Sharp estimates of transition probability density for Bessel process in half-line

In this paper we study the Bessel process R_t^{(\mu)} with index \mu\neq 0 starting from x>0 and killed when it reaches a positive level a, where x>a>0. We provide sharp estimates of the transition probability density p_a^{(\mu)}(t,x,y) for the whole range of space parameters x,y>a and every t>0.


Introduction
Let R (µ) t be the Bessel process with index µ = 0. The transition probability density (with respect to the Lebesgue measure) of the process is expressed by the modified Bessel function in the following way Our main goal is to describe behaviour of densities of the transition probabilities for the process R (µ) t killed when it leaves a half-line (a, ∞), where a > 0. Note that if the process starts from x > a then the first hitting time T (µ) a of a level a is finite a.s. when µ < 0 but it is infinite with positive probability when µ > 0. The density kernel of the killed semi-group is given by the Hunt formula where x, y > a and t > 0. The main result of the paper is given in Theorem 1. Let µ = 0 and a > 0. For every x, y > a and t > 0 we have Here f (t, x, y) µ ≈ g(t, x, y) means that there exist positive constants c 1 and c 2 depending only on the index µ such that c 1 ≤ f /g ≤ c 2 for every x, y > a and t > 0. Since the constants are independent of a > 0, one can pass to the limit with a → 0 + and obtain the well-known estimates of p (µ) (t, x, y). Since the function I µ (z) behaves as a power function at zero and that some exponential term appears in the asymptotic expansion at infinity (see Preliminaries for the details), the behaviour of p (µ) (t, x, y) depends on the ratio xy/t. Note that similar situation takes place in the case of p (µ) a (t, x, y), which depends on xy/t as well. It can be especially seen in the proof of Theorem 1, where different methods and arguments are applied to obtain estimates (1.3), whenever xy/t is large or small. Finally, taking into account the behaviour of p (µ) (t, x, y), one can rewrite the statement of Theorem 1 in the following way p (µ) a (t, x, y) p (µ) (t, x, y) µ ≈ 1 ∧ (x − a)(y − a) t 1 ∨ t xy , x, y > a, t > 0, (1.4) where the expression on the right-hand side of (1.4) should be read as the description of the behaviour of p (µ) a (t, x, y) near the boundary a. There are several ways to define the function p (µ) a (t, x, y) hence our result and its applications can be considered from different points of view. It seems to be the most classical approach to define the heat kernel p (µ) a (t, x, y) as the fundamental solution of the heat equation ∂ t − L (µ) u = 0, where L (µ) is the Bessel differential operator. In the most classical case, i.e. when the operator L (µ) is replaced by the classical Laplacian, the problem of finding description of the heat kernel has a very long history (see for example [18] and the references within) and goes back to 1980s and the works of E.B. Davies (see [7], [4], [5], [6]). However, the known results for Dirichlet Laplacian on the subsets of R n (see [19]) or in general on Riemannian manifolds (see [18] for the references) are only qualitatively sharp, i.e. the constants appearing in the exponential terms in the upper and lower estimates are different. Note that in our result these constants are the same and consequently, the exponential behaviour of the density is very precise. Such sharp estimates seems to be very rare.
Note also that the operator L (µ) plays an important rôle in harmonic analysis. However, since the set (a, ∞) is unbounded, our consideration corresponds to the case when the spectrum is continuous. This operator on the set (0, 1) and the estimates of the corresponding Fourier-Bessel heat kernel were studied recently in [16] and [17], but once again the results presented there are only qualitatively sharp, i.e. the estimates are not sharp whenever |x − y| 2 >> t. Another essential difference between the case of bounded sets and our case is that in the first one, we can limit our considerations to t ≤ 1, by the application of the intrinsic ultracontractivity. However, the most interesting part of Theorem 1 (with difficult proof) seems to be when t is large.
The third and our principal motivation comes from the theory of stochastic processes and the interpretation of p (µ) a (t, x, y) as a transition density function of the killed semigroup related to the Bessel process R (µ) t . From this point of view, the present work is a natural continuation of the research started in [3] (see also [1]), where the integral representation of the density q (µ) were provided together with its some asymptotics description. The sharp estimates of the density for the whole range of parameters with the explicit description of the exponential behaviour was given in [2]. For the in-depth analysis of the asymptotic behaviour of q x (t) see [12], [11], [10]. The case µ = 0 is excluded from our consideration and it will be addressed in the subsequent work. As it is very common in this theory, this case requires different methods and should be considered separately. In particular, some logarithmic behavior is expected whenever xy < t.
The paper is organized as follows. In Preliminaries we introduce some basic notation and recall properties and known results related to modified Bessel functions as well as Bessel processes, which are used in the sequel. In particular, using scaling property and absolute continuity of the Bessel processes we reduced our consideration only to the case µ > 0 and a = 1. After that we turn to the proof of Theorem 1, which is split into two main parts, i.e. in Section 3 we provide estimates whenever xy/t is large and in Section 4 we prove (1.3) for xy/t small. In both cases the result is given in series of propositions.

2.1.
Notation. The constants depending on the index µ and appearing in theorems and propositions are denoted by capitals letters C (µ) 1 , C (µ) 2 , . . .. We will denote by c 1 , c 2 , . . . constants appearing in the proofs and to shorten the notation we will omit the superscript (µ) , however we will emphasize the dependence on the other variables, if such occurs.

2.2.
Modified Bessel function. The modified Bessel function of the first kind is defined as (see [8] 7.2.2 (12)) It is well-known that whenever z is real the function is a positive increasing real function. Moreover, by the differentiation formula (see [8] 7.11 (20)) d dz and positivity of the right-hand side of (2.1) we obtain that z → z −µ I µ (z) is also increasing.
The asymptotic behavior of I µ (z) at zero follows immediately from the series representation of I µ (z) where the behaviour at infinity is given by (see [8] 7.13.1 (5)) Some parts of the proof strongly depends on the estimates of the ratio of two modified Bessel functions with different arguments. Here we recall the results of Laforgia given in Theorem 2.1 in [13]. For every µ > −1/2 we have Moreover, whenever µ ≥ 1/2, the lower bound of similar type holds, i.e. we have In this section we introduce basic properties of Bessel processes. We follow the notation presented in [14] and [15], where we refer the reader for more details. We write P (µ) x and E (µ) x for the probability law and the corresponding expected value of a Bessel process R (µ) t with an index µ ∈ R on the canonical path space with starting point R 0 = x > 0. The filtration of the coordinate process is denoted by F The laws of Bessel processes with different indices are absolutely continuous and the corresponding Radon-Nikodym derivative is described by where x > 0, µ, ν ∈ R and the above given formula holds P at zero depends on µ. Since we are interested in a Bessel process in a half-line (a, ∞), for a given strictly positive a, the boundary condition at zero is irrelevant from our point of view. However, for completeness of the exposure we impose killing condition at zero for −1 < µ < 0, i.e. in the situation when 0 is non-singular. Then the density of the transition probability (with respect to the Lebesgue measure) is given by (1.1).
For x > 0 we define the first hitting of a given level a > 0 by is infinite with positive probability. We denote by q x,a (s) were obtained in [2]. We recall this result for a = 1, which implies the result for every a > 0, due to the scaling property of Bessel processes. More precisely, it was shown that for every x > 1 and t > 0 we have (2.7) The above-given bounds imply the description of the survival probabilities (see Theorem 10 in [2] ) The main object of our study is the density of the transitions probabilities for the Bessel process starting from x > a killed at time T   x,a (s) in the following way x,a (s)ds. (2.10) The scaling property of a Bessel process together with (2.10) imply that Moreover, the absolute continuity property (2.6) applied for µ > 0 and ν = −µ gives These two properties show that it is enough to prove Theorem 1 only for a = 1 and µ > 0. To shorten the notation we will write q (µ) x,1 (s). Since we consider the densities with respect to the Lebesgue measure (not with respect to the speed measure m(dx) = 2x 2µ+1 dx) the symmetry property of p (µ) 1 (t, x, y) in this case reads as follows: (2.12) Finally, for µ = 1/2 one can compute p we obtain where Making the substitution w = 1/s − 1/t and using formula 3.471.15 in [9] we get Hence we have which together with (2.10) and (2.14) give One can also obtain this formula using the relation between 3-dimensional Bessel process (i.e. with index µ = 1/2) and 1-dimensional Brownian motion killed when leaving a positive half-line. Note also that which is exactly (1.3) for µ = 1/2. We end this section providing very useful relation between densities q (µ) x (t) with different indices, which once again follows from the absolute continuity property.

Lemma 1.
For every x > 1 and t > 0 we have
Proof. The second inequality in (2.19) was given in Lemma 4 in [2]. To deal with the right-hand side of (2.19) we use (2.6) to obtain for every δ > 0 and 0 < ε ≤ δ 2 /2 ∧ 1 where, by Strong Markov property By (2.14), for every r ∈ (0, ε) we have where the last inequality follows from ε ≤ δ 2 /2. It implies that F ε (t, x)/ε vanishes when ε goes to zero. Consequently, dividing both sides of (2.20) by ε and taking a limit when ε → 0, we arrive at Since δ was arbitrary, the proof is complete.

Estimates for xy/t large
We begin this Section with the application of the absolute continuity property of Bessel processes and the formula (2.17) which give the upper bounds for µ ≥ 1/2 and lower bounds for ν ≤ 1/2. These bounds are sharp whenever xy ≥ t. Proposition 1. Let µ ≥ 1/2 ≥ ν > 0. For every x, y > 1 and t > 0 we have Proof. From the absolute continuity property (2.6) we get that for every µ ≥ ν > 0 and every Borel set A ⊂ (1, ∞) we have Taking µ ≥ 1/2 and ν = 1/2 gives the left-hand side of (3.1) and taking ν ≤ 1/2 and µ = 1/2 gives the right-hand side of (3.1).
The absolute continuity can also be used to show the estimates for small times t in a very similar way. Note that if t < 1 then we always have xy > t. The proof of the main Theorem will be provided in subsequent propositions without the assumption that t is bounded, but we present this simple proof to show that for xy ≥ t the estimates for small t are just an immediate consequence of the absolute continuity of Bessel processes. Proposition 2. Let µ > 0. For every x, y > 1 and t ∈ (0, 1] we have Proof. Let µ ≥ ν > 0. Taking Borel set A ⊂ (1, ∞) and t ≤ 1 we have Hence we get Now taking µ ≥ 1/2 and ν = 1/2 together with (2.17) and the result of Proposition 1 gives the proof of (3.3) for µ ≥ 1/2. Analogous argument applied for µ < 1/2 ends the proof.
Next proposition together with Proposition 1 provide the estimates for x, y bounded away from 1. Notice that if x, y > c > 1 and xy > t then and consequently the right-hand side of (1.4) is comparable with a constant which means that p > 0 and C (µ) 3 > 1 such that whenever xy ≥ t and the lower bounds holds for x, y > 2 and the upper bounds are valid for x, y > C This ends the proof for small indices. Now let µ ≥ 1/2. Since the modified Bessel function I µ (z) is positive, continuous and behaves like (2πz) −1/2 e z at infinity (see (2.3)) there exists constant c 1 > 1 such that whenever xy ≥ t. One can show that it is enough to take c 1 = (I µ (1)e −1 √ 2π) −1 . Consequently, applying above given estimate to (1.1) we arrive at , xy ≥ t, (3.5) where the first inequality is just (3.2). Moreover, by (2.19), we have and it together with left-hand side of (3.5) and (2.16) imply and taking into account right-hand side of (3.5) and (3.4) we obtain for x, y > C where Taking into account the general estimate Consequently p (µ) Now we turn our attention to the case when x and y are bounded. The next proposition, however, is much more general.
Proof. Without lost of generality we can assume that 1 < x < y. We put b = (x + 1)/2 and take µ ≥ 1/2. Using (2.6) and the fact that T we can write for every From the other side, the scaling property (2.11) and the formula (2.18) give where the last equalities follows from It ends the proof for µ ≥ 1/2. For 1/2 ≥ ν > 0 we similarly write and we obtain This together with the above-given estimates for p (t, x, y) finish the proof.
Since for x, y < C and xy ≥ t, for some fixed C > 1, we have applying the results of Proposition 4 (with m = C −1 ) and Proposition 1 gives whenever x, y < C and xy ≥ t.
Finally, we end this section with two propositions related to the case when one of the space variables is close to 1 and the other is large. We deal with this case separately for µ < 1/2 and µ ≥ 1/2.
Proof. By monotonicity of I ν (z), for every s ∈ (0, t) we have Hence, using the right-hand side of (2.19), we get , 0 ≤ ν < 1/2, s > 0, and the formula (1.1) we get where the last equality follows from (2.15). Using (2.4) we obtain which together with previously given estimates, (2.9) and finally (2.3) give By elementary computation we can see that Here we have used the following inequalities Thus, by the mean value theorem, there exists d = d x,y,t ∈ (1, x) such that Proposition 6. For every µ ≥ 1/2 and c > 1 there exists constant C (µ) 6 (c) > 0 such that for every 1 < x ≤ c and y ≥ 5c(µ + 1) we have whenever xy ≥ t.
The above-given ratio of modified Bessel functions can be estimated from above by using (2.5) as follows Consequently r (µ) Finally observe that 2y − x − 1 > y − 1 and we arrive at This ends the proof.
The proof of (1.3) in the case xy ≥ t can be deduced from above-given propositions in the following way. Let µ ≥ 1/2 and without any loss of generality we assume that x ≤ y. The upper bounds for every x, y > 1 are given in Proposition 1. From Proposition 3 we know that the lower bounds are valid for x, y > C   3 (µ + 1) in Corollary 1 we get the lower bounds in the remaining range of the parameters x and y. The proof for ν ≤ 1/2 is obtained in the same way.

Estimates for xy/t small
In this section we provide estimates of p (µ) 1 (t, x, y) whenever xy < t. Note also that (1.3) can be written in the following shorter way whenever xy < t. The main difficulty is to obtain the estimates when one of the space parameters is close to 1 and the other is large, i.e. tends to infinity. In this case we have to take care of cancellations of two quantities appearing in (2.9) but also not to lose a control on the exponential behaviour. We begin with the upper bounds. x − 1 x y − 1 y whenever xy ≤ t.
Proof. If x, y > 2 the result follows immediately from the general estimate p (µ) 1 (t, x, y) ≤ p (µ) (t, x, y) and (2.2) which gives p (µ) (t, x, y) ≈ y 2 t µ+1/2 1 √ t exp − x 2 + y 2 2t , xy t ≤ 1. (4.1) Note that for every x, y > 0 and t > 0 there exists c 1 > 0 such that If xy < t, then it immediately follows from (4.1) by estimating the exponential term by 1. For xy ≥ t we use the asymptotic behaviour (2.3) to show that In particular, for all z, w > 1 and 1 < y < 2 there exists c 2 > 0 such that