The first passage problem for stable linear delay equations perturbed by power law Lévy noise
Michael A. Högele Ilya Pavlyukevich
















The first passage problem for stable linear delay equations perturbed by power law Lévy noise 

    Michael A. Högele[Departamento de Matemáticas, Universidad de los Andes,
Bogotá, Colombia; ma.hoegele@uniandes.edu.co]   
 and
 Ilya Pavlyukevich[Institut für Stochastik, Friedrich–Schiller–Universität Jena, Ernst–Abbe–Platz 2, 
07743 Jena, Germany; ilya.pavlyukevich@uni-jena.de]

    
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This article studies a linear scalar delay differential equation subject to small
multiplicative power tail Lévy noise. 
We solve the first passage (the Kramers) problem with probabilistic methods 
and discover an asymptotic loss of memory in this non-Markovian system.
Furthermore, the mean exit time increases as the power of the small noise amplitude, whereas
the pre-factor accounts for memory effects. 
In particular, we discover a non-linear 
delay-induced exit acceleration due to a non-normal growth phenomenon. 
Our results are illustrated in the well-known linear delay oscillator  
driven by α-stable Lévy flights.


Keywords:  
linear delay differential equation; 
α-stable Lévy process; 
Lévy flights; 
heavy tails;
first passage times;
first exit;
exit location.



AMS Subject Classification:  
60H10; 60G51; 37A20; 60J60; 60J75; 60G52









§ INTRODUCTION AND MAIN RESULTS


The Kramers problem, that is, 
the escape time and location of a randomly excited deterministic dynamical system 
from the proximity of a stable state at small intensity 
was first stated in the context of physical chemistry in the seminal works of 
<cit.>, <cit.> and <cit.>. The solution of this 
classical problem is ubiquitous nowadays 
and has given since crucial insight in many diverse areas such as statistical mechanics, 
insurance mathematics, climatic energy balance models and led to the discovery of more complex dynamical effects 
such as for instance stochastic resonance in complex systems <cit.>. 

In the mathematics literature, 
for Markovian systems such as of ordinary and partial differential equations with small Gaussian noise 
this problem was studied extensively with the help of large deviations theory 
which goes back to the seminal work by <cit.> and later by <cit.>, and in the 
recent years 
by 
<cit.>. 
It is well-known 
that for -small Brownian diffusion in the potential well, 
the expected exit (excitation) time grows exponentially as

    τ^∼^V̅/^2,

where V̅ is proportional to the height of the potential barrier  
which has to be overcome by the particle. The exit location is determined by 
the deterministic energy minimizing path. 

<cit.>, however, discusses “physical origins of stochastic forcing” and the 
trade off between Gaussian and non-Gaussian white and colored noises. 
Following their lines of reasoning, modelling with Lévy noises 
is the second best choice in complex systems, since they allow for richer effects
such as 
local asymmetry and the presence of large bursts and jumps which are difficult to realize only with Gaussian influences. 
Furthermore <cit.> give evidence for physicality 
of Lévy noises in particular due to the predictability of fat-tail extremes 
while in the limit of small noise intensity, 
the (re-normalized) exit times are exponentially distributed and hence memoryless or unpredictable. 
Heavy-tailed noise has also been found present in many physical systems, for instance in works by 
<cit.>, <cit.> and <cit.>.

Due to a large variety of Lévy processes, i.e. stochastically continuous processes with independent 
stationary increments, there is no general Kramers' theory for Lévy driven systems. 
Besides the well studied Gaussian case, 
a large deviations result for heavy tail Markovian processes was obtained by <cit.>.  
For exponentially light jumps this question has been solved in 
one dimension by <cit.>. 
In addition, there is a large deviations theory for a special class of 
parameter dependent accelerated noise with exponentially light tails 
by <cit.> based on a variational representation. 
All these approaches yield exponential exit rates on the precise noise dependence. 
Further recent results on the first exit and metastability of Lévy driven systems in finite and infinite dimensions 
were obtained by
<cit.>. 
It is worth mentioning that in the case of an overdamped particle subject to -small 
α-stable noise, α∈ (0,2), the expected exit time behaves polynomially 
 
    τ^∼V̅/^α,

whereas the constant V̅ has a very different interpretation. 
It is not the lowest height of any mountain pass the continuous Brownian diffusion 
path has to climb between different potential wells. 
Instead, it quantifies the tunelling effect
of the “large” jumps that instantaneously overcome the 
(horizontal) distance 
between the deterministic stable state 
where the process lingers most of the time between such “large” jumps  
and the exterior of its domain of attraction. 

In this article, we study such small heavy tailed perturbation 
of a beforehand non-Markovian dynamical system given as a linear delay differential equation. 
The simplest deterministic qualitative model of this kind is given by a linear retarded equation  
for the El Niño-Southern Oscillation phenomenon (ENSO) in <cit.>:

/t X(t)   =AX(t)+ B X(t-r), 
where A is the sum of all processes that induce local changes in the SST, 
that is, the horizontal advection, thermal damping, mean and anomalous upwellings on the 
vertical temperature gradients. The coefficient B subsumes the effects 
of the equatorial Kelvin waves. 
Positive A means that the sum of effects of upwelling and thermal advection 
dominate the thermal damping, so that the temperature grows. 
However, negative values of B can induce stable or periodic solutions. 
For instance, the parameter choice A=2.2, B=3.8, and a delay time of r=0.5 as in <cit.> 
leads to unstable oscillations with a period of approximately 3.0 years and
a growth rate of 1.1 while <cit.> argued that changing this to A=2.4,
B=-2.8, and a delay time of r=0.3 leads to a period of approximately 4 years and
a decay rate 1.5.

A more complex non-linear double-well model with an additional cubic term -X^3 was considered by <cit.>, 
followed by a number of papers
by e.g. <cit.>.  

<cit.> studied the following delay equation perturbed by a small Brownian motion W
X^(t)   = 𝒜(X_t^) t+   W(t)

with a nonlinear Lipschitz continuous vector field 
𝒜. 
Applying a control theoretic approach to this equation 
he established a large deviations principle and 
showed that in analogy to the non-delay case discussed above 
the asymptotocs (<ref>) holds true, however, with V̅ being an abstract solution 
of a difficult delay control problem. 
Recently <cit.> extended these results to 
the small noise SDDEs with multiplicative noise  
in the spirit of <cit.> and established the asymptotics of the 
first exit time of the type (<ref>).
<cit.> considered the retarded Gaussian delay equation 
as a Gaussian process and established the respective large deviations principle 
and the optimal exit paths with the help of the very elaborate Gaussian process theory.
In <cit.> the authors study delay systems perturbed by small accelerated 
Lévy noise with light tails in the spirit of <cit.>. 
More on stochastic double-well systems can be found in <cit.> and 
for Lévy noise also in <cit.>, where the authors study 
the asymptotics in the limit of small delay. 
 
In this paper we study the first exit problem from an interval 
of the delay differential equation 
X^(t)   = 𝒜 X^_t t+ F(X^_t)  Z(t),

with a general linear stable finite delay 𝒜 
perturbed by a small multiplicative heavy-tailed Lévy noise Z, 
including α-stable but also more general weakly tempered perturbations.  
The phenomenological reason for our setting is that on the one hand 
we recover the rate (<ref>), however, 
we detect a new non-normal growth effect in the factor V̅, 
which we can calculate explicitly in the case of a retarded system (<ref>). 
This effect accounts for the non-zero probability of small jump increments, 
which leads to an exit due to deterministic motion well after the occurrence of this jump 
and can be seen in the asymptotic distribution of the exit location, 
which in contrast to the non-delay case exhibits a point measure precisely 
on the boundary of the exit interval. 
It is easily seen that this effect vanishes if we send the memory depth to 0. 
In other words, at first sight these results for non-Markovian systems appear surprising, 
however, since the delay time r is negligible w.r.t. the exit time scale 
^-α, the system behaviour is “almost” Markovian.
Nevertheless the memory affects the prefactor in the asymptotics 
of the first exit time and the limiting 
distribution of the exit location.

The methodological reason for considering this equation is 
the adaption of the proof strategy by <cit.> and 
<cit.> which is an elementary but very helpful
application the Markov property which
seems to be suitable for adaptation in different contexts of the physics literature. 
In addition, our setup covers generic non-degenerate potential gradient systems the linearized around their stable state. 

A technical reason to study this particular problem is that 
in order to trace this effect we need a precise 
understanding of the deterministic dynamics.  
In particular, several important properties of this equation 
are readily given in the literature, 
such as existence, uniqueness of solutions and the invariant measure in <cit.> and  
the segment Markov property in <cit.>.


The article is organized as follows. After the general setup we present our main result 
in Theorem 2.2 followed by the discussion and examples, 
where we compare our results to the linear setting of <cit.> 
and explicitly calculate the nonlinear growth factor. 
The rigorous proofs are postponed to the Mathematical Appendix and consist of two parts. In Section <ref> we give general estimates on the deterministic 
relaxation dynamics and then on the stochastic perturbation. 
In Section <ref> in  we show a generic upper and a lower bound 
on the segment Markov process reducing the dynamics 
to four scenarios on a finite interval. 
This finite interval dynamics 
is treated in Section <ref> for each of 
the cases what leads to the proof of the main result. 




§ OBJECT OF STUDY AND RESULTS




 §.§ General setup


The model under consideration is the following linear delay equation 
with finite memory r>0 perturbed by small Lévy noise defined below.
For r>0 fixed and each T>0, we denote by D[-r,T] 
the space of real valued right-continuous functions ϕ [-r, T] → with left limits (the so-called càdlàg 
functions). Analogously we define the space D[-r,∞). For a function ϕ∈ D[-r,∞) we 
we introduce the segment
of ϕ at time t ≥ 0 as the function ϕ_t(·)∈ D[-r,0] defined by
ϕ_t(s)=ϕ(t+s) for s∈[-r,0].
For a function ϕ∈ D[-r,0], we denote its uniform norm by ϕ_r:=sup_t∈[-r,0]|ϕ(t)|.

Let μ be a finite, signed measure on the interval [-r, 0], so-called the memory measure. 
Consider the following underlying deterministic linear delay equation

x(t; ϕ)    = ϕ(0) + ∫_0^t ∫_[-r,0] x(s+u; ϕ)  μ(u)  s,  t≥0,
 
x(t; ϕ)    = ϕ(t),  t∈[-r,0),
where ϕ∈ D[-r, 0]. It is well known that this equation has a unique solution 
which e.g. can be obtained by the method of steps as in <cit.>.
Similarly to the case of linear ODEs or PDEs, 
the solution x(·; ϕ) can be written down explicitly 
as a convolution integral, namely,

x(t;ϕ)    = ϕ(0) x^*(t) + ∫_[-r, 0] ∫_u^0  x^*(t-s+u)ϕ(s)  s   μ(u) ,  t≥0 
 
x(t;ϕ)    = ϕ(t) ,  t∈[-r, 0),  
where the fundamental solution x^*(·) is the unique solition of 
(<ref>) with the initial segment ϕ^*(t)=0, t∈[-r,0), and ϕ^*(0)=1.






Let (Ω, , (_t)_t≥ 0, ) be a stochastic basis satisfying the usual 
conditions in the sense of <cit.> and let Z=(Z(t))_t≥ 0 be an adapted real-valued 
Lévy process
with the characteristic triplet (σ^2, d, ν). The marginal laws of Z are described by the Lévy–Khintchine formula

ln^ıu Z(t)=-σ^2/2u^2t +ı  d u t+t∫(^ıuz-1-ıu z_[-1,1](z)) ν(z),
where the Gaussian variance σ^2≥ 0, the drift d∈, and the Lévy jump 
measure ν satisfies ν({0})=0 and ∫ (z^2∧ 1) ν( z)<∞. 
For the general theory on Lévy processes see <cit.>.
To introduce multiplicative noise into the equation (<ref>), 
we define the “diffusion” coefficient F D[-r,0]→ D[-r,0], which we 
assume to be functional Lipschitz, i.e.there is a constant L>0 such that for all ϕ, ψ∈ D[-r,0] we have

    F(ϕ)  -  F(ψ) _r ≤ L ϕ-ψ_r.

The functional F can be for example of the form F(ϕ)(t)=f(ϕ(t-r_1),…,ϕ(t-r_n)))
for point delays r_i∈ [0,r], and f being Lipschitz in all its arguments, such that
F(X^_t)(0-)=f(X^(t-r_1-),…,X^(t-r_n-)).
Further examples of F can be found in <cit.>.

Under the assumptions formulated above we consider the stochastic 
delay differential equation with an initial condition ϕ∈ D[-r,0] and >0

    X^(t)    = 
    ϕ(0) + ∫_0^t [∫_-u^0 X^(s+u) μ( u)]  s 
    + ∫_0^t F(X^_s)(0-)   Z(s),     t≥ 0,  
    
    X^(t)    = ϕ(t),    t∈ [-r, 0).

and denote X^(·;ϕ)=X^(·) its solution. 

The multiplicative noise term is understood as the Itô stochastic integral which requires 
the predictability of the integrand F(X^_s)(0-)=lim_h ↑ 0 F(X^_s)(h).
Note that in the pure Gaussian continuous setting F(X^_s)(0-)=F(X^_s)(0).

 
Fix ∈ (0,1] and let F be functional Lipschitz. 
Then for any ϕ∈ D[-r,0] there exists a unique solution X^ of equation (<ref>) which satisfies the convolution formula 

    X^(t; ϕ) =  ϕ(0)x^*(t)+∫_[-r,0]∫_s^0 x^*(t+s-u)ϕ(u)  s μ( u) + ∫_0^t x^*(t-s) F(X^(ϕ))(s-)  Z(s), t≥ 0.

The solution X^ is Markovian in the segment space D[-r,0], i.e. for each 0≤ s≤ t and each measurable set 
B⊆ D[-r,0]

(X^_t∈B|_s)=(X^_t∈B|X_s^) a.s.




 §.§ Main results


Our main result characterizes the interplay between the deterministic stability and the power laws 
of the noise. We will need the following Hypotheses.



H_μ:
We assume that the delay equation (<ref>) is stable, i.e.the memory measure μ satisfies 

    -Λ:= sup__ιRe_ι < 0,

for _ι being a solution of the characteristic equation
- ∫_-r^0 ^ u   μ(u) = 0.
Condition (<ref>) implies that 
for each λ<Λ there is a constant K=K(λ)>0 such that

|x^*(t)|≤K^-λt,  t≥0.
Zero is a stable state. For any function ϕ∈ D[-r,0] such that ϕ(t)=0, t∈[-r,0),
we assume that F(ϕ)(0-)=F_0≠ 0.



H_ν: The goal of this paper is to treat the heavy tail phenomena. 
A convenient analytic tool for this is the theory  
of regularly varying functions, i.e. functions which behave asymptotically like power
functions.
Let λ_ denote the tail of the Lévy measure ν,
λ_=∫_|z|>1/ν(z),  >0.
We assume that there exist α>0 and a non-trivial self-similar Radon measure ν̅ on \{0}
such that
for any u>0 and any Borel set A bounded away from the origin, 0∉B̅, 
the following limit holds true:

    ν̅(u B)=lim_→ 0ν(uB/)/λ_ = 
    1/u^αlim_→ 0ν(B/ )/λ_= 1/u^αν̅(B).

In particular, there exists a non-negative function l slowly varying at zero 
such that 
λ_= ^α l()     >0.
The self-similarity property of the limiting measure ν̅ 
implies that is has no atoms, ν̅({z})=0, z≠ 0, and hence, in the one-dimensional case, ν̅ always has 
the power density

   ν̅(z)=c̅_-(z<0)/|z|^1+α z+c̅_+(z>0)/z^1+α z,  α>0,  c̅_±≥0,  c̅_-+c̅_+>0.


For the interval [a,b], a<0<b, 
we define the first exit time
τ^=τ^(ϕ)   =inf{t≥0X^(t;ϕ)∉[a,b]}.


Due to the continuity of the fundamental solution we obtain that the set of jump sizes
[e_-,e_+]={z∈ such that  z·F_0·x^*(t)∈[a,b]  for all t≥0  }
is a closed interval with e_-<0<e_+.
Denote by 

E   =E(a,b)=  E(a,b;A,B,r):={z∈∃  t≥0 such that  z·F_0·x^*(t)∉[a,b]  }

   =[e_-,e_+]^c=(-∞,e_-)∪(e_+,+∞)

the set of jump sizes which cause the exit from the interval [a,b]. Furthermore consider the sets
   E_b={z∈z·F_0·x^*(t) exits from [a,b] into (b,∞) },

   E_a={z∈z·F_0·x^*(t) exits from [a,b] into (-∞,a) },

   E=E_aE_b,

   E_b^j(v)={z∈z·F_0∈(v,∞) },  v> b,

   E_a^j(v)={z∈z·F_0∈(-∞,v) },  v< a.
Recall that the homogeneity of the measure ν̅ guarantees that ν̅({e_±})=0. 



Let Hypotheses H_μ and H_ν hold true. 
Let [a,b] be an interval, a<0<b, and let ϕ∈ D[-r,0] be an initial segment with no exit, i.e.such that 

a< inf_t∈[-r,∞) x(t,ϕ)≤sup_t∈[-r,∞)x(t,ϕ)< b.
For the set E defined in (<ref>),
assume that ν̅(E)>0. 

1. For each u>0 we have 
lim_→0_ϕ(λ_τ^>u)= ^-uν̅(E).  
2. In addition, we have 
lim_→0 λ__ϕτ^=1/ν̅(E). 
3. In the limit → 0, the exit location is given by 
X^(τ^;ϕ)⟶ 
Π_a^j·c̅_-_(-∞,a)(z)/|z|^1+α z
+
Π_a^c ·δ_a(z)
+Π_b^c ·δ_b(z)
+Π_b^j·c̅_+_(b,∞)(z)/z^1+α z,
where
Π_b^j   =lim_v↓bν̅(E_b\E_b^j(v))/ν̅(E),    Π_a^j=lim_v↑aν̅(E_a\E_a^j(v))/ν̅(E),

Π_b^c   =Π_b-Π_b^j,    Π_a^c=Π_a-Π_a^j.
Note that
Π_a^j+Π_a^c+Π_b^c+Π_b^j=1.

We discover the positive weight Π^c = Π_a^c + Π_b^c on the boundary {a, b} 
which represents the probability of an asymptotically continuous exit from the interval [a, b] 
and stemps from the non-normal growth effect of the deterministic delay equation. 






 §.§ Examples and Discussion


We start this section with examples of Lévy processes with regularly varying heavy tails which satisfy Hypothesis H_ν.



Any α-stable Lévy process with the stability index α∈ (0,2), the skewness parameter β∈[-1,1], and the scale
parameter c>0 satisfies Hypothesis H_ν.  
Indeed, such a Lévy process Z has the characteristic function
^ıu Z(1)=                
	  exp( -c|u|^α(1-ıβtanπα/2)u  ),  α∈(0,1)∪(1,2),

	   exp( -c|u| (1+ıβ2/πu ln|u|  ),  α=1,
                   
and (see <cit.>) its jump measure ν has the form
   ν(z)=(c_-(z<0)/|z|^1+α+c_+(z>0)/z^1+α) z

with
c_±=
c·(1±β)/2|Γ(-α)|cos(απ/2),  α∈(0,1)∪(1,2),

c·(1±β)/π,  α=1.

In this case, the limiting measure ν̅ coincides with ν, so that c̅_±=c_± in (<ref>) and 

λ_=c_++c_-/α^α.


Weakly tempered stable Lévy processes form another important class of perturbations with heavy tails. Various ways of tempering 
have been introduced, e.g. in <cit.>. Roughly speaking, small jumps of a weakly tempered 
α-stable Lévy process
look like those of an α-stable process, but the large jumps, and hence the tails of the p.f.d. are of the order |x|^-1-r
for some r>0. It is easy to construct a weakly tempered stable Lévy process with the help of its jump measure defined as
ν(z)
   =(c_-(z<0)/|z|^1+α_1(1+z^2)^α_2/2+c_+(z>0)/z^1+α_1(1+z^2)^α_2/2) z,

   α_1∈(0,2), α_2>0,  c_±≥0,  c_++c_->0.
In this case the limiting measure is
   ν̅(z)= (c_-(z<0)/|z|^1+α+c_+(z>0)/z^1+α) z,  α=α_1+α_2>0,
and λ_ is as in (<ref>).


The Lévy measures from the previous examples can be “contaminated” by some slowly varying function l_±=l_±(z), 
like l(z)=ln(1+|z|) or
any finite nonnegative function l such that there exists limits l_± =lim_z→±∞ l_±(z)∈ (0,∞), e.g.one can consider jump measures of the form
   ν(z)=(l_-(z)(z<0)/|z|^1+α+l_+(z)(z>0)/z^1+α) z.
Moreover, the additional influence of any drift d and a Brownian motion σ W (see (<ref>)) 
is negligible in comparison to the heavy jumps and 
does not change the asymptotic characteristics of the exit time and location.


The sets E, E_a, E_b, E_b^j(v), and E_b^j(v) appearing in Theorem <ref> 
are determined in terms of the characteristics of the fundamental solution 
x^*. Generally, the fundamental solution is not known explicitly, however its maximum and minimum can be obtained numerically.
By the definition of the fundamental solution, its maximum satisfies
M=max_t≥0 x^*(t)≥1,
is well defined and is attained somewhere on [0,∞).
On the other hand the minimum of x^*(·) may not be attained (e.g. for x^*(t)=^-t) and we set
m=inf_t≥0x^*(t)≤0.
Assume for definiteness that F_0>0. Then the values e_± 
and the set E=[e_-,e_+]^c can be calculated explicitly as  

   e_+=sup{z>0  such that  z·F_0·x^*(t)∈(a,b)  for all t≥0}
=|a|/F_0|m| ∧b/F_0M,

   e_-=inf{z<0  such that  z·F_0·x^*(t)∈(a,b)  for all t≥0}
=-(|a|/F_0 M∧b/F_0|m|),
where we set 1/0=+∞. Analogously one can determine the sets E_a, E_b, E_a^j(v), and E_b^j(v)
but the explicit formulae cannot be given here in general since one needs additional information
whether M or m is attained first.

We finish the discussion with the analysis of linear retarded equation. 


Consider the linear retarded equation
Ẋ^(t)   =A X^(t)+B X^(t-r)+ Ż(t),  t≥0,
 
X^(t)   =ϕ(t), t∈[-r,0], 
driven by a symmetric additive α-stable noise Z
with the Lévy measure ν( z)=|z|^-1-α  z, α∈(0,2).
The stability region of the deterministic equation ẋ(t)=A x(t)+B x(t-r) obviously 
coincides with the stability region of the rescaled equation ẏ=Ã y(t)+B̃ y(t-1), 
where Ã=Ar, and B̃=Br; their fundamental solutions satisfy x^*(r t)=y^*(t), t≥ -1.
The stability region of the parameters (Ã,B̃) 
is depicted on Fig. <ref>. It is bounded by the upper straight
line Ã+B̃=0, Ã<1, and the lower line
which is given parametrically as
Ã=ζζ,  B̃=-ζ/sinζ,  ζ∈(0,π),
see <cit.> for more details.





























































































































































































































The most probable exit trajectories of the perturbed delay equation Ẋ^(t)=2.4 X^(t)-2.8 X^(t-0.3)+Ż(t) from
<cit.> are presented on Fig. <ref>. Taking into account the formulae (<ref>) and 
(<ref>) we find that the mean first exit time from an interval [a,b] around zero satisfies 

_ϕτ^≈1/2^α( 
M^α/|a|^α∨|m|^α/b^α
+
|m|^α/|a|^α ∨M^α/b^α
)^-1
where the values M=M(A,B,r) and m=m(A,B,r) depend on the values A, B, and r
in a complex nonlinear way, see Fig. <ref>. 
Denoting by
M̃=M̃(Ã,B̃)=M(Ar,Br,1) and m̃=m̃(Ã,B̃)=m(Ar,Br,1) 
the extreme values of the fundamental solution y^* we get that

_ϕτ^≈1/2^α( 
M̃^α/|a|^α∨|m̃|^α/b^α
+
|m̃|^α/|a|^α ∨M̃^α/b^α
)^-1.
The equation without delay, i.e. for B=0, is stable for A<0 with the 
fundamental solution x^*(t)=^A t, so that M=1 and m=0, and mean exit time has the asymptotics

_ϕτ^≈1/2^α(1/|a|^α+1/b^α)^-1,
see <cit.>. It is interesting to note that M=1 and m=0 holds for parameters Ã=Ar
and B̃=Br from a larger domain. This domain can be determined numerically and is depicted on  Fig. <ref>.
For parameters in this domain, the asymptotics of the mean exit times (<ref>) and 
(<ref>) of the equation with and without delay coincide, and the asymptotic exit location has no atoms in a and b.
Hence, the first exit dynamics of the delay equation is effectively the same as for the equation without delay.


Eventually it is instructive to compare our results with the asymptotics of the exit time in the Gaussian case which 
has been studied for the first time by <cit.>.

[Example 1, <cit.>]
Consider a linear retarded equation driven by the Brownian additive noise
Ẋ^(t)   =A X^(t)+B X^(t-r)+ Ẇ(t),  t≥0,
 
X^(t)   =ϕ(t), t∈[-r,0], 
The stability region of this equation has been described in the previous example. With the help of the 
large deviations theory one obtains that
for any ϕ with no exit (see (<ref>))

^2 ln_ϕτ^∼(a∧b)^2/r G(Ã,B̃),
where the value G is obtained in terms of the fundamental solution y^∗ of ẏ(t)=Ãy(t)+B̃y(t-1):

G(Ã,B̃)=2∫_0^∞(y^*(t))^2 t   =2/π∫_0^∞t/(Ã+B̃cost)^2+(t+B̃sint)^2

   =
B̃ q^-1sinq-1/Ã+B̃cosq ,  B̃+ |Ã|<0, q=√(B̃^2-Ã^2),

1+|Ã|/2|Ã|,   B̃=Ã<0,

B̃q^-1sinhq -1/Ã+B̃coshq ,   Ã+|B̃|<0, q=√(Ã^2-B̃^2),

see <cit.>. For the equation without delay, i.e. with B=B̃=0 one gets from 
(<ref>) and (<ref>) the well known exponentially large Kramers' time
_ϕτ^∼^|A|(a∧b)^2/^2,  A<0.






§ CONCLUSION
 

In this article we solve the first exit time and location 
problem from an interval [a, b], a < 0 < b,  
in a general class of stable linear delay differential equations 
for -small multiplicative, power law noise, such as α-stable Lévy flights. 
In particular, we cover the linearization of gradient systems 
close to a stable state. 

We recover, on the one hand, the asymptotic 
polynomial exit rate of the order ε^-α 
known in different Markovian settings 
which comes with an ε-independent prefactor.  
In the delay case this prefactor depends nonlinearly on the memory depth r>0 
and can significantly reduce the expected exit times 
compared to the non-delay case 
reflecting a non-normal growth phenomena 
of the deterministic delay dynamics 
which changes the stochastic dynamics. 
This mirrors the situation for Brownian perturbations, 
where such an effect has been known since a long time 
as explained in Example <ref>.
The case of retarded equations is explained 
in detail in Example <ref>. 

Secondly, this non-normal growth becomes evident 
in the limiting exit location from the interval [a, b] 
where in contrast to the non-delay case with jump exits 
we detect a point mass on the boundary. 
This mass stems from random trajectories which exit 
essentially due to deterministic motion. 

The method of proof applied in the mathematical 
appendix consists of a series of elementary 
but new estimates valid for general Markov processes 
and well suited to be adapted to other settings. 







§ MATHEMATICAL APPENDIX: PROOF OF THE MAIN THEOREM <REF>
 



 §.§ General estimates


This section reduces the first exit problem 
in three consecutive steps
from a global problem to several 
smaller problems all of which are local in nature. 
We start by showing that 
the perturbed dynamics is 
dominated by the deterministic dynamics 
plus the error term of the perturbation. 
This is carried out for rather general perturbations 
by stochastic processes given as semimartingales, 
which include the stochastic integrals 
∫_0^t F(X^_s)(0-)   Z(s) under consideration in (<ref>). 
In the sequel we show that the solution driven only by bounded jumps, 
that is, before a first large jump happens  
remains close to the deterministic solution. 
Finally we establish upper and lower bounds 
for the distribution tails and the expectation 
of the exit times (including exit locations) 
for general Markov processes provided we have enough control over 
the short term behavior, which is left for Section <ref> 
in order to conclude. 




  §.§.§ Estimates on the perturbed dynamics


For generalities of semimartingales we refer to the book by <cit.>. 
In general, semimartingales are the class of stochastic Itô integral processes. 



Let S be a càdlàg semimartingale, S(0)=0, ϕ∈ D[-r,0] and
let X be a stochastic process satisfying 
X(t,ϕ)   =ϕ(0)+∫_0^t[∫_[-r,0] X(s+u,ϕ) μ(u) ] s + S(t),     t≥0,

X(t,ϕ)   =ϕ(t),  t∈[-r,0).
Then for each λ∈ (0,Λ) there is C=C(λ,μ,r)>0 such that for t≥ 0

    |X(t,ϕ)| ≤ C·(ϕ_r·^-λ t+ sup_s∈[0,t]|S(s)|),
       |X(t,ϕ)-ϕ(0)x^*(t)|≤ C ·(sup_s∈[-r,0)|ϕ(s)| ·^-λ t +   sup_s∈[0,t]|S(s)|),
        |X(t,ϕ)-x(t,ϕ)|≤ C ·sup_s∈[0,t]|S(s)|.


Inequality (<ref>) estimates the growth of the perturbed solution X in terms of the size of the 
initial segment and the perturbation S, whereas (<ref>) quantifies the memory effect in the initial segment and the 
noise. Inequality (<ref>) controls the deviation caused by the noise alone.

Let λ∈ (0,Λ) and the corresponding K=K(λ) in estimate (<ref>) be fixed.

1. By linearity we note that the 
the difference Y(t;ϕ)=X(t;ϕ)-X(t;0) satisfies the homogeneous deterministic delay equation (<ref>), that is, 
Y(t;ϕ)=x(t;ϕ) for all t≥ -r, and hence the convolution formula (<ref>) immediately implies

|Y(t)|   ≤K |ϕ(0)|·^-λt+ K ∫_[-r,0][ ∫_u^0[ ^-λ(t-s+u)·|ϕ(s)| s] |μ(u)|

   ≤K·|ϕ(0)|·^-λt+ K·(e^r-1)/ ·ϕ_r ·|μ|[-r, 0] ·^-λt

   ≤C_1(λ,r,μ) ϕ_r·^-λt,
where |μ| stands for the standard total variation measure of the finite signed measure μ.

Now we apply the stability of the unperturbed system comparing X(·;0) 
to the following Ornstein–Uhlenbeck type process U given by  
U(t) =  
-λ∫_0^t U(s)  s + S(t)      t≥0, 
 
0       t∈[-r, 0).

The process U has the explicit solution 
U(t) = ∫_0^t ^-(t-s) S(s),   t≥0. 
Integration by parts then yields 
U(t)= S(t) - λ^-λ t∫_0^t S(s)^λ s  s, such that  

|U(t)|   ≤2sup_s∈[0,t]|S(s)|.
We fix the notation V(t) := X(t;0) - U(t), t≥ -r, and note that X(·;0) = V + U. 
It remains to estimate V. Collecting the absolutely continuous parts as 
A(t) := ∫_0^t [∫_[-r, 0] U(s+u)  μ(u)]  s - ∫_0^t U(s)  s
we obtain the following equation for V 
V(t) 
   = X(t;0) - U(t) 

   = ∫_0^t ∫_[-r, 0] (X(s+u;0) - U(s+u))  μ(u)  s 
+ ∫_0^t ∫_[-r, 0] U(s+u)  μ(u)  s - ∫_0^t U(s)  s

   = ∫_0^t ∫_[-r, 0] V(s+u)  μ(u)   s + A(t), 
the solution of which has the explicit convolution representation  

V(t) = ∫_0^t x^*(t-s) A'(s)  s, 
where 

A'(t) = ∫_[-r, 0] U(t+u)  μ(u) - U (t). 
Equation (<ref>) and inequality (<ref>) then yield the estimate 
|V(t)| ≤K ∫_0^t ^-(t-s) |A'(s)|  s,  t≥0.

Furthermore, (<ref>) implies for some C_2=C_2(λ,μ,r)>0
|A'(t)| 
   ≤|μ|[-r, 0] ·sup_u∈[-r, 0] |U(t+u)|  + |U(t)| 

   ≤(|μ|[-r, 0] +) sup_u∈[-r, 0] |U(t+u)|

   ≤C_2 ·sup_s ∈[0, t] |U(s)|.
Combining the two preceding inequalities results in a constant C_3 = C_3(, K, μ, r)>0 such that 

|V(t)|    = K·C_2  ·sup_s ∈[0, t] |U(s)| ·∫_0^t ^-(t-s) s
≤K·C_2/λ·sup_s ∈[0, t] |U(s)| =C_3 ·sup_s ∈[0, t] |U(s)|
and (<ref>) follows as a combination of (<ref>), (<ref>) and (<ref>).

2. We denote the perturbed fundamental solution with initial segment ϕ(0)_{0} by X̃. It satisfies  
   X̃(t)=ϕ(0)+∫_0^t∫X̃(s+u) μ(u) s+S(t),

   X̃(t)=0,  t∈[-r,0).
Then
X(t;ϕ)-ϕ(0)x^*(t)=X(t;ϕ)-X̃(t)+ X̃(t)-ϕ(0)x^*(t) 
and Ỹ(t) :=X(t,ϕ)-X̃(t) is the solution of 
   Ỹ(t)=∫_0^t[∫_[-r,0]Ỹ(s+u) μ(u)] s,

   Ỹ(t)=ϕ(t),  t∈[-r,0).
Hence arguing as for the term A in part 1. we get 
|Ỹ(t)|≤C_1·sup_s∈[-r,0)|ϕ(s)|·^-λt
with the same constant C_1 as in (<ref>).
Analogously to the identification of Y in part 1. we observe
X̃(t)-ϕ(0)x^*(t)=X(t;0)=V(t)+U(t), 
where the processes V and U are already estimated in (<ref>) and (<ref>) of part 1. 
Inequality (<ref>) then follows by 
   |X(t,ϕ)-ϕ(0) x^*(t)| ≤|Ỹ(t)|+ |V(t)|+|U(t)|
≤C_1sup_[-r,0)|ϕ(s)|^-λt+2(C_3+1)sup_s∈[0,t]|S(s)|.
3. Since X(t;ϕ)-x(t;ϕ)=X(t;0)=V(t)+U(t), estimate (<ref>) follows immediately by the previous results.




  §.§.§ Estimates on the stochastic perturbation


Let us rewrite the underlying Lévy process Z 
as a sum of a compound Poisson process
η^ρ(t)    =∑_s≤tΔZ(s)(|ΔZ(s)|>ρ),     t≥0, 
whose jumps  are larger than some threshold ρ> 1 in absolute value, and an independent 
Lévy process ξ^ρ=Z-η^ρ with bounded jumps. This is always possible and can be easily seen 
by comparison of the Lévy–Khintchine formula (<ref>) for Z 
with those for η^ρ and ξ^ρ

   ln^ıu η^ρ(t)=t∫_|z|>ρ (^ıuz-1) ν(z),

   ln^ıu ξ^ρ(t)
=-σ^2/2u^2t +ı d_ρu t+t∫_|z|≤ρ(^ıuz-1-ıu z) ν(z),

where the new drift d_ρ = d + ∫_1<|z|≤ρ z ν( z). We denote the jump times and sizes of η^ρ by {τ_k}_k≥ 1,
and  {J_k}_k≥ 1 respectively and recall that they are independent. Moreover, the interjump times τ_1,τ_2-τ_1,…, are iid
exponentially distributed random variables with the parameter
β_ρ=∫_|z|>ρν(z),
and the jump sizes {J_k}_k≥ 1 are also iid with the probability law 
(J_k∈A)=1/β_ρ∫_|z|>ρ_A(z) ν(z),  A∈().


Let X^=X^(·;ϕ) be the solution to the delay SDDE (<ref>), and
consider the stochastic integral process 

S^,ρ(t)=∫_0^t F(X^_s)(0-) ξ^ρ(s).  



In this article we are interested in the first exit time of X^(t) from the interval [a, b], 
which coincides by definition by the first exit time of X^_t in segment space from 
the segment interval a, b _r. By the Lipschitz continuity (<ref>) we have 

    sup_ϕ∈ a, b _rF(ϕ)_r ≤  L (F(0)_r + sup_ϕ∈ a, b _rϕ_r).

That is, before the first exit t≤τ 
the coefficient of (<ref>) is bounded by 

    F(X^)(t-)_r ≤ L(F(0)_r + max{|a|, b}) < ∞.
 
Therefore it is without a loss of generality if we assume that F is uniformly bounded, that is, 
sup_ϕ∈ D[-r, 0]F(ϕ)_r ≤ C_F < ∞ for some global constant C_F>0.




Let F be uniformly bounded by a constant C_F>0. 
Then for any ρ>1, T>0, δ>0 and any p>1 there is a constant C_S>0 such that
for any 0≤ s≤ t≤ T and >0 sufficiently small we have 
(sup_u∈[0,t-s]|S^,ρ(s+u)- S^,ρ(s)  |>δ)≤C_S^p. 


Note that ξ̃^ρ(t)=ξ^ρ(t)-d_ρ t, t≥ 0, is a martingale with bounded jumps,
as well as for fixed s≥ 0 the stochastic integral process S̃^,ρ(s+u)- S̃^,ρ(s)  
=∫_s^s+u F(X^_v)(0-) ξ̃^ρ(v), u≥ 0.
Then the triangle inequality, the Markov inequality, and the classical Burkholder-Davis-Gundy inequality for p>1 
(see for instance <cit.>, Theorem 4.8) 
yield a constant C_p>0 such that for 
>0 being sufficiently small it follows 
(sup_u∈[0,t-s]   |S^,ρ(s+u)- S^,ρ(s)  |>δ)

   ≤( ·C_F·|d_ρ|·T>δ/2)
+ (  sup_u∈[0,t-s]|∫_s^s+u F(X^_v)(0-) ξ̃^ρ(v) |>δ/2)

   ≤0+  ^p·2^p /δ^p  sup_u∈[s, t-s]|∫_s^s+u F(X^_v)(0-) ξ̃^ρ(v)|^p

   ≤^p·2^pC_p/δ^p T^p/2·C_F^p·(σ^2+ ∫_|z|≤ρ z^2 ν(z))^p/2
=C_S·^p.




 §.§ General segment Markov estimates


Let X be a càdlàg segment Markov process as given in (<ref>) 
and denote for a<0<b the first exit time of X from [a, b] by 
τ=τ(ϕ)=inf{t≥0X(t)∉[a,b]}.




If for some T≥ r>0, m_1, m_2>0 satisfying m_1 T≤ m_2 T<1 and p_1(B), p_2(B)>0 
we have the short term estimates   

    inf_ϕ∈ a,b_r_ϕ(τ≤ T)≥ m_1 T,
       inf_ϕ∈ a,b_r_ϕ(τ≤ T, X(τ)∈ B)≥ m_1 T· p_1(B),
       sup_ϕ∈ a,b_r_ϕ(τ≤ T, X(τ)∈ B)≤ m_2 T· p_2(B),

then for each u≥ 0 we have 

    sup_ϕ∈ a,b_r_ϕ(τ>u)
    ≤^-u m_1/1- m_1 T,
       sup_ϕ∈ a,b_r_ϕ(τ>u,X(τ)∈ B)
    ≤^-u m_1/(1- m_1 T)^2·(m_2/m_1p_2(B)-m_1Tp_1(B)),
       sup_ϕ∈ a,b_r_ϕ(X (τ )∈ B)
    ≤ m_2 /m_1 · p_2(B)

and 
sup_ϕ∈a,b_r _ϕτ   ≤1/m_1.


We first show (<ref>). For each u>0 denote k=⌊u/T⌋. Then kT≤ u and 
_ϕ(τ>u, X(τ)∈B) ≤_ϕ(τ> kT , X(τ)∈B).The segment Markov property 
and (<ref>) yield for any initial condition ϕ∈ a,b_r 


_ϕ(τ> kT , X(τ)∈B)
   =_ϕ[ (τ> kT, X(τ)∈B)·(τ>(k-1)T)   ]

   =_ϕ[ (τ> kT, X(τ)∈B)·(τ>(k-1)T) |_(k-1)T  ]

   =_ϕ[ (τ>(k-1)T)·[ (τ> kT, X(τ)∈B)·|_(k-1)T  ]]

   =_ϕ[ (τ>(k-1)T)·_X_(k-1)T[ (τ> T, X(τ)∈B)]]

   ≤_ϕ(τ>(k-1)T))·sup_ψ∈a,b_r _ψ(τ> T, X(τ)∈B)

   ≤[1-inf_ϕ∈a,b_r _ϕ(τ≤T)]^k-1 
·sup_ψ∈a,b_r _ψ(τ> T, X(τ)∈B)

   ≤(1-m_1T)^k-1 
·sup_ψ∈a,b_r _ψ(τ> T, X(τ)∈B).


While the first term satisfies 

(1-m_1T)^k-1 = (1- m_1 k T/k)^k/1-m_1T ≤(1- m_1 u/k)^k/1-m_1T ≤^-m_1 u/1-m_1T.
The last inequality is given by ^-m_1 u - (1-m_1 u/k)^k≥ 0 for all m_1 u≤ k. 
The latter condition is satisfied since m_1 k T ≤ m_1 u ≤ k, which is a consequence of m_1 T < 1 in the statement. 
We rewrite the last term on the right side of (<ref>) as 
_ϕ(τ> T, X(τ)∈B) = _ϕ(X(τ)∈B) - _ϕ(τ≤T, X(τ)∈B).While the second term is estimated by (<ref>) 
we calculate the first summand, which is (<ref>), 
with the segment Markov property and (<ref>) 

_ϕ(X(τ)∈B)   =∑_k=1^∞_ϕ(X (τ)∈B,(k-1)T <τ≤kT)

   =∑_k=1^∞_ϕ[ ((k-1)T <τ, X(τ)∈B,τ≤kT)|_(k-1)T]]

   =∑_k=1^∞_ϕ[ ((k-1)T <τ, X(τ)∈B,τ^≤kT)|_(k-1)T]]

   =∑_k=1^∞_ϕ((k-1)T <τ)·_X_(k-1)T (X(τ)∈B,τ^≤T)]

   ≤sup_ψ∈a,b_r _ψ(X(τ)∈B,τ≤T)·∑_k=1^∞_ϕ((k-1)T <τ)

   ≤m_2·T·p_2(B)  ·∑_k=1^∞(1- m_1T)^k-1

   = m_2 /m_1  ·p_2(B).
The estimates (<ref>) and (<ref>) yield  

 _ϕ(τ> T, X(τ)∈B) ≤m_2T·p_2(B)·1/m_1T -m_1Tp_1(B) 
and (<ref>)-(<ref>) with (<ref>) imply (<ref>).
Setting B= estimate (<ref>) follows directly from (<ref>)-(<ref>).
Eventually 
_ϕτ   ≤T ∑_k=1^∞k·_ϕ((k-1)T <τ≤kT)=T ∑_k=0^∞k·_ϕ( τ>Tk)≤T ∑_k=0^∞(1-m_1T)^k =1/m_1. 





Let B be a Borel set. If for some T≥ r>0, m_1, m_2>0 satisfying m_1 T ≤ m_2 T <1, 
p_1(B), p_2(B) ≤ 1 and δ∈(0,|a|∧ b) we have the short term estimates 

    sup_ϕ_r≤δ_ϕ(τ≤ T,X(τ)∈ B )≤ m_1 T· p_1(B),
       inf_ϕ_r≤δ_ϕ(τ≤  T , X(τ)∈ B)≥ m_2 T· p_2(B),
       inf_ϕ_r≤δ_ϕ(τ > T , X_T_r≤δ)≥ 1-m_1 T,

then for each u≥ 0 we get 

    inf_ϕ_r≤δ_ϕ(τ>u,X(τ)∈ B)≥ (1-m_1 T)^-u/Tln(1- m_1 T)·(m_2/m_1· p_2(B) - m_1 T p_1(B)),
       inf_ϕ_r≤δ_ϕ(X(τ)∈ B)≥m_2/m_1 p_2(B).
       inf_ϕ_r≤δ_ϕτ≥1-m_1 T/m_1 .



For each u>0 denote k=⌈u/T⌉. In particular (k-1)T < u ≤ kT. Then
the segment Markov property and (<ref>) 
yield for any initial condition ϕ_r≤δ

_ϕ(τ>u) ≥_ϕ(τ> kT)    ≥_ϕ[ (τ> kT)·(τ>(k-1)T)·( X_(k-1)T_r≤δ)   ]

   = _ϕ[ (τ> kT)·(τ>(k-1)T)·( X_(k-1)T_r≤δ)  |_(k-1)T]

   = _ϕ[  (τ>(k-1)T)·( X_(k-1)T_r≤δ) [ (τ> kT) |_(k-1)T]]

   = _ϕ(τ>(k-1)T)·( X_(k-1)T_r≤δ) _X_(k-1)T [ (τ> kT)]


   ≥_ϕ(τ>(k-1)T,X_(k-1)T_r≤δ)
·inf_ψ_r≤δ _ψ(τ> T)
    
   ≥[inf_ψ_r≤δ _ψ(τ> T,X_T_r≤δ)]^k-1
·inf_ψ_r≤δ _ψ(τ>T )    
 
   ≥(1-m_1T)^k
and (<ref>) follows by 
_ϕτ   ≥T ∑_k=1^∞(k-1)·_ϕ((k-1)T <τ≤kT)

   =T ∑_k=1^∞_ϕ( τ>Tk)≥T ∑_k=1^∞(1-m_1T)^k =1-m_1T/m_1 . 
Furthermore, inequality (<ref>) is a result from 
_ϕ(X(τ)∈B)   =∑_k=1^∞_ϕ(X (τ)∈B,(k-1)T <τ≤kT)

   ≥∑_k=1^∞_ϕ[ (τ> (k-1)T )·(X_(k-1)T_r≤δ) [(X(τ)∈B,τ≤kT)|_(k-1)T]]

   ≥inf_ψ_r≤δ _ψ( X(τ)∈B,τ≤T)
·∑_k=1^∞_ϕ( τ>(k-1)T,X_(k-1)T_r≤δ)  

   ≥m_2T ·p_2(B)·∑_k=1^∞(1-m_1T)^k-1

   = m_2 /m_1  ·p_2(B). 
Finally, repeating the chain of inequalities (<ref>) with the additional event {X(τ)∈ B}, we get 
_ϕ(τ>u,    X(τ)∈B)≥_ϕ(τ> kT ,X(τ)∈B)

   ≥_ϕ[ (τ> kT,X(τ)∈B)·(τ>(k-1)T)·( X_(k-1)T_r≤δ)   ]

   = _ϕ[ (τ> kT,X(τ)∈B)·(τ>(k-1)T)·( X_(k-1)T_r≤δ)  |_(k-1)T]




   ≥_ϕ(τ>(k-1)T,X_(k-1)T_r≤δ)
·inf_ψ_r≤δ _ψ(τ> T,X(τ)∈B )
    
   ≥[inf_ψ_r≤δ _ψ(τ> T,X_T_r≤δ)]^k-1
·inf_ψ_r≤δ _ψ(τ>T,X(τ)∈B)    
 
   ≥(1-m_1T)^k·( m_2/m_1·p_2(B)- m_1T·p_1(B)   ),
and (<ref>) follows directly.
 






 §.§ Proof of the main estimates


The main result will follow directly from the following six inequalities.




For any ϰ>0  there is  T>0 and _0>0 such that for all ∈(0,_0]

    inf_ϕ∈ a,b_r_ϕ(τ^≤ T)≥λ_ν̅(E) T(1-ϰ).





For any ϰ>0 there are δ>0, T>0, and _0>0 such that for all ∈(0,_0]

    inf_ϕ_r≤δ_ϕ(τ^ > T , X_T^_r≤δ)≥ 1 - λ_ν̅(E) T(1+ϰ).





For any ϰ>0 and v>b there is T>0 and _0>0 such that for all ∈(0,_0]

    inf_ϕ∈ a,b_r_ϕ(τ^≤ T, X^(τ^)>v)≥λ_ T·ν̅(E_b^j(v))(1-ϰ).





For any ϰ>0 and v>b there is T>0 and _0>0 such that for all ∈(0,_0]

    sup_ϕ∈ a,b_r_ϕ(τ^≤ T, X^(τ^)>v)≤λ_ T·ν̅(E_b^j(v))(1+ϰ).





Analogously to Lemmas <ref> and <ref> one proves the estimate for the exit into the set (-∞,a).

 For any ϰ>0 and v<a there is T>0 and _0>0 such that for all ∈(0,_0]

    inf_ϕ∈ a,b_r_ϕ(τ^≤ T, X^(τ^)<v)≥λ_ T·ν̅(E_a^j(v))(1-ϰ),
       sup_ϕ∈ a,b_r_ϕ(τ^≤ T, X^(τ^)<v)≤λ_ T·ν̅(E_a^j(v))(1+ϰ).



Before passing to the proof of the Lemmas we make several preparatory comments. Let ϰ>0 be an arbitrary small number.

1. For γ > -min{|a|,b} denote 
E(γ)={z∃  t≥0 such that  z ·F_0 ·x^*(t)∉[a-γ,b+γ] }
=[e_-(γ),e_+(γ)]^c,  E(0)=E.
Assume that ν̅(E)>0.
Due to the continuity of the fundamental solution,
lim_γ→0 e_±(γ)=e_±and for any ϰ>0 with the help of (<ref>) we get

ν(E(γ)/)/λ_= 
 ν(E(γ)/)/λ_ν̅(E(γ))·ν̅(E(γ))/ν̅(E)≥1-ϰ/20
for γ and  sufficiently small. 

2. Let λ∈(0,Λ) and K=K(λ) according to (<ref>) be fixed.
For any δ>0 we can choose R>r such that 
max{|a|,b}·K·^-λ(R-r)< δ.
In particular, for any ϕ∈ a,b_r and t≥ R we have.

|x(t,ϕ)|< δ. 
Note that R also bounds the time horizon of a non-normal growth exit so that a deterministic
exit can occur only before the time instant R.

3. For δ and R chosen, we can fix T>0 such that
2R/T≤ϰ/20.
4. Finally, for δ>0 and 0≤ s≤ t≤ T denote 
_s,t=_s,t(δ)={sup_ u∈[0,t-s]   |S^,ρ(s+u)-S^,ρ(s)  |≤δ}.
With the help of Lemma <ref> with p>α we get (_s,t(δ))=o(λ_), and in particular

(_s,t(δ))> 1-ϰ/20T λ_ν̅(E)
for  small enough.



  §.§.§ Proof of Lemma <ref>


We show that with high probability the exit from [a,b] is to occur imminently after the first large jump. 




For δ=δ(ϰ,γ)>0, T=T(δ) and R=R(δ) to be chosen later and for any ϕ∈ a,b_r 
we exclude the following error events which eventually turn out to have 
small probability and estimate

_ϕ(τ^≤T)   ≥_ϕ(τ^≤T,J_1 ∈E(γ), 
R≤τ_1≤T-R,τ_2>T,_0,τ_1(δ),_τ_1,T(δ)).
Now we show that with a proper choice of the parameters, the set of conditions 
{ J_1 ∈ E(γ)}∩{R≤τ_1≤ T-R}∩{τ_2>T}∩_0,τ_1∩_τ_1,T
will imply that the exit occurs before the time T, τ^≤ T. 

Indeed, at the time instant τ_1 we have
X^(τ_1)=X^(τ_1-)+ F(X^_τ_1)(0-) J_1 .
To estimate X^(τ_1-) we note that (<ref>) together with (<ref>) 
guarantee that for all δ>0, on the event _0,τ_1(δ) we have
sup_t∈[τ_1-r,τ_1)|X^(t,ϕ)|< 2 C δ.
Then obviously if 2LCδ<F_0/2 then the event
|J_1| (F_0-2 L C δ)≥2max{|a|,b}
implies that τ^=τ_1≤ T.
Hence from now on we assume without loss of generality that J_1∈ E(γ) and |J_1|≤ C_J for 
C_J=4 max{|a|,b}/F_0.

Then according to (<ref>) on the event _τ_1,T(δ)
sup_t∈[τ_1,τ_1+R]|X^(t,ϕ)-  X^(τ_1)·x^*(t-τ_1)   |
≤C ·( 2Cδ+ δ)=(2C+1)Cδ.
Finally, comparing
|X^(τ_1) - F_0 J_1|≤|X^(τ_1-) |+ |J_1| ·|F(X^_τ_1)(0-)-F_0 | 
≤2 C δ+ |J_1| ·L ·2Cδwe obtain that 
|X^(τ_1)·x^*(t )- J_1 F_0·x^*(t)|≤K|X^(τ_1)- J_1F_0| ≤2 CK δ+ |J_1| ·2KLCδ.
This means that  if J_1∈ E(γ) and |J_1|≤ C_J then either 
sup_t∈[τ_1,τ_1+R]X^(t,ϕ)    ≥sup_t∈[0,R]X^(τ_1)·x^*(t) -(2C+1)Cδ

   ≥sup_t∈[0,R] J_1 F_0·x^*(t) - 2 CK δ- |J_1| ·2KLCδ-(2C+1)Cδ

   ≥b+γ-C(2 K - 2C_J 2KL  +2C+1)δ≥b+γ/2
for δ>0 small enough
or analogously
inf_t∈[τ_1,τ_1+R]X^(t,ϕ) ≤a-γ/2
for the same δ. Hence, from now on δ>0 as well as R and T are fixed. 

It is left to show that estimate (<ref>) yields the required accuracy in the limit → 0.
First we choose the large jump threshold ρ>0 such that ^-β_ρ T(δ)≥ 1-ϰ/20. 



Hence for  small

_ϕ(τ^≤T)   ≥_ϕ( J_1 ∈E(γ), R(δ)≤τ_1≤T(δ)-R(δ),τ_2>T(δ)
,_0,τ_1(δ) ,_τ_1,T(δ) )

   ≥( J_1 ∈E(γ))·(R(δ)≤τ_1≤T(δ)-R(δ),τ_2>T(δ))
- 2ϰ/20T(δ)λ_ν̅(E)   

   ≥ν(E(γ)/)/β_ρ·∫_R(δ)^T(δ)-R(δ) 
∫_T(δ)-t_1^∞β_ρ^2·^-t_1 β_ρ·^-βt_2 t_2 t_1
-ϰ/10T(δ)λ_ν̅(E)  

   =ν(E(γ)/) ·(T(δ)-2R(δ))^-β_ρT(δ)   -ϰ/10T(δ)λ_ν̅(E)

   =T(δ)λ_ν̅(E) ν(E(γ)/)/λ_ν̅(E) ·T(δ)-2R(δ)/T(δ)^-β_ρT(δ) 
-ϰ/10T(δ)λ_ν̅(E)

   =T(δ)λ_ν̅(E) ·(1-ϰ/20)^3 
-ϰ/10T(δ)λ_ν̅(E)

   ≥T(δ)λ_ν̅(E)·(1-ϰ).






  §.§.§ Proof of Lemma <ref>

 
Passing to the complements we get
_ϕ(τ^≤T or X_T^_r> δ)
≤_ϕ(τ^≤T)+_ϕ(τ^> T and X^_T_r> δ).
Step 1. We consider the following decomposition
Then
_ϕ(τ^≤T )
   = _ϕ(τ^≤T,τ_1 > T)
+_ϕ(τ^≤T,τ_1≤T<τ_2)
+_ϕ(τ^≤T,τ_2≤T)

   =p_1+p_2+p_3
and show that the main contribution to the exit probability is made by the first (and virtually only the first) jump J_1, i.e. by
the term p_2.

1. To estimate p_1 we write
p_1   ≤_ϕ(τ^≤T,τ_1 > T,_0,T)+(_0,T^c)=p_11+p_12

On the event {τ_1>T}∩_0,T(δ), due to (<ref>) in Lemma <ref>
sup_t∈[-r,T]|X^(t,ϕ)|≤2Cδ,
which is incompatible with {τ^≤ T} for δ sufficiently small, and hence
p_11=0. By Lemma <ref>, p_12≤ Tλ_ν̅(E)ϰ/20 for  small enough.

2. Show that the probability p_2 is the essential one. Take into account that τ_1 and 
τ_2-τ_1 are i.i.d. exponentially distributed r.vs. with
the parameter β_ρ:=∫_|z|>ρν( y) (ρ will be chosen large to guarantee that β_ρ is small).
Note that (τ_1|τ_1≤ T<τ_2) is uniform on [0,T], see e.g. <cit.>.
Then we decompose and disintegrate
_ϕ(τ^≤T,τ_1≤T<τ_2)
   ≤_ϕ(τ^≤T,τ_1≤T<τ_2,_0,τ_1 ,_τ_1,T )
+1/T∫_0^T(_ϕ(_0,t^c)+_ϕ(_t,T^c)) t

   =
_ϕ(τ^≤T,R≤τ_1≤T-R, τ_2>T,_0,τ_1 ,_τ_1,T )

   +_ϕ(τ^≤T,τ_1< R< T<τ_2,_0,τ_1 ,_τ_1,T )

   +_ϕ(τ^≤T,T-R< τ_1≤T<τ_2,_0,τ_1 ,_τ_1,T )
+ p_24 

   =p_21+p_22+p_23 +p_24.
a) We show that for any γ>0 small enough we can choose δ>0 such that
on the event {R≤τ_1≤ T-R,τ_2>T}∩_0,τ_1(δ)∩_τ_1,T(δ) we have
{τ^≤T}⊆{J_1∈E(-γ)}
or equivalently we show that if the jump size J_1 is not large enough, no exit occurs.

The set of conditions in the probability p_11 guarantees with the help of (<ref>)
that 
sup_[-r,τ_1)|X^(t;ϕ)|≤2Cδ.
At the time instant τ_1 we have
X^(τ_1)=X^(τ_1-)+ F(X^_τ_1)(0-) J_1 
and 

|F(X^_τ_1)(0-)-F_0|≤L ·2CδFor t∈[τ_1,T]
|X^(t;ϕ) - J_1 F_0·x^*(t-τ_1)|    ≤|X^(t;ϕ)-X^(τ_1)·x^*(t-τ_1)|

   + 
|X^(τ_1) - J_1 F_0  | ·|x^*(t-τ_1)|
Then according to (<ref>) on the event _τ_1,T(δ)

sup_t∈[τ_1,T]|X^(t,ϕ)-  X^(τ_1)·x^*(t-τ_1)   |
≤C ( 2Cδ+ δ)=(2C+1)Cδ.
Finally, noting that if J_1∉ E(-γ), then | J_1|≤C̃_J=2max{|a|,b} we compare

|X^(τ_1) - F_0 J_1|≤|X^(τ_1-) |+ |J_1| ·|F(X^_τ_1)(0-)-F_0 | 
≤2 C( 1 + C̃_J   L )δ.
Hence combining (<ref>) and (<ref>) we obtain 
sup_[τ_1,T]|X^(t;ϕ) - J_1 F_0·x^*(t-τ_1)| ≤C̃δfor some C̃>0. Choosing δ small such that C̃δ<γ/2 we get that if J_1∉ E(-γ) 
then either 
sup_t∈[τ_1,T]X^(t,ϕ)    ≤sup_t∈[0,T] J_1 F_0·x^*(t) +C̃δ≤b-γ/2.
or analogously
inf_t∈[τ_1,T]X^(t,ϕ) ≥a+γ/2.
Now choose ρ such that ^-β_ρ T(δ)≥ 1-ϰ/20. 
Then analogously to (<ref>) for small  we get
p_21   ≤(
J_1∈E(-γ),R≤τ_1≤T-R )
=(J_1∈E(-γ))·(R≤τ_1≤T-R )



   ≤ν(E(-γ)/)/β_ρ·β_ρ(T-2R)
≤λ_T(δ)ν̅(E)(1+ϰ/20)


b)
To estimate the probability p_22 we note that on [0,τ_1)
the process X^(·,ϕ) belongs to the 2Cδ-neighborhood of zero.
Recalling (<ref>) we choose δ>0 small enough such that |F(X_τ_1^)(0-)|≤ 2|F_0|
Hence, to guarantee the exit, the jump size J_1 must obviously satisfy 
2· |F_0|· |J_1|≥δ' for some δ'>0, since otherwise applying (<ref>) 
to the perturbation process
S(t)=∫_0^t F(X_s-^)(0-) ξ^ρ(s) + F(X_τ_1^)(0-) J_1·_[τ_1,∞)(t)

which satisfies on _0,τ_1∩_τ_1,T 
sup_t∈[0,T]|S(t)|< 2δ+ δ'
we obtain that the X^(·) does not leave the C(3δ+δ')-neighborhood of zero on [0,T]
so the exit would be impossible.
Hence choosing T large we obtain for  small enough that

p_22≤(|J_1|≥δ'/2|F_0|,τ_1≤R)
   = 1/β_ρ∫_|z|≥δ'/2|F_0| ν(z)·β_ρ·∫_0^R ^-β_ρt t

   ≤2C' ·R/T ·λ_T≤ϰ/20 ·ν̅(E)·λ_T,
where we used that due to (<ref>)
lim_→01/λ_∫_|z|≥δ'/2|F_0| ν(z)= 
C'=ν̅(|z|> δ'/2|F_0| )∈(0,∞).


c) Analogously to b) we estimate the probability p_23: 
p_23≤(|J_1|≥δ'/2|F_0|,T-R≤τ_1≤T)


   ≤2C' ·R/T ·λ_T≤ϰ/20·ν̅(E) ·λ_T.
d) Obviously due to Lemma <ref>
p_24≤2ϰ/20·ν̅(E) ·T ·λ_.
3. To estimate p_3 we argue analogously that either the first large jump J_1 has to 
satisfy 2· |F_0|· |J_1|>δ' or, if this is not the case, the 
second jump J_2 has to satisfy the same condition, since otherwise the solution X^(·;ϕ) will stay in 
some small neighborhood of zero which size is depends on δ, δ'.
Together with the condition that at least two jumps occur on [0,T] this yields
p_3   ≤_ϕ( τ^≤T,τ_2≤T ,_0,τ_1(δ),_τ_1,τ_2(δ), _τ_2,T (δ)  )
+_ϕ(_0,T^c(3δ))

   ≤2 _ϕ(2·|F_0|·| J_1|>δ',τ_2≤T)+(_0,T^c(3δ_1))

   ≤2/β_ρ∫_|z|≥δ'/2 |F_0|ν(z)·β_ρ^2  
·T
≤4C'·β_ρ·T ·λ_.


We choose ρ>1 such that 
4C'β_ρ<ν̅(E)·ϰ/20 to obtain p_2≤ϰ·ν̅(E)· T ·λ_/20.

Step 2.
Now we estimate
_ϕ(τ^> T and X^_T_r> δ)
   =_ϕ(τ^> T and X^_T_r> δ,τ_1>T)

   +_ϕ(τ^> T and X^_T_r> δ,τ_1≤T< τ_2 )

   +_ϕ(τ^> T and X^_T_r> δ,τ_2≤T )=q_1+q_2+q_3.


Since we have to control the behavior of X^ at the end segment of the interval [0,T], we have to exploit 
the exponential stability of the deterministic delay system in order to inhibit the uncontrolled accumulation of small errors 
over time.


1. Let δ>0 be chosen in Step 1 and fixed. We choose δ_1<δ/2C
and T large such that C·^-λ(T-r)<1/2 and then (<ref>) 
guarantees that on the event _0,T(δ_1), the trajectory X^ belongs to the 
δ-neighborhood of zero, hence for small 
q_1≤_ϕ(τ^> T and X^_T_r> δ,τ_1>T, _0,T(δ_1) )+(_0,T^c(δ_1) )
≤0+ ϰ/20Tλ_ν̅(E).



2. Estimate the probability q_2. Take into account that τ_1 and τ_2-τ_1 are iid exponentially distributed r.vs. with
the parameter β_ρ:=∫_|z|>ρν( y) (ρ will be chosen large to guarantee that β_ρ is small).
Then we desintegrate
_ϕ   ( τ^> T and X^_T_r> δ,τ_1≤T< τ_2 )

   ≤_ϕ(τ^> T and X^_T_r> δ,_0,τ_1(δ_1),_τ_1,T(δ_1))
+_ϕ(_0,T^c(2δ_1))

   =
_ϕ(τ^> T,τ_1< R,τ_2>T,X^_T_r> δ,_0,τ_1(δ_1),_τ_1,T(δ_1))

   +_ϕ(τ^> T,T-R< τ_1≤T, τ_2>T,X^_T_r> δ,_0,τ_1(δ_1),_τ_1,T(δ_1))

   +_ϕ(τ^> T,R≤τ_1≤T-R,τ_2>T,X^_T_r> δ,_0,τ_1(δ_1),_τ_1,T(δ_1))

   +_ϕ(_0,T^c(2δ_1))

   =q_21+q_22+q_23+q_24.
a) As in Step 1 b) if 2· |F_0|· |J_1|≤δ' for δ'>0 and δ_1 sufficiently small then
X^_T_r≤δ. Hence,
q_21≤( 2·|F_0|·|J_1|>δ',τ_1< R)≤ϰ/20·ν̅(E)·λ_T
as in (<ref>).

b) Analogously, to estimate q_22 note that right before the jump τ_1, X_τ_1-_r≤δ/2, so that
if 2· |F_0|· |J_1|≤δ' then X^_T_r≤δ.
Hence
q_22≤(2|F_0|·|J_1|> δ',T-R<τ_1≤T)≤ϰ/20 ·ν̅(E) ·λ_T.
c) Finally, again, if 2· |F_0|· |J_1|≤δ' then X^_T_r≤δ, so that 
we can assume that 2· |F_0|· |J_1|> δ'.
On the other hand, right before the jump τ_1, X_τ_1-_r≤δ/2. Since on the set of events of q_23
T-r-τ_1>T-R>T-r, we choose the difference R-r large enough such that the solution X^ satisfies 
X^_T_r≤δ. Thus
q_23=0.

d) As usual, for  small enough q_14≤ϰ·ν̅(E) ·λ_ T/ 20.
3. To estimate q_3 we argue analogously that at least one of the jumps J_1 or J_2 should satisfy 
2· |F_0|· |J_i|>δ'. 
Together with the condition that at least two jumps occur on [0,T] this yields as in Step 1.3
q_3   ≤2 _ϕ(F·| J_1|>δ/3,τ_2≤T)+_ϕ(_0,T^c(3δ_1))

≤ϰ/20 ·ν̅(E)·T ·λ_with the same choice of ρ>1.

Eventually, collecting the estimates from the Steps 1 and 2, we find
_ϕ(τ^≤T or X^_T_r>δ)≤Tλ_ν̅(E)(1+ϰ).




  §.§.§ Proof of Lemmas <ref> and <ref>




The proof of Lemma <ref> goes along the lines with the proof upper estimate for 
the probability _ψ(τ^≤ T) in Step 1 in Lemma <ref>.
The only difference is the weaker condition on the initial segment ψ∈ a,b_r 
and an additional condition on the exit location X^(τ^)>v for v>b.
We estimate
_ψ(τ^≤T,X^(τ^)>v)
   =_ψ(τ^≤T,τ_1>T, X^(τ^)>v)

   +_ψ(τ^≤T,τ_1≤T<τ_2,X^(τ^)>v)

   +_ψ(τ^≤T,τ_2≤T,X^(τ^)>v)

   =p_1+p_2+p_3.
We consider the term p_1 in detail and show how to adapt the argument. Indeed, 
p_1≤_ψ(X^(τ^)>v,τ^≤T,τ_1>T,_0,T(δ))+_ψ(_0,T^c(δ))=p_11+p_12.
We show that p_11=0 for δ small. Indeed, due to Lemma <ref>

|X^(t,ψ)-x(t;ψ)|≤Cδ.
For ψ∈ a,b_r, let us consider two cases.

a) Let ψ be such that
sup_t∈[0,T]x(t;ψ)≤ v-2Cδ.
Hence (<ref>) implies that sup_t∈[0,T]X^(t;ψ)≤ v-Cδ <v and hence X^(τ^)≤ v-Cδ<v.

b) On the other hand, assume that there is t^* such that
t^*=t^*(ψ)=inf{t≥0x(t;ψ)> v-2Cδ}≤T.
Hence, applying (<ref>) again we get 
X^(t^*;ψ)≥ v-3Cδ >b and thus τ^< t^* and X^(τ^)≤ v-Cδ<v.

Therefore, p_11=0, and p_1≤ p_12 which is known to be of the order o(λ_), see (<ref>).

The probabilities p_2 and p_3 are treated analogously by taking into account that no 
exit with X^(τ^)>v can occur before or after the first jump τ_1.
 
 
 
The proof of Lemma <ref> goes along the lines with the proof of the lower estimate for 
the probability _ψ(τ^≤ T) in Lemma <ref> with the same obvious modifications: the 
condition X^(τ^)>v on the exit location can be satisfied (with high probability) only if τ^=τ_1 and   
the jump size J_1 is large enough so that it belongs to a set E_b(v+γ).
 
 



  §.§.§ Proof of the main Theorem <ref>

 
Let the initial segment ϕ satisfy
(<ref>),
in particular ϕ∈ a,b_r.

1. Combining Lemma <ref> with Lemmas <ref>, <ref> and <ref>, we immediately obtain 
estimates from above for the probabilities _ϕ(λ_τ^>u), u>0, _ϕ(X^(τ^)>v), v>b, and the mean value 
λ__ϕτ^.

2. On the other hand, or each initial segment ϕ_r≤δ, for δ>0 small enough,
Lemmas <ref>, <ref>, <ref>, and <ref>
yield the  estimates from below. 
Hence, it is left to relax the condition on the initial segment ϕ. This can be done easily.

Let ϕ be an initial segment with no deterministic exit which satisfies (<ref>) and let u>0.
Denote δ_ϕ:=dist(x(·;ϕ),[a,b])>0.
Choose R>r large enough and δ_0>0 so that on _0,R(δ_0) we have X^(R;ϕ)_r≤δ. Let 
ϰ>0.
Then
the segment Markov property yields
_ϕ(λ_τ^>u)   ≥_ϕ(λ_τ^>u,_0,R(δ_0)) 
=_ϕ[( λ_τ^>u,_0,R(δ_0)  )|_R

   =_ϕ(_0,R(δ_0) )·[( λ_τ^>u )|_R

   =_ϕ[(_0,R(δ_0) )·_X_R^(λ_(τ^-R)>u)]

   ≥inf_ψ_r≤δ_ψ(λ_τ^>u+λ_R) -  _ϕ(^c_0,R(δ_0) )

   ≥inf_ψ_r≤δ_ψ(λ_τ^>u)·(1-ϰ)
in the limit as → 0.

Analogously, for any ϰ>0 and  small we estimate the mean value:
λ__ϕτ^   ≥λ__ϕ[τ^·(τ_>R)·(_0,R(δ_0)] 
=λ__ϕ[[  τ^·(τ_>R)·(_0,R(δ_0)  )|_R

   = λ__ϕ[(τ_>R)·(_0,R(δ_0) )·[τ^|_R

   = λ__ϕ[(τ_>R)·(_0,R(δ_0) )·_X_R^(τ^-R)]

   ≥λ__ϕ[(τ_>R)·(_0,R(δ_0) )·inf_ψ_r≤δ _ψ (τ^-R)]

   = λ_·inf_ψ_r≤δ _ψ (τ^-R) ·_ϕ[(τ_>R)·(_0,R(δ_0) )]

   ≥λ_·inf_ψ_r≤δ _ψ (τ^-R) 
·(1-_ϕ(τ_≤R)-_ϕ(_0,R^c(δ_0) ))

    ≥λ_·inf_ψ_r≤δ _ψ τ^-λ_R-
λ_·inf_ψ_r≤δ _ψ τ^·(_ϕ(τ_≤R)+_ϕ(_0,R^c(δ_0) ))

   ≥λ_(1-ϰ) ·inf_ψ_r≤δ _ψ τ^.
The probability _ϕ(λ_τ_>u, X^(τ_)>v), v>b, is treated analogously.








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