Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case

In this work we consider a one-dimensional Brownian motion with constant drift moving among a Poissonian cloud of obstacles. Our main result proves convergence of the law of processes conditional on survival up to time $t$ as $t$ converges to infinity in the critical case where the drift coincides with the intensity of the Poisson process. The complements a previous result of T. Povel, who considered the same question in the case where the drift is strictly smaller than the intensity.


Introduction
The investigation of stochastic processes in a random environment has along history and is still an active area of research. A very thoroughly studied model is the one of a diffusive particle in a Poissonian environment of obstacles. For a detailed description of the general framework and the mathematical details we refer to the very readable account of this presented in [16]. Our starting point is the following model composed of a one-dimensional Brownian particle (X t ) t≥0 , starting from 0, with a constant drift h = 0 and law W h , which moves in an environment given by an independent Poisson process in R with intensity ν whose law is denoted by P. Further we denote by C t = sup s≤t X s − inf s≤t X s = M t − m t the range of the process. Let W h t be the restriction of the Wiener measure W h to C (0, t). The Brownian particle starts from zero and gets killed upon hitting a point of the Poisson process, i.e. the killing time is denoted by T . In this work we will focus on the expected survival time W h ⊗ P {T > t} = E h e −νCt and in particular on the behaviour of the conditioned law Our special emphasis is on the case where |h| = ν which due to symmetry can be reduced to h = ν. Before we state our results we recall what is known if h = ν and indicate why this model is of interest. Motivated by previous heuristic arguments by physicists and simulation studies (see [9] and [5]) it was shown in [4] that even for the higher dimensional analogue this model exhibits a phase transition in the sense that Thus there is a critical parameter regime given by |h| = ν. In [15] it was later demonstrated that in dimension one lim t→∞ 1 t 1 3 log e h 2 2 t E h e −νCt = 1 2 (|h| − ν) 2 ) if |h| < ν, (1.2) which gives a much more precise version for the subcritical case |h| < ν than (1.1). The correct scaling exponents for the higher dimensional problems have also been derived in [15]. The one-dimensional situation was further investigated in more detail by T. Povel in [10], where he in particular proved the following result Theorem 1.1 (Theorem A in [10]). Let |h| ∈ (0, ν).
1. The limiting distribution of t − 1 3 X ·t 2 3 under Q t as t goes to infinity is given as the taboo measure starting from 0 with taboo interval (0, c 0 ) where c 0 = π 2 ν−|h| 1 3 . 2. The limiting distribution of the process X · under the measure Q t converges as t → ∞ to a mixture of Bessel-3-processes under which X starts in 0 and never hits a random levelã and the density of the mixture is given by h 2ã e −|h|ã ,ã > 0.
The taboo measure in [10, Theorem A i)] is defined for a ∈ (0, c 0 ) and B ∈ F t by where X is a zero-drift Brownian motion, T (0,c) = inf {t ≥ 0 : X t / ∈ (0, c)} and φ is the first eigenfunction for the operator 1 2 d 2 dx 2 with Dirichlet boundary condition. For a = 0 the taboo measure can be defined as a weak limit of P (0,c0) a (·) as a → 0.
EJP 22 (2017), paper 14. As is explained nicely in [10] item 2 of Theorem 1.1 describes the microscopic behaviour of the model and as a matter of fact the limit result is different in the case h = 0. Thus the presence of the drift has an influence on the microscopic limit.
Analogues results for the case of a random walk with drift instead of a Brownian motion with drift have been established in [18] and in the case of a Brownian motion with drift moving among soft Poissonian obstacles the precise analogue of Theorem 1.1 is offered in [14]. In those works a similar exclusion of the critical case is supposed.
As mentioned in this document we study the critical value model in Povel i.e. we focus on the case h = ν.
Even though this type of problem has been intensively investigated we have not been able to locate results covering this case in the present literature and it is the aim of the present work to fill in this gap. It will turn out that the macroscopic behaviour is the same as the one in the case |h| < ν but the details of the proof tend to be much more demanding. Our starting point will be the same as the one of Povel [10] but we are forced to work a long a different route as already one of his first steps breaks down in the case h = ν. In fact the first task consists in controlling the behavior of the normalization constant E h e −νCt as t → ∞. In order to establish this Povel [10] relies on an application of the classical Laplace method, which is not applicable in our setting namely the case h = ν. Remark 1.2. In contrast to [10] our method of analysing the asymptotic behaviour of Q (h) t will essentially rely on some facts from the theory of Mellin transforms and on some ideas around the Poisson summation formula.
The topic of our work can be considered to be a further contribution to the general topic of penalizations of diffussion processes (compare e.g. [11], [12] and [13]). Let us also emphasize that the range of diffusion processes has been of considerable interest in the probability literature (see e.g. [6], [2] and [17]) but our main result does not seem to follow easily from these studies.
Let us end this introduction with some remarks concerning the structure of this work. In the subsequent section 2 we summarize our main results concerning the asymptotic behaviour of the survival distribution and the limit of the conditioned measure Q (h) t . These results are proved in sections 3 and 4, wherein we make use of the asymptotic properties of certain functions appearing naturally during the proof. The investigation of those functions is deferred to sections 5, 6, 7 and 8.
During the preparation of the revision of the manuscript Hugo Panzo informed us that he considered a strongly related problem, namely the case of Brownian motion with positive drift reflected at zero penalized by the maximum of this process. He announced very precise results for this model.

Notation and conventions
Throughout the paper we use f ∼ g to denote that lim f /g = 1 and f g to imply the existence of two positive constants C 1 < C 2 such that C 1 f ≤ g ≤ C 2 f . Throughout the paper we consider a one-dimensional Brownian motion X with drift h ∈ R. We write W h respectively W h t for the Wiener measure on C (0, ∞) respectively the restriction of the Wiener measure on C (0, t). When h = 0 we drop the superscript. Similarly, we denote by E h x [·] the expectation of the Brownian motion with drift h = 0 started from x. When h = 0 we omit the superscript and write instead E x [·].
We use the m t , M t , 0 ≤ t ≤ ∞ to denote the running minimum, running maximum of X, i.e. m t = inf s≤t X s and M t = sup s≤t X s . We use C t for the running range of the process, i.e. C t = M t − m t .
Due to symmetry throughout the paper we assume that h = ν > 0.

Asymptotic expansion for the Laplace exponent
As mentioned before the first crucial quantity to be understood is E (h) 0 e −νCt when ν = h since it is the normalizing constant in the conditioned measure (1.3). When the drift ν = h, [10, p.223, (4) and (5)] discusses the precise rate of asymptotic. Furthermore, 2 e −hCt does not grow exponentially and in our one-dimensional situation it is also not difficult to see that E Obviously, for our purpose a much stronger control on the rate of decay is necessary and therefore as a first step we provide in Lemma 2.1, which is proved in section 3.2, a complete asymptotic expansion for the behaviour of E (h) 0 e −hCt as t → ∞. Lemma 2.1. Let X be a one-dimensional Brownian motion with drift h > 0. We have the following asymptotic expansion: namely for any n ∈ N + , as t → ∞, (2.1)

Remark 2.2.
It is useful to compare the assertion of this Lemma with the results (1.1) and (1.2). In the one-dimensional situation the critical case the asymptotic behaviour of the expected survival distribution differs significantly from the subcritical and the supercritical cases in the fact that the decay has only additionally a polynomial decay factor. Therefore, from this point of view it is not clear, whether the behaviour of in the critical case is similar to the subcritical and the supercritical, respectively, or different from both regimes.

End point limiting behaviour
The next result shows that the minimum m t under the limiting measure is a nondegenerate random variable and thus in the limit the process is pushed away from −∞.
In the following theorem we study the joint law of the maximum M t = sup s≤t X s up to time t and the X t as t → ∞.
where M ∞ has a distribution function which does not depend on h > 0 and is given by the expression where the function G(·, ·) is defined in (5.3).

Remark 2.5. This result shows that under Q
(h) t the process is sub-ballistic and estimates its escape rate, i.e.

√
t. This is in contrast with higher dimensional discrete models of the same type where at criticality the process is ballistic, see [7], and the fact that in one dimension all models but this one are ballistic too, see [8]. Let us emphasize, that results which analogously to Theorem 2.4 do identify the scale t 1/2 seem to be missing in the subcritical case.
Remark 2.6. Thus in the case h = ν the position X t at time t and its maximum up to time t properly rescaled exhibit the same behaviour and even converge to a fully dependent pair of random variables. Moreover, the limiting distribution does not depend on h. This eventually follows from the scaling property of the Brownian motion, see section 3.5.
The same will be valid for the minimum process m t and X t provided h < 0.

Remark 2.7.
It is interesting to note that variants of (2.4) appear throughout the review paper [1]. Thus, our random variable M ∞ is a transformation of various quantities such as the maximum of a Brownian bridge, etc., but since we have no further probabilistic explanation as to why these relationships hold we do not discuss the matter further.

Limiting process
Next we consider the convergence of the process X under the measures Q (h) t . Thus, we will specify how the beginning of the process X is affected in the limit by the conditional measures (1.3). We have the following result.
the process X converges to the process Y which is a mixture of (shifted) three dimensional Bessel processes. In more detail, Y is a Brownian motion started from zero and not allowed to hit independent random level −ã whose density is given by h 2 ae −ha da, a > 0.

Useful analytical and spectral computations
We start the proofs by deriving useful formulae and introducing suitable notation.
Recall that T (0,c) = inf {t ≥ 0 : X t / ∈ (0, c)} is the first exit time for the process X from the interval (0, c) , c > 0. First using Girsanov's theorem and then following [10, p.226] our first claim expresses E  3.1. Let X be a one-dimensional Brownian motion with drift h > 0. We have that, for any 0 < y ≤ ∞, Proof. Using the Girsanov's theorem and then the scaling property of the Brownian motion we rewrite (3.1) as follows (3.2) As in [10, p.226] we re-express the quantity From (3.1) of Lemma 3.1 it is obvious that it suffices to work with the case h = 1.
Before proceeding further we evaluate the quantities involved in Lemma 3.1 using some tools from spectral theory. In the sequel we denote by a ∧ b = min{a, b} and a ∨ b = max{a, b}.

4)
In more detail when y = ∞ we have that and therefore Proof. The semigroup of Brownian motion with zero drift killed at the double exit time T (0,c) = inf{s ≥ 0 : B s / ∈ (0, c)} is a compact selfadjoint semigroup and the transition density has the following eigenfunction expansion for all x, y ∈ (0, c), where λ j = − π 2 j 2 2c 2 , j ≥ 1, are the eigenvalues and with v ≤ c, we derive immediately (3.4) and, plugging y = ∞ in (3.4) then (3.5) follows.
with v = c and the application of the Fubini theorem is immediate since the series (3.5) is clearly uniformly convergent for a ∈ [0, c], for any fixed c, t > 0. This is precisely (3.6). Finally, (3.7) follows by substitution in (3.1) of (3.6) with h = 1. This completes the proof.

Proof of Lemma 2.1
Proof. To prove Lemma 2.1 we rewrite (3.7) as follows: first thanks to (3.1) we work with h = 1 and we change variables We then write with the following definition of the integrand and we put Note that immediately then we get that Study of I 1 (t): From section 6, see (6.13), we deduce that, for any n ∈ N + , where the function G ( v 2 , 0) under the integral is computed from (5.3) of section 5. Clearly then the asymptotic relation (5.6), which yields G v and thus, for any n ∈ N + , Therefore using (3.14) and (3.13) in (3.12) lead to our claim (2.1) where we just recall that when h = 1 we use relation (3.1) which is basically a rescaling of the time.

Preliminary estimates
The proof of Theorem 2.3 and subsequent results hinge on the following key statements.
where we refer to (7.9) and (8.1) with ν = ∞ for the expressions of J 1 (t, a), J 2 (t, a) and recall that for s ∈ {0, 1} the function H is defined by πj π 2 j 2 uρ + 1 e − π 2 j 2 2 uγ sin (πjs + πjah) . Our next lemma improves the result above in a sense that it allows to truncate the integral from ln(t). This will be useful when we wish to remove the dependence on a at the lower limit of the integrals in (3.15) and (3.16). Proof. We observe that from the spectral expansion (3.8) sup Now we derive an easy estimate by splitting the supremum over (0, c) and giving appropriate estimates for the cases c ∈ (0, 1) and c ∈ (1, ln(t)) sup . Therefore using the elementary bound e Xt ≤ e c valid on t < T (0,c) we obtain that and the claim follows.

Proof of Theorem 2.3
Now we are ready to start with the proof of Theorem 2.3.

Proof of Theorem 2.3:
The result is an easy consequence of Lemma 3.3. Recall that by the definition of Q (h) and then (3.2), for any A > 0, Shifting the starting point from 0 → a ∧ Ah for the zero mean Brownian motion under This is valid for any A > 0 and we note that he −ha da is the probability density of Exp(h).
This concludes our claim.

Proof of Theorem 2.4
Proof of Theorem 2.4: Choose ν > 0 and we consider as in the proof of Theorem 2.3 Therefore we note that any possible limit will be invariant with respect to h. Hence, assume that h = 1. An easy computation involving the representation (3.3) and Note that and henceforth To study the first term in (3.24) we use (3.5) . Therefore from (3.25) and (3.26) we deduce in (3.24) that tO t (ν) = o (1) and hence However, we proceed with the same steps leading to (3.17) in the proof of Lemma 3.3 to get with obvious modification coming from integrating between (a, a + ν √ t) in the inner where J 1 (t, a, ν), J 2 (t, a, ν) are defined and studied in sections 7.2 and 8. From (8.2) and DCT we obtain that e −a |J 1 (t, a, ν)| da = o(1) We then observe that for any ν > ϑ > 0 Then without loss of generality put h = 1. However, using (3.3) to express e −Ct and the same computation as in (3.23) we get that Exactly as the proof of tO t (ν) = o (1) we get tO t (ν, ϑ) = o (1), namely that the second integral is irrelevant for the asymptotic. However, noting that and this shows that, for any pair ν > ϑ > 0, we have that This proves that lim t→∞

Proof of Theorem 2.8 4.1 Preliminaries and notation
We recall that a three dimensional Bessel process Y a started from a ≥ 0 is a stochastic process with continuous paths. It describes the radial part of a three dimensional Brownian motion started from a and can be identified with a Brownian motion started from a ≥ 0 conditioned not to cross zero. We denote by P † a the canonical measure induced by Y a on the space C (0, ∞). We recall that the scaling property of the Bessel process translates as follows: for any bounded functional F : where T (0,∞) is the first exit from the half-line (0, ∞), see [3, (8.3.2) p.83] which applies with h(x) = x in the case of zero drift Brownian motion.

Proof of Theorem 2.8
Proof of Theorem 2.8. Fix u > 0 and a bounded, continuous functional F := F u : C (0, u) → R + with ||F || ∞ its supremum norm. Choose B > 0 and let in the sequel  Like in any of the previous proofs and especially (3.21) we have that in the sense of measures Moreover, to evaluate the latter we follow with immediate modifications (3.22) to get where for the sake of brevity we have put Assuming the validity of Lemma 4.1 below and using the asymptotic relation (  This concludes the proof as B > 0 is arbitrary and this holds for any bounded positive measurable functional F and any u > 0. However, the last expression corresponds to a shifted to zero three dimensional Bessel process Y e h − e h , started from independent random variable with distribution P (e h ∈ dx) = h 2 xe −hx dx, x > 0. This concludes the proof of the theorem.
To study the measure th 2 U 1 t,h (dx, A) we prove the following proposition. Lemma 4.1. We have that for any Proof. Recall (4.7) for the definition of O(X). We consider and estimate in the sense of  Therefore, it remains to study the remaining portion of the integral, or the measure  where we note that a ≥ 0 ∨ (−hx) since otherwise we have that the impossible inequality m t > X u , u ≤ t must hold, namely the running minimum to exceed the value of the process. However, according to Lemma 3.4 and henceforth However (4.2) allows us to deduct that We show using (4. it remains to show that lim t→∞S (t, dx, A) is the zero measure. First note that for any fixed 0 ≤ a ≤ A and c > ln(t) we have that in sense of measures This for the first inequality together with the estimate (4.12) for the second givẽ
Then the following result is a standard consequence of the Poisson summation. Proof. To justify (5.4) we apply the Poisson summation for f ( The relations (5.5) and (5.6) are a result of differentiation of (5.4) which applies due to the uniform convergence of (5.3) in any small enough neighbourhood of v > 0.
(6.5) Therefore Mf 2 is invertible on its region of definition.
So it remains to consider the integral term. Invoking (6.9) we note that Upon integration with respect to v and changing variables x = u + v, y = u we get where we have used that c = −{c} − n and B(·, ·) is the classical Beta function.
and even We start with (7.8). Clearly, when u ≥ 1, a trivial bound using (7.4) gives (7.8) and using this in (7.3) the following estimate is obtained and we get upon changing Splitting the integration in v at the point where b j = j − 1 3 , namely v j = 50a 2 j − 1 3 − 1, This proves (7.8).
Finally from (3.11) we recognize that the last sum is simply F (u, 0) which according to (5.3) leads to F (u, 0) = G u, 1 2 . Therefore since G ∞, 1 2 = 0, see (5.3), Moreover, (7.6) ensures that even and therefore the DCT applies and yields our claim namely (7.10). Even more this uniform bound on the derivative gives (7.11) and subsequently the uniform convergence in (7.10) for a-compact sets.
When ν < ∞ we are able to give some other useful estimates. where G is defined in (5.3).