The End of Stability

x∈R R

x∈R R

x∈R R

F(x, νt)

G(x, tν)

νt= 0 νt∈(0,1) νt >1

(F+G)(x, νt)

Figure 10: Illustration of Example 5.32 part a). For the replacement system, a symmetric pitchfork bifurcation manifests attν= 2. Here,ε >0is some positive constant to illustrate the principle shape change of the potentialF+G.

The rest of the Assumption 5.30 is mostly a comfort of notation

• As we have seen in (5.2.12), using aν-adiabatic solution as reference provides a cor-rection term at most of order ν in the dierential law, see (5.2.13), (5.2.14). An additional term of that size can be dealt with similar to the treatment of 2σh√

t∆

within the proof of Theorem 5.35. This would lead to a modication of the family of successive upper bounds(β_i^?)i∈I at most of order|logν|. Setting x^adν =x^? ofν oblit-erates this minor inaccuracy. In other words, theν-adiabatic solution, which is dened with respect to the replacement system, from now on satises the delay dierential law when neglecting nonlinear terms.

• Moreover, the inaccuracy in the denition of time points, see e.g. Remark 5.3, comes to an end.

Example 5.32. The following two examples serve as paragons for our study concerning the transition phase.

a) An instance of a delayed symmetric pitchfork bifurcation that satises all requirements of Assumption 5.30 except for c₂), and also uniformly vanishing quadratic nonlinearity, that we will consider in Subsection 5.3.4, is given by

F(x, tν) :=

Z x 0

f(y, tν)dy=x⁴+ 2x², G(x, tν) :=

Z x 0

g(y, tν)dy=−(1 +νt)x²











forx∈R, t∈[0, T].

Here, x^adν(·) = 0 is an adiabatic solution that follows the equilibrium branch x^?(·) = 0 over the whole time interval. The equilibrium branch x^? is uniformly stable up to any timet∈(0,1) and becomes unstable atνt= 1. See Figure 10 for an illustration.

b) Another example is given by the linear nonautonomous journey through the stability boundary which we have seen in Figure 4. Consider







dx(t) =−a(t)x(t)dt+b(t)x(t−r)dt+σdW(t) for all t∈[0, T /ν], x₀= Υ,

where (a(t), b(t))_{[0,T /ν]} leaves the area S˜ := {x, y ∈ R : x < −|y|} at t = T₂/ν. See Figure 11.

−ar br

(r⁻¹,−r⁻¹)

S ˜

S

−4

−4

−3

−3

−2

−2

−1

−1

1 1

2 2

(a(t), b(t))

Figure 11: Illustration for Example 5.32 b). The arrow-headed lines represent possible shapes of the coecient-combination paths((a(t)r, b(t)r)t∈[0,T /ν] through the boundary of S˜. The pale yellow area and the labels have been taken over from Figure 4 for comparability. The parameter-combination journey, that corresponds to the dotted double-arrow headed line, is covered by our results, while the actual analytic area of stabilityS is not left in the case of the dotted line, when the combination escapes fromS.˜

We introduce a time-dependent noise amplier

F : [T1/ν, T2/ν]→[0,1]adapted and bounded by1.

The reason for this slight extension is that it simplies the work on the upcoming time interval, when the system turns slowly increasingly unstable. The Gaussian nature of lin-earizations stays untouched by this. Further, as a notational update, for this subsection we will denote

a+:= sup

t∈[T1/ν,T2/ν]

a(t)≥ sup

t∈[T1/ν,T2/ν]

b(t) =:b+.

Starting fromT₁/ν, we keep denoting the deviation process asy, given as the unique solution of











dy(t) = −a(t)y(t) +b(t)y(t−r) +Rf(y(t), νt) +Rg(y(t−r), νt)

dt+σF(t)dW(t) fort≥T1/ν, y_T1/ν= Υ,

for some appropriateΥ∈ C(J,R). And we keep thinking of yas the deviation process of a solution of (5.1.3) from the ν-adiabatic solutionx^adν = 0. Due to the previous subsection we conveniently assume thatkΥk ∈ O(√

ν). For somen∈N, let (θi :i∈ {0,1, . . . , n−1}) withT₁/ν=θ₀< θ₁< . . . < θ_n=T₂/νdenote the equidistant partition of[T₁/ν, T₂/ν]into n pieces with step width t∆ = ^T2_nν^−T1, where n is chosen big enough such that t∆ < r at least. Informally speaking, in timeT₁/ν, there are ^T2√−T_ν ¹ time units left before the stability

is lost and the system has completed the transition to a possibly critical or unstable regime, i.e., ^|b(s)|_a(s) ≥1. To be precise at that point, criticality or instability of the linearized system frozen in T2/ν is only reached in case of positive delayed feedbacka(T2/ν) =b(T2/ν)>0, represented through the pale yellow overhang in Figure 11. But the deduced estimates apply just as well for the case b(T2/ν) =−a(T2/ν)<0. For a nondecreasing family of constants 0< β0≤β1≤. . .≤βn, let

τβ(x) = inf (

t∈[T1/ν, T2/ν] :|y(t)|>

n−1

X

i=0

βi1[θi,θi+1](t) )

. (5.3.4)

We continue to denoteI:={0, . . . , n−1}. For an appropriate choice of(β_i)_i∈I we are going to show that

|y(t)| ≤βi for allt∈[θi, θi+1] andi∈I with high probability.

For alli∈Iwe dene the linear nonautonomous piecewise approximation(Y⁽ⁱ⁾(t))_{t∈[θi−r,θi+1]} as the unique solutions of







dY⁽ⁱ⁾(t) =−a(t)Y⁽ⁱ⁾(t) +b(t)Y⁽ⁱ⁾(t−r)dt+σF(t)dW(t) fort∈[θ_i, θ_i+1], Y_θ⁽ⁱ⁾

i =y_θi, (5.3.5)

each of which admits a representation through the variation-of-constants formula, see The-orem 3.5. To that end, let ( ˇY(t, u) : u ∈ [T₁/ν−r, T₂/ν], t ∈ [u−r, T₂/ν]) denote the fundamental solution corresponding to the dierential law of (5.3.5) initiated at some u, and evaluated at t. Within a regime with a(·) > |b(·)|, through a simple contradiction argument as in Lemma 4.1, we may deduce that

|Yˇ(t, u)| ≤1 for allu∈[T1/ν−r, T2/ν], t∈[u−r, T2/ν]. (5.3.6) Let further denote(T^det(t, u) :u∈[T1/ν, T2/ν], u−r≤t)denote the solution semi group, that maps from J to R, of the deterministic counterpart of system (5.3.5), initiated atu and evaluated in t. Then, we may rewrite the approximation for eachi∈I as

Y⁽ⁱ⁾(t) =T^det(t, θi)yθ_i+σξ⁽ⁱ⁾(t) with ξ⁽ⁱ⁾(t) :=

Z t θ_i

Yˇ(t, u)F(u)dW(u) for allt∈[θ_i, θ_i+1].

(5.3.7)

The rst summand is the solution of the deterministic version of (5.3.5) and as long as a(·)>|b(·)|, analogue to (5.3.6), it is easy to check that

|T^det(t, θi)yθ_i| ≤ kyθ_ik= sup

u∈[−r,0]

|y(θi+u)| ≤ sup

u∈[T1/ν−r,θi]

|y(u)| for allt∈[θi, θi+1].

The second summand(ξ⁽ⁱ⁾(t))_{t∈[θi,θi+1]} solves







dξ⁽ⁱ⁾(t) = −a(t)ξ⁽ⁱ⁾(t) +b(t)ξ⁽ⁱ⁾(t−r)

dt+σF(t)dW(t) fort∈[θ_i, θ_i+1], ξ⁽ⁱ⁾_θ

i = 0,

and actually forms a Gaussian process for everyi∈I. Then, an application of the Fernique

inequality provides the following lemma.

Lemma 5.33. For h >p

1 + 4 logp where p∈N with √

p logp≥4(a+t∆+ 1), (5.3.8) we have that

P (

sup

t∈[θi,θi+1]

|ξ⁽ⁱ⁾(t)|> h2√ t_∆

)

≤ 5p² 2 e^−h

2

2 for alli∈I.

Proof. We x i∈I, writeY⁽ⁱ⁾ =Y, and focus on ( ˇY(t, u) :u∈[θi, θi+1], t≥u−r). The Fernique parametersQ=Q_ξ(i) andΓ = Γ_ξ(i) are easily obtained. We nd that

kΓk= sup

s∈[0,t∆]

E

ξ⁽ⁱ⁾(θi+s)²

= sup

s∈[0,t∆]

Z s 0

Yˇ²(θi+s, θi+u)F²(θi+u)du

≤ Z t_∆

0

F²(θi+u)du≤t∆.

Further, rewriting s=s−θ_i, t=t−θ_i and formally substitutingv =u−θ_i, it is easy to see that

E

"

Z t θi

Yˇ(t, u)F(u)dW(u)− Z s

θi

Yˇ(s, u)F(u)dW(u) ²#

= Z s

θi

Yˇ(t, u)−Yˇ(s, u)²

F²(u)du+ Z t

s

Yˇ²(t, u)F²(u)du

≤ Z s

0

Yˇ(θi+t, θi+v)−Yˇ(θi+s, θi+v)² dv+

Z t s

Yˇ²(θi+t, θi+v)dv for alls, t,∈[θi, θi+1], i.e., s, t∈[0, t∆].

Then, with (5.3.6) we nd that Z s

0

Yˇ(θi+t, θi+v)−Yˇ(θi+s, θi+v)² dv

= Z s

0

Z t−s 0

−a(θi+s+u) ˇY(θi+s+u, θi+v) +b(θi+s+u) ˇY(θi+s+u−r, θi+v)du

2

dv

≤(a++b+)²t∆(t−s)²≤4a²₊t∆(t−s)² for alls, t∈[0, t∆], s≤t.

And it is easy to see that Z t

s

Yˇ²(θ_i+t, θ_i+v)dv≤t−s for alls, t∈[0, t_∆], s≤t.

Therefore, with a glimpse at the previously used notation in (3.4.10) and (3.4.11), and the formulas from the Appendix A.2, we obtain that

Q1≤2a+

√t∆

Z ∞ 1

t∆p^−u2du≤ 2a₊t^3/2_∆

2plogp and Q2≤ Z ∞

1

√t∆p^−u

2 2 du≤

√t∆

√plogp.

Finally, we may deduce from the condition on the minimal size of p, which we stated in (5.3.8), that

pkΓk +Q(t∆)≤√

t∆ + 2 +√ 2

√t∆

√plogp+ a+t

3 2

∆

plogp

!

=√ t_∆

1 + (2 +√ 2 )

1

√plogp+ a₊t_∆ plogp

≤2√ t_∆. And the claim follows through an application of the Fernique inequality.

Let us denote τ_2h^(i)√_t

∆ := infn

t∈[θi, θi+1] :|ξ⁽ⁱ⁾(t)|>2h√ t∆

o for alli∈I, h >0, τ_2h^√_t

∆(ξ) := minn τ⁽ⁱ⁾

2h√

t∆ :i∈Io

for allh >0. (5.3.9) Then obviously, forhandpsatisfying (5.3.8), we have that

P

τ_2h^√_t∆(ξ)< T2/ν ≤n5p² 2 exp

−h² 2

, (5.3.10)

which makes the below Corollary is just as obvious. It states the result if we plug in the previous corollary into representation (5.3.7).

Corollary 5.34. For h >0 andp∈N satisfying (5.3.8), we have that

|Y⁽ⁱ⁾(t)| ≤ sup

u∈[T1/ν−r,θi]

|y(u)|+h2σ√ t∆

for allt∈[θi, θi+1], t≤τ_2h^√_t∆(ξ), i∈I.

AndP{τ_2h^√_t∆(ξ)< T /ν} ≤3np²exp(−h²/2). Further, for alli∈Iwe dene

Z⁽ⁱ⁾(t) :=y(t)−Y⁽ⁱ⁾(t) for allt∈[θi, θi+1]. (5.3.11) whichP-almost surely solves







dZ⁽ⁱ⁾(t)

dt =−a(t)Z⁽ⁱ⁾(t) +b(t)Z⁽ⁱ⁾(t−r) +Rf(y(t), νt) +Rg(y(t−r), νt)fort∈[θi, θi+1), Z_θ⁽ⁱ⁾

i = 0.

By Theorem 3.5 the pieces(Z⁽ⁱ⁾(t))_{t∈[θi,θi+1]} also admit respective representations through the variation-of-constants formula in terms of the previously dened fundamental solution Zˇ(t, u) = ˇY(t, u), u∈ [θ_i, T /ν], t ∈ [u−r, T /ν], characterized through its dierential law (5.3.5). Namely,(Z⁽ⁱ⁾(t))_{t∈[θi,θi+1]} may be written as

Z⁽ⁱ⁾(t) = Z t

θ_i

Z(t, u)ˇ

Rf(y(u), νu) +Rg(y(u−r), νu)

du for allu∈[θ_i, θ_i+1), i∈I.

Here, the fact thatf andg are supposed to be odd functions atT2/ν comes into play and

serves through (5.3.3) that

Z⁽ⁱ⁾(t)<

Z t θi

√ν|logν|N˜fy²(u) +√

ν|logν|N˜gy²(u−r) +Mfy³(u) +Mgy³(u−r)du

≤t_∆√

ν|logν| sup

u∈[T1/ν−r,θi+1]

|y(u)|

!2

+t_∆M sup

u∈[T1/ν−r,θi+1]

|y(u)|

!3

for allt∈[θ_i, θ_i+1], t≤τ_2h^√_t

∆(ξ), i∈I.

And therefore,

|y(t)|< sup

u∈[T1/ν−r,θi]

|y(u)|+ 2hσ√ t∆ +√

ν|logν|N˜ sup

u∈[T1/ν−r,θi+1]

|y(u)|

!2

t∆

+M sup

u∈[T₁/ν−r,θ_i+1]

|y(u)|

!³ t∆

for allt∈[θi, θi+1], t≤τ_2h^√_t∆(ξ), i∈I.

(5.3.12) Theorem 5.35. In the situation of Assumption 5.30 let (θi : i ∈ {0, . . . , n−1}) denote the equidistant partition of [T1/ν, T2/ν] with step widtht∆= _n^∆√1,2_ν|logν|< r. Assume that there isk >0such thatky_T1/νk< k√

ν. Assume further forh=O log_∆

√1,2

ν |logν| that ν andσ < _|log^ν_ν|2 are small enough such that there is K > k, independent ofν, with

e^{16νt∆K( ˜N|logν|+M K)}

∆1,2 t∆

√ν|logν|

k+2σh√ t_∆

√ν

∆_1,2 t∆

√ν |logν|

< K. (5.3.13)

Dene the family of increasing constants(β^?_i)i∈{−1,0,1,...,n−1} inductively through







β₋₁^? :=k√ ν ,

β_i^?:=β^?_i−1 1 + 16νt∆K( ˜N|logν|+M K)

+ 2σh√

t∆ fori∈I, (5.3.14) and suppose thatσ andν are small enough such that

4( ˜N|logν|+M K)√

ν t_∆(β_i−1^? + 2hσ√

t_∆)<1 for alli∈I, (5.3.15) K√

ν < 1

2( ˜N|logν|+M K)√ ν t∆

, (5.3.16)

2hσ√

t_∆ ≤k√

ν . (5.3.17)

Then, the following assertions hold true:

a) We have that β_n−1^? ≤K√ ν.

b) For all i∈I we have thatβ_i^?≥β_i−1^? + 2hσ√

t∆ + ˜N√

ν|logν|(β_i^?)²+M(β_i^?)³.

c) The family (β_i^?)i∈{−1,0,1,...,n−1} constitutes an upper bound for y, i.e.|y(t)| ≤β_i^? for all t∈[θi, θi+1] and{−1, . . . , n−1} as long ast < τ_2hσ^√_t∆ and (5.3.10) holds true.

Proof. a) The simple recursion formula for(β_i^?)_i∈I can be explicitly solved and serves β_i^?=β₋₁^?

1 + 16νt_∆K( ˜N|logν|+M K)ⁱ⁺¹ + 2hσ√

t∆ i

X

j=0

(1 + 16νt∆K( ˜N|logν|+M K))^j for alli∈I.

Using that(1+x)≤e^x, we receive an upper boundary forβ_n−1^? forn= _t^∆1,2

∆

√ν|logν|through β_i^?≤β_n−1^? ≤β₋₁^? e^{16νt∆K( ˜N|logν|+M K)}

∆1,2 t∆

√ν|logν|

+ ∆_1,2 t∆

√ν |logν|2hσ√

t_∆e^{16νt∆K( ˜N|logν|+M K)}

∆1,2 t∆

√ν|logν|

. And therefore,

β^?_n−1<√

ν e^{16νt∆K( ˜N|logν|+M K)}

∆1,2 t∆

√ν|logν|

k+2σh√ t∆

√ν

∆1,2

t∆

√ν |logν|

. Note that this was assumed to be less or equal to K√

ν in (5.3.13).

b) In a rst step, we show thatβ^?_i ≥β_i−1^? + 2hσ√

t_∆ + ( ˜N|logν|+M K)√

ν(β^?_i)²t_∆. For i∈Irewriting the desired inequality with the notation R:= ( ˜N|logν|+M K)yields

β_i^?≥β_i−1^? + 2hσ√

t∆ +R√

ν(β_i^?)²t∆

⇔(β_i^?)²− 1 R√

ν t∆

β^?_i + 1 R√

ν t∆

β_i−1^? + 2hσ√ t_∆

≤0. (5.3.18) Then through inequality (5.3.15) on the size of σ, inequality (5.3.18) is true for β_i^? ∈ [β_i^(?1), β_i^(?2)], where

β^(?1)_i = 1 2R√

ν t∆

− s 1

4R²νt²_∆ − 1 R√

ν t∆

β_i−1^? + 2hσ√ t_∆

= 1

2R√ ν t∆

− 1

2R√ ν t∆

s

1− 4R²νt²_∆ R√

ν t∆

β_i−1^? + 2hσ√ t_∆

, and

β_i^(?2)= 1 2R√

ν t_∆ + s 1

4R²νt²_∆ − 1 R√

ν t_∆ β_i−1^? + 2hσ√ t∆

=O 1

√ν t_∆

.

We use the fact that√

1−x >1−^x₂ −^x₂² forx∈(0,1) and obtain that β^(?1)_i < 1

2R√ ν t∆

− 1

2R√ ν t∆

1−2R√

ν t∆(β_i−1^? + 2hσ√ t∆)

−8R²νt²_∆(β_i−1^? + 2hσ√ t∆)²

=β_i−1^? + 2hσ√

t_∆ + 4R√

ν t_∆(β_i−1^? + 2hσ√ t_∆)².

Then, through (5.3.17) we know thatσis small enough such that 2hσ√

t∆ ≤β₋₁^? ≤β^?_i−1 for alli∈I.

Hence, the squared-parentheses term satises (β_i−1^? + 2σh√

t∆)²≤4(β_i−1^? )²≤4K√ ν β_i−1^? . Then, we receive that

β_i^(?1)< β_i−1^? + 2hσ√

t∆ + 16KRνt∆β^?_i−1

=β_i−1^? (1 + 16K( ˜N|logν|+M K)νt_∆) + 2hσ√

t_∆ =β_i^?. And by assumption (5.3.16), we have that

β_i^?≤β_n−1^? ≤K√

ν < 1 2R√

ν t∆

≤β^(?2)_i for alli∈I.

Therefore,β_i^? satises the desired inequality (5.3.18).

c) Letτ_β?(y)be dened as in (5.3.4). From (5.3.12) we know that

|y(t)|< β_i−1^? + 2hσ√

t∆ + ˜N|logν|√

ν(β_i^?)²t∆+M(β^?_i)³t∆

for allt < τ_β^?

i(y), t∈[θ_i, θ_i+1], t≤τ_2h^√_t

∆(ξ), i∈I, while

β_i−1^? + 2hσ√

t∆ + ˜N|logν|√

ν(β_i^?)²+M(β_i^?)³t∆≤β^?_i fort∈[θi, θi+1], t≤τ_2h^√_t∆(ξ), i∈I, which actually shows thatτβ^?(y)> τ_2h^√_t∆(ξ)providing that

|y(t)| ≤β^?_i for allt∈[θi, θi+1], t≤τ_2h^√_t∆(ξ), i∈I.

The three assumptions (5.3.15), (5.3.16), (5.3.17) do not raise any further issue. The rst one is in principle justied through a) sinceνis considered to be small. Assumption (5.3.16) is also naturally fullled for suciently smallν, and so is Assumption (5.3.17), because we assumed thatσ < ν.

Remark 5.36. In particular, |y(t)| will not exceed a size of order √

ν before stability is ultimately lost (with high probability) if σ≤ _|log^ν_ν|2.

Im Dokument Concentration Inequalities for Nonautonomous Stochastic Delay Differential Equations (Seite 118-126)