Bessel process killed upon leaving a half-line
Kamil Bogus
Wrocław University of Technology
Probability and Analysis, 5 May 2015, Bedlewo
˛
Kamil Bogus
Bessel process killed upon leaving a half-line
Articles
Kamil Bogus, Jacek Małecki
Sharp estimates of transition probability density for Bessel
process in half-line,
Potential Analysis (2015), DOI 10.1007/s11118-015-9461-x.
Kamil Bogus, Jacek Małecki
Heat kernel estimates for the Bessel differential operator in
half-line,
arXiv:1501.02618, submitted.
Kamil Bogus
Bessel process killed upon leaving a half-line
Bessel processes
For µ ∈ R the one-dimensional diffusion process on (0, +∞) with
infinitesimal generator 12 L(µ) , where
L(µ) =
d2
2µ + 1 d
,
+
dx 2
x
dx
is called Bessel process with index µ.
Kamil Bogus
Bessel process killed upon leaving a half-line
Bessel processes
For µ ∈ R the one-dimensional diffusion process on (0, +∞) with
infinitesimal generator 12 L(µ) , where
L(µ) =
d2
2µ + 1 d
,
+
dx 2
x
dx
is called Bessel process with index µ.
Example
B(t) = (B1 (t) + x, B2 (t), . . . , Bn (t))
BM in Rn
P–
n
2
X (t) = ||B(t)|| = ((B1 (t) + x) + k =2 Bk2 (t))1/2
X (t) – Bessel process with index µ = n2 − 1, starting from x ≥ 0.
Notation:
n – dimension,
µ – index,
n = 2µ + 2.
Kamil Bogus
Bessel process killed upon leaving a half-line
Modified Bessel function of the first kind
For µ ∈ R and z ∈ C\(−R+ ) we define modified Bessel function of
the first kind as
Iµ (z) =
∞
z µ X
2
k =0
z 2k
1
.
k !Γ(µ + k + 1) 2
Asymptotic behaviour of Iµ (z) can be described as follows
(
zµ
µ+2
) for z → 0+
2µ Γ(1+µ) + O(z
Iµ (z) =
.
z
√e
(1 + O(1/z)) for z → ∞
2πz
Example
r
I1/2 (z) =
Kamil Bogus
2
sinh(z)
πz
Bessel process killed upon leaving a half-line
Transition probability density
Transition probability density for Bessel process with index µ
(with respect to the reference measure m(µ) (dy ) = y 2µ+1 dy ) is given
by
2
xy
1
x + y2
−µ
(µ)
I|µ|
,
p (t, x, y ) = (xy ) exp −
t
2t
t
where x, y , t > 0.
Example
p(1/2) (t, x, y ) = √
1
1
2πt xy
(x − y )2
(x + y )2
exp −
− exp −
2t
2t
Kamil Bogus
Bessel process killed upon leaving a half-line
Hitting times of Bessel process
We denote the first hitting time of a given level a > 0 by
(µ)
Ta
(µ)
= inf{t ≥ 0 : Rt
(µ)
= a}
(µ)
and we write qx,a (t) for the density of Ta .
Example
(1/2)
qx,1 (t) =
x −1 1
(x − 1)2
√
exp −
,
x
2t
2πt 3
Kamil Bogus
x > 1, t > 0
Bessel process killed upon leaving a half-line
Hitting times of Bessel process - sharp estimates
Theorem (Byczkowski, Małecki, Ryznar, 2013)
Let µ 6= 0. For every x > 1 and t > 0 we have
µ
(µ)
qx,1 (s) ≈
and
(0)
qx,1 (t)
1
(x − 1) 1 ∧ 2µ
x
2
e−(x−1) /(2t)
x 2|µ|−1
t 3/2
(t + x)|µ|−1/2
x −1 1
(x − 1)2
,
≈ √
exp −
2t
x t 3/2
t < 2x,
and for t ≥ 2x
(0)
qx,1 (t)
x −1
1 + ln x
1
(x − 1)2
≈
exp −
x (1 + ln(t + x))(1 + ln(1 + t/x)) t
2t
µ
Here f ≈ g means that there exists constant c = c(µ) > 0 such that
c −1 g ≤ f ≤ cg
.
Kamil Bogus
Bessel process killed upon leaving a half-line
Killed Bessel process
We define Bessel process killed upon leaving the half-line
˜ (µ) = {R
˜ t : t ≥ 0} , corresponding to the
(a, +∞) as a process R
Bessel process R (µ) , where
(
(µ)
(µ)
dla t < Ta
Rt
(µ)
˜
Rt =
(µ) .
ξ
dla t ≥ Ta
Here state ξ ∈
/ (0, ∞) means "cemetary" and we assume that
(µ)
R0 = x > a ≥ 0.
Main goal
Sharp estimate of transition probability density for the killed Bessel
process.
Kamil Bogus
Bessel process killed upon leaving a half-line
Transition probability density for killed process
The transition probability density of the Bessel process killed
upon leaving the half-line (a, +∞) can be expressed by the Hunt
formula in the following way
(µ)
(µ)
pa (t, x, y ) = p(µ) (t, x, y ) − ra (t, x, y )
where x, y > a, t > 0.
Here
Z
(µ)
ra (t, x, y ) =
t
(µ)
qx,a (s)p(µ) (t − s, a, y )ds.
0
Example
(1/2)
p1 (t, x, y )
1
1
=√
2πt xy
(x − y )2
(x + y − 2)2
exp −
− exp −
2t
2t
Kamil Bogus
Bessel process killed upon leaving a half-line
(µ)
Motivation to investigate pa (t, x, y )
PDE
p(µ) (t, x, y ) is a solution of the heat equation
1 (µ)
∂t − L
u=0
2
where L(µ) :=
d2
dx 2
+
2µ+1 d
x dx
is the Bessel differential operator.
Harmonic analysis
Operator L(µ) on (0, 1) plays an important rôle in research
Fourier-Bessel heat kernels.
Stochastic process
(µ)
pa (t, x, y ) is transition probability density of killed semigroup
associated with Bessel process R (µ) .
Kamil Bogus
Bessel process killed upon leaving a half-line
Main theorem - case µ 6= 0
Theorem (K. Bogus, J. Małecki, 2015)
Let µ 6= 0. For every x, y > 1 and t > 0 we have
µ
(µ)
p1 (t, x, y ) ≈
(x − 1)(y − 1)
1∧
t
t
1∨
p(µ) (t, x, y ),
xy
where
µ 1
(x − y )2 xy |µ|+1/2
−µ−1/2
p(µ) (t, x, y ) ≈ √ (xy )
exp −
.
1∧
2t
t
t
Kamil Bogus
Bessel process killed upon leaving a half-line
Main theorem - case µ = 0
Theorem (K. Bogus, J. Małecki, 2015)
Let µ = 0. For every x, y > 1 and t > 0 we have
(0)
p1 (t, x, y ) ≈ ln x ln y
ln
3t
3t
√ ln
√
x+ t y+ t
−1
2
1
x + y2
exp −
t
2t
whenever xy ≤ t, and
(0)
p1 (t, x, y ) ≈
1
(x − 1)(y − 1)
(x − y )2
√
1∧
exp −
t
2t
xyt
whenever xy ≥ t.
Kamil Bogus
Bessel process killed upon leaving a half-line
The result of Q.S. Zhang
Theorem (Q.S. Zhang, 2002)
For every C 1,1 -bounded set D ⊂ Rn there exists constants
c1 , c2 , T > 0 depending only on D such that
c1
G(x, t; y , 0)
1
c2 |x − y |2
|x − y |2
≤
≤
exp
−
exp
−
t
f (t, x, y )
c2 t
t n/2
c1 t n/2
h
i
)
where f (t, x, y ) = ρ(x)ρ(y
∧ 1 and ρ(x) = dist(x, ∂D).
t
Here G stands for the heat kernel of Dirichlet Laplacian on set
D ⊂ Rn .
Kamil Bogus
Bessel process killed upon leaving a half-line
The result of P. Gyrya and L. Saloff-Coste
Theorem (P. Gyrya, L. Saloff-Coste, 2011)
Let D ⊂ X be a unbounded uniform domain and X be a Harnack –
type Dirichlet space satisfying some technical assumptions.
Then there exist constants c1 , c2 , c3 , c4 ∈ (0, +∞) such that for all
t > 0 and x, y ∈ D
c1 Px (TD > t)Py (TD > t)p(c2 t, x, y )
≤
pD (t, x, y )
≤
c3 Px (TD > t)Py (TD > t)p(c4 t, x, y ),
where p(t, x, y ) is a global heat kernel.
In our case: For µ > 0 and x, y > a > 0,
(µ)
µ
(µ)
(µ)
pa (t, x, y ) ≈ Px (Ta
(µ)
(µ)
> t)Py (Ta
t >0
> t)p(µ) (t, x, y ),
It is not true for xy ≥ t!
Kamil Bogus
Bessel process killed upon leaving a half-line
xy ≤ t
Methods used in the proof
Hunt formula,
Absolute continuity property of Bessel processes with different
indices
Strong Markov Property
Scalling property of Bessel processes.
Chapmann-Kolmogorov equations
Inequalities involving ratios of Iµ (z)
(µ)
Sharp estimates of qx,a (s) and survival probabilities of Bessel
processes
Exact formula for the transition probability density for the Bessel
process, with index µ = 1/2, killed on first hitting time in level
a>0
Kamil Bogus
Bessel process killed upon leaving a half-line
What is next ?
(µ)
Asymptotics of pa (t, x, y ) for t → ∞, t → 0, . . .
Kamil Bogus
Bessel process killed upon leaving a half-line
What is next ?
(µ)
Asymptotics of pa (t, x, y ) for t → ∞, t → 0, . . .
Bessel process killed on exiting interval (a, b)
Kamil Bogus
Bessel process killed upon leaving a half-line
What is next ?
(µ)
Asymptotics of pa (t, x, y ) for t → ∞, t → 0, . . .
Bessel process killed on exiting interval (a, b)
Killed Bessel bridges
Kamil Bogus
Bessel process killed upon leaving a half-line
What is next ?
(µ)
Asymptotics of pa (t, x, y ) for t → ∞, t → 0, . . .
Bessel process killed on exiting interval (a, b)
Killed Bessel bridges
Applications in harmonical analysis and PDE
Kamil Bogus
Bessel process killed upon leaving a half-line
What is next ?
(µ)
Asymptotics of pa (t, x, y ) for t → ∞, t → 0, . . .
Bessel process killed on exiting interval (a, b)
Killed Bessel bridges
Applications in harmonical analysis and PDE
...
Kamil Bogus
Bessel process killed upon leaving a half-line
The End
Thank You for attention !
Kamil Bogus
Bessel process killed upon leaving a half-line
References 1/5
Mathematical tables of integral, series,. . .
M. Abramowitz, I. A. Stegun,
Handbook of Mathematical Functions with Formulas. Graphs and
Mathematical Tables,
Dover, New York, 9th edition (1972).
I. S. Gradshteyn, I. M. Ryzhik,
Table of Integrals, Series, and Products,
Academic Press, California, 7th edition (2007).
Stochastic processes
A. N. Borodin and P. Salminen,
Handbook of Brownian Motion - Facts and Formulae,
Birkhauser Verlag, Basel, 2. edition (2002).
K. L. Chung, Z. Zhao,
From Brownian Motion to Schrodinger’s Equation,
Springer-Verlag, New York (1995).
D. Revuz, M. Yor,
Continuous Martingales and Brownian Motion,
Springer, New York, (1999).
Kamil Bogus
Bessel process killed upon leaving a half-line
References 2/5
Modified Bessel functions
A. Baricz,
Bounds for modified Bessel functions of the first and second
kinds,
Proc. Edinb. Math. Soc. 53(3) (2010) 575-599.
E. K. Ifantis, P. D. Siafarikas,
Bounds for modifed Bessel functions,
Rendiconti del Circolo Matematico di Palermo, Vol. 40, Issue 3
(1991), 347-356.
A. Laforgia, P. Natalini,
Some inequalities for modifed Bessel functions,
Journal of Inequalities and Applications (2010), art. 253035, 10
pages.
I. Nasell,
Rational Bounds for Ratios of Modified Bessel Function,
SIAM J. Math. Anal., vol. 9(1) (1978), 1-11.
Kamil Bogus
Bessel process killed upon leaving a half-line
References 3/5
Hitting times of Bessel processes
T. Byczkowski, M. Ryznar,
Hitting distibution of geometric Brownian motion.
Studia Math., 173(1) (2006), 19-38.
Y. Hamana, H. Matsumoto,
The probability densities of the first hitting times of Bessel
processes,
J. Math-for-Ind. 4B (2012) 91-95.
Y. Hamana, H. Matsumoto,
The probability distribution of the first hitting time of Bessel
processes,
Trans. Amer. Math. Soc., 365 (2013), 5237-5257.
T. Byczkowski, J. Małecki, M. Ryznar,
Hitting times of Bessel processes,
Pot. Anal. vol. 38 (2013), 753-786.
Kamil Bogus
Bessel process killed upon leaving a half-line
References 4/5
Heat kernel estimates
E. B. Davies,
Heat kernels and spectral theory,
Cambridge Tracts in Mathematics, Cambridge University Press,
vol. 92, Cambridge (1990).
E. B. Davies",
Intrinsic ultracontractivity and the Dirichlet Laplacian",
J. Funct. Anal., vol. 100, (1991), 162–180.
P. Gyrya, L. Saloff-Coste,
Neumann and Dirichlet heat kernels in inner uniform domains,
Asterisque (2011).
L. Saloff-Coste,
The heat kernel and its estimates.,
Adv. Stud. Pure Math., vol. 57 (2010), p. 405–436.
Q. S. Zhang,
The boundary behavior of heat kernels of Dirichlet Laplacians,
J. Differential Equations, vol. 182 (2002), p. 416-430.
Kamil Bogus
Bessel process killed upon leaving a half-line
References 5/5
Bessel process killed upon leaving a half–line or interval
K. Bogus, J. Małecki
Sharp estimates of transition probability density for Bessel
process in half-line,
Potential Analysis (2015), DOI 10.1007/s11118-015-9461-x.
K. Bogus, J. Małecki
Heat kernel estimates for the Bessel differential operator in
half-line,
arXiv:1501.02618, submitted (2015).
˙
J. Małecki, G. Serafin, T. Zórawik,
Fourier-Bessel heat kernel estimates. arXiv:1503.02226,
submitted (2015).
A. Nowak, L. Roncal,
Sharp heat kernel estimates in the Fourier-Bessel setting for a
continuous range of the type parameter,
Acta Math. Sin. (Engl. Ser.), vol. 30 p.437–444 (2014).
Kamil Bogus
Bessel process killed upon leaving a half-line