choose a positive-definite matrix Di such
that
√
3
T
kDi k|x|2 ∀x ∈ Rn .
(2.13)
x Di x ≥
2
Obviously, there are many such matrices. Let σ be a constant
large enough for
4β
σ 2 > Pm
.
2
i=1 kDi k
Set Ai = σDi .
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Then
m
X
2
|Ai x| ≤ σ
i=1
and
m
X
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
2
m
X
kDi k2 |x|2
i=1
m
3σ 2 X
kDi k2 |x|4 .
4
|x T Ai x|2 ≥
i=1
i=1
Thus, Corollary 3 shows that the solution obeys
log(|x(t)|)
lim sup
≤
log t
t→∞
3σ 2 Pm
2
i=1 kDi k
4
P
m
σ2
2
i=1 kDi k − 2β
2
Xuerong Mao
Exponential Growth
a.s.
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
Theorem
The potentially exponential growth of the solution to a nonlinear
system ẏ (t) = f (y (t), t) can be suppressed by Brownian
motions provided that the condition
hx, f (x, t)i ≤ α + β|x|2 ,
∀(x, t) ∈ Rn × R+ ,
is satisfied. Moreover, one can even use only a scalar Brownian
motion to suppress the exponential growth.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Consider the 2-dimensional linear ODE
dy (t)
= q + Qy (t)
dt
with initial value y (0) = y0 ∈ R2 , where
1
−2
1
q=
, Q=
.
2
2 −3
The solution obeys
lim sup |y (t)| ≤
√
5.
t→∞
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
The 2-dimensional linear SDE
dx(t) = [q + Qx(t)]dt + ξdB1 (t) + Dx(t)dB2 (t)
with initial value x(0) = x0 ∈ R2 , where
2
0 σ
ξ=
, D=
2
−σ 0
with σ 2 = 32. The solution obeys
lim inf
t→∞
1
log(|x(t)|) ≥ 1.5
t
a.s
This shows that the noise expresses the exponential growth.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Theorem
Assume that there are non-negative constants c1 –c6 such that
c5 > c1 ,
c6 > c2 + 2c4 ,
− 2hx, f (x, t)i ≤ c1 + c2 |x|2 ,
(3.1)
|x T g(x, t)|2 ≤ c3 |x|2 + c4 |x|4
(3.2)
and
|g(x, t)|2 ≥ c5 + c6 |x|2
(3.3)
for all (x, t) ∈ Rn × R+ . Set
a = c5 −c1 ,
b = c6 −c2 −2c4 ,
Xuerong Mao
c = c5 −c1 +c6 −c2 −2c3 . (3.4)
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Theorem
(i) If, furthermore,
c ≥ 2(a ∧ b),
(3.5)
then the solution of equation (2.1) obeys
lim inf
t→∞
1
log(|x(t)|) ≥ 12 (a ∧ b) a.s.
t
(3.6)
(ii) If (3.5) does not hold but
ab >
1 2
c ,
4
(3.7)
then the solution of equation (2.1) obeys
lim inf
t→∞
n
1
1
ab − 0.25c 2 o
log(|x(t)|) ≥ min a, b,
a.s. (3.8)
t
2
a+b−c
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Remark 1:
Condition (3.2) can be satisfied by a large class of functions.
For example, if both f and g obey the linear growth condition
|f (x, t)| ∨ |g(x, t)| ≤ K (1 + |x|),
then
−2hx, f (x, t)i ≤ 2|x||f (x, t)| ≤ 2K |x| + 2K |x|2 ≤ K + 3K |x|2
and
|x T g(x, t)|2 ≤ |x|2 |g(x, t)|2 ≤ K 2 |x|2 (1+|x|)2 ≤ 2K 2 (|x|2 +|x|4 ),
that is f and g obey (3.2). Nevertheless, instead of using the
linear growth condition, the forms described in (3.2) enable us
to compute the parameters c1 –c4 more precisely.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Remark 2: In particular, linear SDEs always obey (3.2).
However, as shown in the previous section, there are many
linear SDEs whose solutions will grow at most polynomially but
not exponentially with probability one.
This of course indicates that condition (3.3) is very critical in
order to have an exponential growth.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Outline
1
Introduction
2
Noise Suppresses Exponential Growth
Motivating Examples
Polynomial Growth of SDEs
Noise Suppresses Exponential Growth
3
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Consider an n-dimensional ODE
dy (t)
= f (y (t), t),
dt
t ≥ 0.
(3.9)
Assume that f is sufficiently smooth and, in particular, it obeys
− 2hx, f (x, t)i ≤ c1 + c2 |x|2 ,
∀(x, t) ∈ Rn × R
for some non-negative numbers c1 and c2 .
Xuerong Mao
Exponential Growth
(3.10)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Our aim here is to perturb this equation stochastically into an
SDE
dx(t) = f (x(t), t)dt + g(x(t))dB(t)
(3.11)
so that its solutions will grow exponentially with probability one.
We will design g to be independent of t so we write g(x, t) as
g(x) in this section.
Moreover, we shall see that the noise term g(x(t))dB(t) can be
designed to be a linear form of x(t).
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Case 1: The dimension n of the state space is even.
Choose the dimension of the Brownian motion m to be 2.
Design g : Rn → Rn×2 by
g(x) = (ξ, Ax),
where ξ = (ξ1 , · · · , ξn )T ∈ Rn and, of course, ξ 6= 0, while


0 σ
−σ 0





.
..
A=




0 σ
−σ 0
with σ > 0.
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
So the stochastically perturbed system (3.11) becomes


x2 (t)
 −x1 (t) 




..
dx(t) = f (x(t), t)dt + ξdB1 + σ 
 dB2 (t).
.


 xn (t) 
−xn−1 (t)
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Exponential Growth
(3.12)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
For x ∈ Rn , compute
|x T g(x)|2 = (x T ξ)2 + (x T Ax)2 = (x T ξ)2 ≤ |ξ|2 |x|2
and
|g(x)|2 = |ξ|2 + |Ax|2 = |ξ|2 + σ 2 |x|2 .
That is, conditions (3.2) and (3.3) are satisfied with
c3 = |ξ|2 ,
c4 = 0,
c5 = |ξ|2 ,
c6 = σ 2 .
(3.13)
Consequently, the parameters defined by (3.4) becomes
a = |ξ|2 − c1 ,
b = σ 2 − c2 ,
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c = σ 2 − c1 − c2 − |ξ|2 .
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
If we choose
|ξ|2 > c1
and σ 2 = c1 + c2 + |ξ|2
(3.14)
then b = |ξ|2 + c1 ≥ a, c = 0 and
|ξ|4 − c12
a(|ξ|2 + c1 )
ab − 0.25c 2
≤
< a,
=
a+b−c
2|ξ|2
2|ξ|2
and hence, by Theorem 5, the solution of equation (3.12) obeys
lim inf
t→∞
|ξ|2 − c12
1
log(|x(t)|) ≥
t
4|ξ|2
Xuerong Mao
Exponential Growth
a.s.
(3.15)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Alternatively, if we choose
|ξ|2 > c1
and σ 2 = 3|ξ|2 + c2 − c1
(3.16)
then b = 3|ξ|2 − c1 = 2|ξ|2 + a > a, c = 2a and hence, by
Theorem 5, the solution of equation (3.12) obeys
lim inf
t→∞
1
log(|x(t)|) ≥ 12 (|ξ|2 − c1 )
t
Xuerong Mao
Exponential Growth
a.s.
(3.17)
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Case 2: The dimension n of the state space is odd and n ≥ 3.
Choose the dimension of the Brownian motion m to be 3 and
design g : Rn → Rn×3 by
g(x) = (ξ, A2 x, A3 x),
where ξ =
6 0, while

0 σ
−σ 0


..

.
A2 = 

0 σ


−σ 0










 , A3 = 






0
with σ > 0.
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
0
Exponential Growth
0 σ
−σ 0





..

.

0 σ
−σ 0
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
So the stochastically perturbed system (3.11) becomes




0
x2 (t)
 x2 (t) 
 −x1 (t) 




 −x3 (t) 


..




.
dx(t) = f (x(t), t)dt+ξdB1 +σ 
 dB3 (
 dB (t)+σ 
..

 xn−1 (t)  2

.




 xn (t) 
−xn−2 (t)
−xn−1 (t)
0
(3.18)
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
For x ∈ Rn , compute
|x T g(x)|2 = (x T ξ)2 + (x T A2 x)2 + (x T A3 x)2 = (x T ξ)2 ≤ |ξ|2 |x|2
and
|g(x)|2 = |ξ|2 + |A2 x|2 + |A3 x|2 ≥ |ξ|2 + σ 2 |x|2 .
Hence, conditions (3.2) and (3.3) are satisfied with the same
parameters specified by (3.13).
If we choose ξ and σ as (3.14) then the solution of equation
(3.18) obeys (3.15).
If we choose ξ and σ as (3.16) then the solution of equation
(3.18) obeys (3.17).
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Case 3: Scalar case (n = 1). consider a linear ordinary
differential equation
ẏ (t) = p − qy (t),
t ≥ 0,
(3.19)
where p and q are both positive numbers. It is known that for
any given initial value, the solution of this equation obeys
lim y (t) =
t→∞
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p
.
q
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
However, the solution of the corresponding linear SDE
dx(t) = (p − qx(t))dt +
m
X
(ui + vi x(t))dBi (t)
(3.20)
i=1
obeys
lim sup
t→∞
log(|x(t)|)
≤1
log t
a.s.
(3.21)
That is, the solution of this SDE will grow at most polynomially
with probability one.
In
other words, the linear stochastic perturbation
Pm
i=1 (ui + vi x(t))dBi (t) may not force the solution of a scalar
system ẏ (t) = f (y (t), t) to grow exponentially. endframe
Xuerong Mao
Exponential Growth
Introduction
Noise Suppresses Exponential Growth
Noise Expresses Exponential Growth
Motivating Examples
Exponential growth of SDEs
Noise Expresses Exponential Growth
Theorem
Any nonlinear system ẏ (t) = f (y (t), t) can be stochastically
perturbed by Brownian motions into the SDE (3.11) whose
solutions will grow exponentially with probability one provided
that the condition
−2hx, f (x, t)i ≤ c1 + c2 |x|2 ,
∀(x, t) ∈ Rn × R
is satisfied and the dimension of the state space is greater than
1. Moreover, the noise term g(x(t))dB(t) in (3.11) can be
designed to be a linear form of x(t).
Xuerong Mao
Exponential Growth