Kamil Bogus
Wrocªaw University of Technology, Poland
Bessel process killed upon leaving a half-line
Joint work with Jacek Maªecki.
(µ)
In this talk we study the Bessel process Rt with index µ ∈ R, starting from
x > 0. The transition probability density (with respect to the reference measure
m(µ) (dy) = y 2µ+1 dy ) of the process is expressed by the modied Bessel function of
the rst kind in the following way
2
xy x + y2
1
−µ
I|µ|
,
x, y > 0, t > 0.
p(µ) (t, x, y) = (xy) exp −
t
2t
t
The transition density function of this process killed when it reaches a positive level
a (where x > a) is given (with respect to m(µ) (dy)) by the Hunt formula
p(µ)
a (t, x, y)
(µ)
=p
Z
(t, x, y) −
t
(µ)
p(µ) (t − s, a, y)qx,a
(s)ds,
x, y > a, t > 0,
0
(µ)
(µ)
(µ)
where qx,a (s) is the density of the rst hitting time Ta = inf{t > 0 : Rt = a}.
(µ)
Our purpose is to derive sharp two-sided estimates of pa (t, x, y). The main result
is given in the following
Theorem 1
(KB, J. Maªecki, 2015).
and t > 0 we have
µ
p(µ)
a (t, x, y) ≈
1∧
(x − a)(y − a)
t
Let µ 6= 0 and a > 0. Then for every x, y > a
1
√ (xy)−µ−1/2 ×
t
xy µ−1/2
(x − y)2 1∧
.
× exp −
2t
t
Moreover if µ = 0 and a > 0, then we have for every x, y > a and t > 0 that
p(0)
a (t, x, y)
3t
3t
√ ln
√
≈ ln(x/a) ln(y/a) ln
ax + a t ay + a t
−1
2
1
x + y2
exp −
t
2t
when xy ≤ t, and
p(0)
a (t, x, y)
when xy > t.
(x − a)(y − a)
1
(x − y)2
√
≈ 1∧
exp −
t
xyt
2t
µ
(µ)
(µ)
Here f (t, x, y) ≈ g(t, x, y) means that there exist positive constants c1 and c2
(µ)
(µ)
depending only on the index µ such that c1 ≤ f /g ≤ c2 for every x, y > a and
µ
t > 0. We write ≈ instead of ≈ if these constants do not depend on any parameter.
Notice that the constants appearing in the exponential terms in the upper and
lower estimates are the same and consequently the exponential behaviour of the density is very precise. We will also discuss our results in context of classical estimates
of Dirichlet heat kernel for Laplacian and estimates of FourierBessel heat kernel.
This presentation is based on joint work with Jacek Maªecki ([1] and [2]).
References
[1] K. Bogus, J. Maªecki, Sharp estimates of transition probability density for Bessel
process in half-line, Potential Analysis, DOI 10.1007/s11118-015-9461-x, p.1-22,
2015.
[2] K. Bogus, J. Maªecki, Heat kernel estimates for the Bessel
in half-line, preprint, arXiv:1501.02618, p. 1-10, 2015.
dierential operator