Some asymptotic formulars for a Brownian motion whose dimension is variable with a regular variation from a parabolic domain to research the variation of biological species

Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R d +1 , d ≥ 1, let τ D denote the ﬁrst time the Brownian motion exits from D . Estimates with exact constants for the asymptotics of log P ( τ D > T ) are given for T → ∞ , depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the ﬁrst exit time and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.


Introduction and main result
In the natural world, it is very important to maintain ecological stability. It is fatal to nature that the ecological chain is very likely to be destroyed when the ecology is affected by natural disasters or a series of external factors. It will lead to changes in the entire biological chain when the extinction or addition of a species in the ecological chain. In view of this phenomenon in the biological world, this paper studies the first exit time of Brownian motion. Each dimension of Brownian motion can be regarded as a change in the number of species, while the dimension of the Brownian motion can be seen as the species of the species in the ecology and it changes over time.
Throughout the paper, let {B(t), t ≥ 0} be a standard d-dimensional Brownian motion and {W (t), t ≥ 0} be a standard one-dimensional Brownian motion, such that B and W are independent, and x := ( d i=1 x 2 i ) 1/2 be the Euclidean norm of x := (x 1 , . . . , x d ) ∈ R d .
In the last few years, in view of various motivations, two different types of asymptotic behaviours are studied. One type was usually called "the large deviation", and the other was usually called "the small ball estimates". The "large deviation" has been studied by a large number of mathematicians and applied in mathematics. The "small ball estimates" receives also much research interest, yet relatively little is known. For the sake of clarity, let us recall some well-known results of small ball estimates. A general situation is: where · s is a semi-norm in the space of real of functions on [0, 1], and α > 0, β > 0 are finite constants. Under some special norms, the well known results were obtained, such as L 2 -norm, uniform sup-norm, Hölder-norm and so on. All the results related to small ball estimates can be seen in Kuelbs (1999), applying a Gaussian correlation inequality into the small deviation, Li obtained a well-known asymptotic estimates for Brownian motion under weighted sup-norms as follows: where j (d−2)/2 is the smallest positive root of the Bessel function J (d−2)/2 . Although these above probabilities are more general than (??), f (t) is still a nonrandom function. Li considered the first exit time problem and study the probability in [? ] (2003) as follows: where f (x) is a nondecreasing lower semicontinuous convex function on [0, ∞) with f (0) finite. The upper and lower estimates of log-probability in (??) were obtained by Gaussian technique firstly. However, in view of the generality of f (x), the upper and lower estimates were not asymptotically equivalent. Lifshits and Shi in [? ] (2002) considered this problem and gave further restrictions and assumed f (t) = t p , where p > 1. They obtained the lower and upper estimates of the following probability and also provided the lower and upper estimates are asymptotically equivalent, thus improving Li's estimates in this case. It is the results of Lifshits, Shi and Li that motivate our study and we thank them for giving a copy of Li's paper and useful discussions in their letters.
Returning to our first exit time problem in this paper, we develop the probabilities in (??) and (??), and consider more general case, namely, where u 0 > 0, h(x) is a continuous strictly positive function and also regularly varying at ∞ of index r, namely, lim x→∞ h(λx)/h(x) = λ r , λ > 0 and 0 < r < p/2.
For example h(x) = x p/4 or (x log(1 + x)) p/3 and so on. The probability (??) is very useful. It is more general and complicated than (??). Studying it aims to understand the further relationship between the estimates of this probability and the regular variation h. Next, we give the main result of our paper as follows: (1.7) 2. If lim t→∞ t p/(p+1) /h(t) = 0, and .
where Γ(·) denotes the usual gamma function, and c 1 and c 2 are strictly positive constants which are independent of p and t.
It is easy to see the relationship among (??), (??) and Theorem ??. If lim t→∞ t p/(p+1) /h(t) = ∞, the Brownian motion W (s) is the dominant term in (??), and the asymptotic behavior of log P (τ D > t) is the same to the one in (??). Otherwise, the function h(s) is the dominant term in (??), and the asymptotic behavior is the same to the one in (??). Since there is not an exact constant in (??) when ∞ 0 f −2 (s)ds = ∞, we do not obtain a precise estimate for 1 < p ≤ 2, p/2 ≤ r ≤ 1. More work needs to be done in this direction, and it also seems a challenging problem.
The rest of the paper is organized as follows. In Section 2, we present several estimates of exit probabilities for moving boundaries. They are necessary for the proof of Theorem ??. In Section 3, combining the Lemmas in Section 2, the detailed proof of the upper estimates in Theorem ?? is given. Finally, In Section 4, we give the detailed proof of the lower estimates in Theorem ??.

Exit probability for moving boundary
To prove Theorem ??, in this section, we need the following results due to Shao

From Li [? ]
, it is easy to see, for any x > 0, where K is a various positive constant, and j v is the smallest positive zero of the Bessel function J v , v = (d − 2)/2. However, up to now, we don't know any result for the dimension d(t). So, in this section, we give two lemmas for getting upper and lower bounds of the probability P ( sup 0≤s≤t B(s) ≤ x).
for all s, t ≥ 0. Then there exists 0 < c 1 < ∞ depending only on α and d such that for any 0 < x < 1.
The argument of Lemma ?? was given in Shao and Wang [? ]. And they also gave a lower bound of small ball probability of Gaussian fields.
In fact, Shao in [? ] obtained estimate of the probability P (sup 0≤s≤t B(s) ≤ x) with exact constants. However, we don't care about the precise coefficients in this paper. So we just use the above Lemmas.By Lemma ?? and Lemma ??, we get the following Propositions. .
x, then the inequality (??) can be written as Proposition 2.2. Let g(t) be a continuous strictly positive function such that g (t) ≤ 0 is continuous and g (t) ≥ 0. Then here c 2 is a sufficiently large constant. In particular, under the additional condition for and δ > 0 and t large. . (2.14) The rest proof is similar to the proof of Theorem 2.2 in Li [? ].
is called regularly varying at ∞ of index ρ. Furthermore, h also satisfies the following conditions: is a positive measurable function and said to be slowly varying, namely, satisfying The study of functions h(x), l(x) and their properties such as (??-??), together with their wide-ranging applications, constitutes the theory of regular variation, instituted by Karamata in [? ] (1930) and subsequently developed by him and many others. All above results can also be easily found in [? ] (1987).
For any p > 1 Here whereġ denotes the Radon-Nikodym derivative of g and Φ(x) is the distribution function of a standard normal random variable.
The argument of Lemma (??) given by Novikov in [? ] (1979) follows from wellknown theorems concerning the equivalence of measures of Gaussian processes.
Before giving the last Lemma, let us introduce some notation appeared in [? ] (2002). Here and in the following, where A ↑ θ,p is the set of all nondecreasing functions in the set A θ,p defined by A θ,p := g : [0, 1] → R + , g(0) = 0, g absolutely continuous,

Proof of upper bounds
Our upper bounds argument is a modification of the one appeared in [? ] (2002). We need to pay special attention to the regular variation h(t). In view of completeness, we give the detailed proof. For the sake of brevity, we write For any ε ∈ (0, 1), let t 0 = 0, t i = (1 − ε) N −i t, 1 ≤ i ≤ N . From Theorem 1.5.3 in [? ] (1987), we know that any function h varying regularly with non-zero exponent is asymptotic to a monotone function. In particular l is slowly varying and r > 0, x r l(x) is asymptotic to a nondecreasing function. Without loss of generality, in this argument, we assume that h(t) is nondecreasing for t large. Thus, for t large, we have By Anderson's inequality and the fact that W has stationary increments, we have for a N ≥ 0, Plugging the inequality (??) into (??), by induction, we have (3.4) By Proposition ??, we know that By conditioning on the Brownian motion W , and combining (??) and (??), we have Combining the monotonicity of t → M (t) and t → h(t) for t large, we have In the case of lim t→∞ t p/(p+1) /h(t) = ∞, we split the expectation in (??) into two parts and on H c , (3.14) For I, using Lemma ??, we can easily obtain inf δ>0,θ>0 For II, using u 0 t −1/2 ≤ εt −1/2 h(t) for t large, we have In view of the fact lim t→∞ t p/(p+1) /h(t) = ∞, it is easy to see that .
Then, we finish the proof of the upper estimates in (??). Next, in the case of lim t→∞ t p/(p+1) /h(t) = 0 and r < p/2, we still split the expectation in (??) into two parts, Next, dealing with III and IV , respectively. On the one hand, combining the monotonicity of M (t) and h(t) for t → ∞, It is convenient to recall the reflection principle of the Brownian motion Then, the derivative of (??) with respect to x is Combining (??) and (??), we have Note that, for t → ∞, Plugging (??) into (??) yields, for any δ > 0, On the other hand, using the monotonicity of M (t) and h(t) for t → ∞ again, where we apply the condition 1 of Lemma (??) into the last inequality in (??). If lim t→∞ h(t · s)/h(t) = s r , and this convergence is uniform in s ∈ [(1 − ε) N −1 , 1], then, for any δ > 0 and t large, we have Next, on the one hand, using . .

Proof of lower bounds
Our lower bounds argument is based on the calculus of variations and Li's profile function method from [? ] (2003). We also need to pay special attention to the regular variation h(t). Let g ∈ Λ 2 , where Λ 2 is the Hilbert space of all absolutely continuous functions x such that x(0) = 0, 1 0ẋ we have In the second inequality of (??), we can remove the regular variation h(t) in the first probability without loss of the correct asymptotic rate, since t p/(p+1) g(s/t) is the dominant term from the fact (??). Next, dealing with the above two probabilities in (??), respectively. Using (??) in Proposition ??, we can easily obtain which shows thatġ has a continuous derivative and alsoġ(1) = 0,ġ(t) > 0. Multiplying (??) withġ(t) and integrating, partial integration of the right-hand side term yields Differentiating (??), multiplying withġ(t) and integrating again yields .
Separation of variables and integration yields so that Now, on the one hand, integrating both sides in (??), .
Next, in the case lim t→∞ t p/(p+1) /h(t) = 0, r < p/2, (4. 16) it is similar to the proof above, the key step is to find an appropriate profile function. From (??), we assume that there exists m such that p/(p + 1) ≤ r < m < p/2. We have for g ∈ Λ 2 For the second inequality in (??), we can remove t 1−m/p g(s/t) in the first probability without loss of the correct asymptotic rate, since h(s) is the dominant term in this probability from the fact (??). By scaling, we get the second probability. Next, dealing with the two probabilities above in (??), respectively. Using Proposition ??, we have

Conclusion
This paper proves that the estimation of the upper and lower bounds during the first exit time of the Brownian motion is meaningful for the species ecological chain problem in the biological world.

Consent for publication
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Funding
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Availability of data and material
This article is intended to prove the theoretical study of species in the ecosphere, without data.

Competing interests
The authors declare that they have no competing interests.

Authors' contributions
The first author(Xiaoming Li) conceived the innovation of this paper and provided relevant lemmas to prove the theory. The second author(Wenbin Che) and the third author(Jingjun Zhang) gave detailed proof of the main conclusion of the paper.