Asymptotic behaviour of the Bessel heat kernels

We consider Dirichlet heat kernel $p_a^{(\mu)}(t,x,y)$ for the Bessel differential operator $L^{(\mu)}=\frac{d^2}{dx^2}+\frac{2\mu+1}{2x}$, $\mu\in\mathbb{R}$, in half-line $(a,\infty)$, $a>0$, and provide its asymptotic expansions for $xy/t\rightarrow\infty$.


Introduction
Let µ ∈ R and a > 0. We consider Dirichlet heat kernel p (µ) a (t, x, y) in half-line (a, ∞) (with respect to the reference measure m (µ) (dy) = m (µ) (y)dy = y 2µ+1 dy) for the Bessel differential operator This function is a fundamental solution of the heat equation 2∂ t = L (µ) with Dirichlet boundary condition at the point a. Our main objective was to derive the precise asymptotic expansion of p (µ) a (t, x, y) for xy/t → ∞. The main result of the paper can be stated as follows.
It is worth emphasizing, that in some régime of the space parameters we are able to give more precise (long-time) estimates of the error term. In particular, when x, y are bounded away from the boundary, i.e. level a > 0, we can write explicitly the first n (for given n ∈ N) terms in the expansion of E µ (t, x, y). Moreover, even when one of the space variables is located near the boundary, we can give one term more in an expansion of the error term in comparison with (1.2). However, for this purpose, additional assumption on the quantity (x − a)(y − a)/t is made. Theorem 2. Let µ ∈ R, a > 0, n ∈ N and t 0 (µ) is defined as in (1.3). We have In order to understand these results better, notice that the case a < x, y < 2a, t ≥ t 0 (µ) is excluded from our consideration. Indeed, it holds then xy/t < 4a 2 /t ≤ 4a 2 /t 0 (µ), which is a contradiction with the condition xy/t → ∞. Moreover, since p (µ) a (t, x, y) is a symmetric function of space variables x, y, i.e. p (µ) a (t, x, y) = p (µ) a (t, y, x), x, y > a > 0, t > 0, (1.5) one can assume, without loss of the generality, that x < y. In consequence, it is enough to consider (for t ≥ t 0 (µ)) two cases: when a < x < 2a < y and when x, y > 2a. Note that one-term, long-time expansion of E µ (t, x, y) was derived just in one situation, namely when a < x < 2a < y and 0 < (x−a)(y−a) t < 1, while in the other cases we are able to provide longer expansions of the error term.
Studying the behaviour of heat kernels (for second-order differential operators) is of primary importance in many areas of the mathematical analysis, especially in the theory of partial differential equations, harmonic analysis, as well as in the potential theory. However, even for Laplacian, the problem of providing precise description of the behaviour of the corresponding Dirichlet heat kernels is very difficult. Exact formulas, in terms of elementary functions, of these heat kernels for Laplace operator are available only in a few special cases (e.g. half-line, R d or half-space). As a consequence, providing estimates of the heat kernels is one of the most important approach to describe behaviour of these objects. Some classical results involving short-time estimates of the Dirichlet heat kernel for Laplacian were derived by S. R. S. Varadhan in [19] and [20]. It is also worth mentioning that estimates of heat kernels for Laplace operator in bounded C 1,1 domains were proved by E. B. Davies in [6] (upper bound) and Q. S. Zhang in [21] (lower bound) -see also [22] for the case of domains with bounded complements. These results are only qualitatively sharp, which means that exponential terms in upper and lower bound are different. Notice that estimates of the Dirichlet heat kernels for Laplacian in a ball, with the same exponents in upper and lower bounds (so-called sharp estimates), were obtained very recently by J. Małecki and G. Serafin in [13]. For more information (including more general second-order differential operators and domains e.g. manifolds) in this topic we refer the Reader to the monographs [7], [9] and references therein.
From the probabilistic point of view, p (µ) a (t, x, y) is a transition probability density of the Bessel process with index µ ∈ R starting from x > a and killed upon leaving a half-line (a, ∞), a > 0. In this context, results in this article are a continuation of the research related to the first hitting times at a given level by the Bessel process. Namely, sharp estimates of the density of these hitting times were obtained recently in [4] and [10]. The next natural step was to describe the transition probability density p (µ) a (t, x, y). T. Takemura in [16] derived the integral representations involving highly oscillating functions. Then, in papers [2] and [3], sharp estimates of p (µ) a (t, x, y) in a full range of the variables x, y > a and t > 0 were obtained. In this context, providing asymptotic expansion of the considered Bessel heat kernels is a natural improvement of these results. It is worth noting that the analogous expansions for the density of the first hitting time of Bessel processes were derived recently in [17] and [15].
To compare the main results of this paper with the known ones for the heat kernels of the Bessel operator in half-line, let us invoke the estimates from [2,3] related to the case xy/t → ∞. In particular, it was shown that for µ ∈ R we have whenever x, y > a, t > 0 and xy ≥ t. Here f ∧ g denotes the minimum of f and g. In fact, it should be stressed that the estimates of p a (t, x, y) given in [2] and [3] are valid for the whole range of the parameters x, y > a, t > 0 and µ ∈ R. However, our results are more precise (but on limited range of parameters) and the estimate (1.6) for xy/t large enough is a consequence of those given in Theorem 1. To make it clearer, let us point out the main differences between above mentioned results. At first, notice that the leading term given in (1.6), i.e.
is of a correct asymptotic order, but it is not of the exact asymptotic form. Secondly, the function t describes the behaviour of the heat kernel near the boundary (the level a > 0) more precise than the expression 1 ∧ (x−a)(y−a) t from (1.6). This difference is the most apparent when (x − a)(y − a)/t is small due to the expansion of z → 1 − e −z near zero. Finally, our third remark is that, in opposite to (1.6), Theorem 1 contains estimates of the error term. This accuracy strongly depends (in the case xy/t → ∞) on the régime of the time parameter (see (1.2)). As it was already mentioned, under stronger assumptions on the space-variables x, y, even more precise bounds can be obtained -see Theorem 2.
The paper is organized as follows. In Preliminaries basic notation and known results on modified Bessel functions as well as on Bessel processes are introduced. The next two sections are devoted to the proofs of Theorems 1-2. In particular, Section 3 contains short-time estimates of p f are simultaneously satisfied, then we write f µ ≈ g. The notation f = O µ (g) denotes |f | µ g. However, in proofs we omit writing µ over the signs and ≈. Similarly, the dependence of the constants on the parameters is not indicated in proofs throughout this article. To simplify the notation, we introduce the function where t > 0 and the coefficients A, B are given by Moreover, the following function (see Theorem 1) where x, y > 1, t > 0 and µ ≥ 0, will be used in the paper.

Modified Bessel function.
The modified Bessel functions of the first and second kind are denoted by I µ (z) and K µ (z), respectively, and defined as follows for µ ∈ N; otherwise we take a limit when µ → n ∈ N of the last expression. Their asymptotic behaviour at infinity can be described as follows (see 9.7.1 and 9.7.2 in [1]) where the coefficients c On the other hand, the asymptotic behaviour of I µ (z) at zero is given by (see 9.6.7 in [1]) Finally, we recall the following two-sided bounds for the ratio of the modified Bessel function of the first kind, provided by A. Laforgia in [11] x y µ e y−x ≤ I µ (y) Here the upper bound holds for µ > −1/2 , while the lower estimate is true for µ ≥ 1/2. Moreover, the inequality I µ (x) ≤ I µ (y) is valid whenever µ ≥ 0 and y ≥ x > 0 (see formula 8.445 in [8]).

Bessel processes.
We work on the canonical path space C([0, ∞), R), equipped with the filtration {F t } t≥0 generated by trajectories up to time t. Then, for a given trajectory R ∈ C([0, ∞), R), the first hitting time T a at given level a > 0 is defined as Moreover, we define the Bessel process BES (µ) (x), with index µ ∈ R and starting point x > 0, as a one-dimensional diffusion on [0, ∞) with infinitesimal generator where the killing condition at zero for µ < 0 is imposed. The probability law and the corresponding expected value of the Bessel process BES (µ) (x) are denoted by P (µ) x and E (µ) x , respectively. Notice that the laws of Bessel processes with different indices are absolutely continuous and their Radon-Nikodym derivative is given by where the equality holds P For a = 1 we write simply q (µ) x (s), where x > 1 and s > 0. Sharp estimates of this function were obtained in [4], i.e. for x > 1 and s > 0 we have and However, some more precise estimates of q (µ) x (s) are also available. In particular, the following asymptotic expansion (see Theorem B in [18]) will be used in our article This result can be strengthened by giving more precise estimates of the error term for 0 ≤ µ < 1/2 (see Lemma 4 in [4]), namely we have Some improvement is available also in the case µ ≥ 1/2, i.e. Lemma 1 in [2] states that where µ ≥ 1/2 and x > 1, s > 0.
Recall now that m (µ) (dy) = m (µ) (y)dy = y 2µ+1 dy. The crucial fact is that the considered Bessel heat kernel (for a = 0) is the transition probability density p (µ) (t, x, y) of the Bessel process BES (µ) (x) Moreover, this function can be expressed in terms of the modified Bessel function of the first kind in the following way (2.12) In particular, using (2.2), one can write where x, y > 0, t > 0 and xy/t → ∞. On the other hand, in the case a > 0 the considered Bessel heat kernel is the transition probability density of the Bessel process starting from x > a and killed upon leaving a half-line (a, ∞) We are able to represent the considered heat kernel p (µ) a (t, x, y) in terms of the above mentioned densities p (µ) (t, x, y) and q (µ) x,a (s)ds. (2.14) In fact, the last formula is a starting point for deriving estimates of the Bessel heat kernels. Notice that putting µ = −ν ≥ 0 in (2.6) gives a (t, x, y), x, y > a, t > 0, (2.15) which shows us that it is enough to consider indices µ ≥ 0. Moreover, we can focus only on the case a = 1 due to the following scaling property where x, y > a > 0 and t > 0. Observe that the function p  Therefore, we will often refer to the formula (2.19), especially in proofs of propositions in the next two sections.
As it was already mentioned, it is enough to find estimates of p (µ) a (t, x, y) only for a = 1 and 1 < x < y (see (2.16) and (1.5)). Moreover, due to (2.15), one can assume that µ ≥ 0. We begin with the part of Theorem 1 related to the case 0 < t < t 0 (µ).

Long-time estimates.
In this section we provide estimates of the Bessel heat kernel p (µ) a (t, x, y) for large times t. As previously, we will assume that a = 1, 1 < x < y and µ ≥ 0. Moreover, as it was mentioned in Introduction, we need to investigate only two cases: x, y > 2 and 1 < x < 2 < y. We start with the first one. In fact, using below given Proposition 2, the part (i) of Theorem 2 is also proved. Proposition 2. Let µ ≥ 0 be fixed. Then we have for x, y > 2 and t > 0 whenever xy/t → ∞.
Proof. Since p (µ) 1 (t, x, y) ≤ p (µ) (t, x, y), it is enough to focus on the proof of an appropriate lower bound for the considered heat kernel. We begin with the case x ≥ t. Notice that, due to the asymptotic behaviour of the modified Bessel function at zero (2.4) and at infinity (2.2), we have for w > 0 I α (w) w α e w , α ∈ {−1/2, µ}. (4.1) Combining this inequality (for w = y/(t − s) and α = −1/2) with the following bound (see (2.11) and (2.10)) The last line is a consequence of the exact formula for t 0 f A,B,t (s)ds given in Lemma 1 and estimates of t 0 sf A,B,t (s)ds from Lemma 2. Both of these lemmas can be found in Appendix. Since xy/t → ∞ we can use (2.13) to justify the existence of C 1 > 0 such that where the inequality y > t was used (since y > x ≥ t). Moreover, because of x, y > 2 the last exponent can be bounded from above by exp (−xy/(4t)). Hence, the Hunt formula (2.14) gives us finally Therefore, it remains to deal with the situation when x < t. To achieve this, we split integral defining the expression r 1 (t, x, y) as follows and then we estimate separately each of these integrals, starting with J (µ) 1 (t, x, y). Let 0 < s < x. Taking α = −1/2, w = y t−s in (4.1) and using the following inequality (see (2.7) and (2.8)) we obtain The last estimate is a consequence of the exact formula for t 0 f a,b (s)ds given in Lemma 1 from Appendix. Since xy/t → ∞ we can use (2.13) to justify the following inequalities where the second one is a consequence of the assumption x, y > 2. To deal with the part J (µ) 2 (t, x, y) we use (4.1) with w = y t−s , α = µ and the following upper bound (see again (2.7) and (2.8)) Namely, we infer that Moreover, making substitution w = 1 s − 1 t , one can rewrite this inequality as follows Therefore, estimating (1 + tw) 2µ ≤ (t/x) 2µ for w < 1 x − 1 t and extending the range of integration to (0, ∞), provide us Here the second line is a consequence of (5.1) and (5.2) (see Appendix), since (x−1)(y−1) t ≥ xy 2t → ∞ for x, y > 2 and xy/t → ∞. On the other hand, the last one follows from asymptotic behaviour of p (µ) (t, x, y) for xy/t → ∞ (see (2.13)). Since we assumed xy/t → ∞ and 2 < x < y, we get Summarizing, in view of (4.2) and (4.3), the usage of the Hunt formula (2.14) immediately ends the proof.

Proof of the part (i) of Theorem 2.
Let us fix n ∈ N. Due to the asymptotic behaviour of p (µ) (t, x, y) for xy/t → ∞ (see (2.13)) we can rewrite the result obtained in Proposition 2 as follows where x, y > 2, t ≥ 4 and xy/t → ∞. Notice that for x, y > 2 we have Thus we can replace the term exp − (x−y) 2 2t in (4.4) by the expression Moreover, since the exponential growth dominates the power one, the product of the expressions in square brackets in (4.4) can be simply replaced by the first of these. Therefore, in this way we have proved the part (i) of Theorem 2.
Proof. Using (2.5) to estimate ratios I µ y t−s /I µ y t and I µ y t /I µ xy t from above, we get Combining this together with the inequality q (µ) , valid for s > 0 and µ ≥ 1/2 (see (2.11)), we obtain with the function f A,B,t (s) defined in (2.1). We split the last integral into three parts and estimate them separately. To simplify the notation let us denote To deal with K 1 it is sufficient to use Lemma 1 from Appendix. Indeed, we have Now we focus our attention on the expression K 2 . Observe that for s ∈ (0, t/2) we have t/(t − s)) µ+1/2 − 1 ≈ s/(t − s). Hence, making substitution w = 1 s − 1 t in the integral K 2 , we can write where we have substituted r = b/(t 2 w). Since b/t ≈ y 2 /t → ∞ for y > 2 > x and xy/t → ∞, the last integral behaves like a constant. It implies (4.10) To estimate K 3 from above, let us observe that it holds (t/(t − s)) µ+1/2 − 1 ≈ (t/(t − s)) µ+1/2 , whenever t/2 < s < t. Hence, putting once again w = 1 s − 1 t we get Note that since a/t 1 we have e −aw ≈ 1. Thus, the substitution r = b/(t 2 w) leads to The last integral is the upper incomplete Gamma function Γ(µ, b/t), which behaves like (b/t) µ−1 e −b/t since b/t ≈ y 2 /t → ∞ (see 8.357 in [8]). Therefore, we have Combining together (4.9), (4.10) and (4.11) we obtain r (µ) for some constants C 1 , C 2 > 0. Hence, from the Hunt formula (2.14), we obtain Let us decompose the last expression as a sum B 1 + B 2 − C 2 B 3 , where Observe that the term B 1 contribute to the leading part for the expansion of p 1 (t, x, y). Hence we start with estimating B 2 . One can see that it holds since (x−1)(y−1) t < 1 and 1 < x < 2 < y. Thus, there exists constant C 3 > 0 such that On the other hand, we can estimate B 3 as follows with the same justification as for B 2 . This estimate implies that for some C 4 , C 5 > 0 we have (4.13) Collecting together (4.12) and (4.13) we arrive at p (µ) (4.14) On the other hand, because of (2.13) for n = 1, there exists C 6 > 0 such that where xy/t → ∞. Putting the last inequality to (4.14) we get the desired result with the constant C (µ) 2 := C 3 + C 5 + C 6 > 0.

Appendix
In this section we collect three lemmas, which play crucial rôle in the proofs of the main results of the paper. To make the paper more self-contained we recall the following dc.
Since the integrand on the left-hand side (LHS) is positive, we can use the Fubini-Tonelli theorem to change the order of integration as follows On the other hand, making substitution w = √ c+ √ d √ t in the right-hand side (RHS) we have Comparing LHS with RHS give us thesis of the first part of Lemma 2. The second part follows from the asymptotic behaviour of the error function at infinity, namely we have 1 − erf(z) ≈ e −z 2 /z as z → ∞.
The last lemma contains some estimates of exponents, which play important rôle especially in the proof of Proposition 4. and simply estimate denominator from below by using the inequality e z − 1 ≥ z, which is true for z > 0.