Bessel process killed upon leaving a half-linebcc.impan.pl/15PandA/uploads/files/Kamil-Bogus.pdf · Articles Kamil Bogus, Jacek Małecki Sharp estimates of transition probability

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 MałeckiSharp estimates of transition probability density for Besselprocess in half-line,Potential Analysis (2015), DOI 10.1007/s11118-015-9461-x.

Kamil Bogus, Jacek MałeckiHeat kernel estimates for the Bessel differential operator inhalf-line,arXiv:1501.02618, submitted.


Bessel processes

For µ ∈ R the one-dimensional diffusion process on (0,+∞) withinfinitesimal generator 1

2 L(µ), where

L(µ) =d2

dx2 +2µ+ 1

xddx,

is called Bessel process with index µ.

Example

B(t) = (B1(t) + x ,B2(t), . . . ,Bn(t)) – BM in Rn

X (t) = ||B(t)|| = ((B1(t) + x)2 +∑n

k=2 B2k (t))1/2

X (t) – Bessel process with index µ = n2 − 1, starting from x ≥ 0.

Notation:n – dimension,µ – index,n = 2µ+ 2.


Bessel processes

For µ ∈ R the one-dimensional diffusion process on (0,+∞) withinfinitesimal generator 1

2 L(µ), where

L(µ) =d2

dx2 +2µ+ 1

xddx,

is called Bessel process with index µ.

Example

B(t) = (B1(t) + x ,B2(t), . . . ,Bn(t)) – BM in Rn

X (t) = ||B(t)|| = ((B1(t) + x)2 +∑n

k=2 B2k (t))1/2

X (t) – Bessel process with index µ = n2 − 1, starting from x ≥ 0.

Notation:n – dimension,µ – index,n = 2µ+ 2.


Modified Bessel function of the first kind

For µ ∈ R and z ∈ C\(−R+) we define modified Bessel function ofthe first kind as

Iµ(z) =(z

2

)µ ∞∑k=0

1k !Γ(µ+ k + 1)

(z2

)2k.

Asymptotic behaviour of Iµ(z) can be described as follows

Iµ(z) =

{zµ

2µΓ(1+µ) + O(zµ+2) for z → 0+

ez√

2πz(1 + O(1/z)) for z →∞

.

Example

I1/2(z) =

√2πz

sinh(z)


Transition probability density

Transition probability density for Bessel process with index µ(with respect to the reference measure m(µ)(dy) = y2µ+1dy ) is givenby

p(µ)(t , x , y) =1t

(xy)−µ exp(−x2 + y2

2t

)I|µ|(xy

t

),

where x , y , t > 0.

Example

p(1/2)(t , x , y) =1√2πt

1xy

(exp

(− (x − y)2

2t

)− exp

(− (x + y)2

2t

))


Hitting times of Bessel process

We denote the first hitting time of a given level a > 0 by

T (µ)a = inf{t ≥ 0 : R(µ)

t = a}

and we write q(µ)x,a (t) for the density of T (µ)

a .

Example

q(1/2)x,1 (t) =

x − 1x

1√2πt3

exp(− (x − 1)2

2t

), x > 1, t > 0


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

q(µ)x,1 (s)

µ≈ (x − 1)

(1 ∧ 1

x2µ

)e−(x−1)2/(2t)

t3/2

x2|µ|−1

(t + x)|µ|−1/2

and

q(0)x,1(t) ≈ x − 1√

x1

t3/2 exp(− (x − 1)2

2t

), t < 2x ,

and for t ≥ 2x

q(0)x,1(t) ≈ x − 1

x1 + ln x

(1 + ln(t + x))(1 + ln(1 + t/x))

1t

exp(− (x − 1)2

2t

)Here f

µ≈ g means that there exists constant c = c(µ) > 0 such that

c−1g ≤ 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(a,+∞) as a process R(µ) = {Rt : t ≥ 0} , corresponding to theBessel process R(µ), where

R(µ)t =

{R(µ)

t dla t < T (µ)a

ξ dla t ≥ T (µ)a

.

Here state ξ /∈ (0,∞) means "cemetary" and we assume thatR(µ)

0 = x > a ≥ 0.

Main goal

Sharp estimate of transition probability density for the killed Besselprocess.


Transition probability density for killed process

The transition probability density of the Bessel process killedupon leaving the half-line (a,+∞) can be expressed by the Huntformula in the following way

p(µ)a (t , x , y) = p(µ)(t , x , y)− r (µ)

a (t , x , y)

where x , y > a, t > 0.Here

r (µ)a (t , x , y) =

∫ t

0q(µ)

x,a (s)p(µ)(t − s,a, y)ds.

Example

p(1/2)1 (t , x , y) =

1√2πt

1xy

(exp

(− (x − y)2

2t

)− exp

(− (x + y − 2)2

2t

))


Motivation to investigate p(µ)a (t , x , y)

PDE

p(µ)(t , x , y) is a solution of the heat equation(∂t −

12

L(µ)

)u = 0

where L(µ) := d2

dx2 + 2µ+1x

ddx is the Bessel differential operator.

Harmonic analysis

Operator L(µ) on (0,1) plays an important rôle in researchFourier-Bessel heat kernels.

Stochastic process

p(µ)a (t , x , y) is transition probability density of killed semigroup

associated with Bessel process R(µ).


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

p(µ)1 (t , x , y)

µ≈(

1 ∧ (x − 1)(y − 1)

t

)(1 ∨ t

xy

)p(µ)(t , x , y),

where

p(µ)(t , x , y)µ≈ 1√

t(xy)−µ−1/2 exp

(− (x − y)2

2t

)(1 ∧ xy

t

)|µ|+1/2.


Main theorem - case µ = 0

Theorem (K. Bogus, J. Małecki, 2015)

Let µ = 0. For every x , y > 1 and t > 0 we have

p(0)1 (t , x , y) ≈ ln x ln y

(ln

3tx +√

tln

3ty +√

t

)−1 1t

exp(−x2 + y2

2t

)whenever xy ≤ t , and

p(0)1 (t , x , y) ≈

(1 ∧ (x − 1)(y − 1)

t

)1√xyt

exp(− (x − y)2

2t

)whenever xy ≥ t .


The result of Q.S. Zhang

Theorem (Q.S. Zhang, 2002)

For every C1,1-bounded set D ⊂ Rn there exists constantsc1, c2,T > 0 depending only on D such that

c1

tn/2 exp(−c2|x − y |2

t

)≤ G(x , t ; y ,0)

f (t , x , y)≤ 1

c1tn/2 exp(−|x − y |2

c2t

)where f (t , x , y) =

[ρ(x)ρ(y)

t ∧ 1]

and ρ(x) = dist(x , ∂D).

Here G stands for the heat kernel of Dirichlet Laplacian on setD ⊂ Rn.


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 allt > 0 and x , y ∈ D

c1Px(TD > t)Py (TD > t)p(c2t , x , y) ≤ pD(t , x , y)

≤ c3Px(TD > t)Py (TD > t)p(c4t , x , y),

where p(t , x , y) is a global heat kernel.

In our case: For µ > 0 and x , y > a > 0, t > 0

p(µ)a (t , x , y)

µ≈ P(µ)

x (T (µ)a > t)P(µ)

y (T (µ)a > t)p(µ)(t , x , y), xy ≤ t

It is not true for xy ≥ t !


Methods used in the proof

Hunt formula,Absolute continuity property of Bessel processes with differentindicesStrong Markov PropertyScalling property of Bessel processes.Chapmann-Kolmogorov equationsInequalities involving ratios of Iµ(z)

Sharp estimates of q(µ)x,a (s) and survival probabilities of Bessel

processesExact formula for the transition probability density for the Besselprocess, with index µ = 1/2, killed on first hitting time in levela > 0


What is next ?

Asymptotics of p(µ)a (t , x , y) for t →∞, t → 0, . . .

Bessel process killed on exiting interval (a,b)

Killed Bessel bridgesApplications in harmonical analysis and PDE. . .


What is next ?





What is next ?



Killed Bessel bridges

Applications in harmonical analysis and PDE. . .


What is next ?



Killed Bessel bridgesApplications in harmonical analysis and PDE

. . .


What is next ?





The End

Thank You for attention !


References 1/5Mathematical tables of integral, series,. . .

M. Abramowitz, I. A. Stegun,Handbook of Mathematical Functions with Formulas. Graphs andMathematical 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 processesA. 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).


References 2/5

Modified Bessel functionsA. Baricz,Bounds for modified Bessel functions of the first and secondkinds,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, 10pages.I. Nasell,Rational Bounds for Ratios of Modified Bessel Function,SIAM J. Math. Anal., vol. 9(1) (1978), 1-11.


References 3/5

Hitting times of Bessel processesT. 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 Besselprocesses,J. Math-for-Ind. 4B (2012) 91-95.Y. Hamana, H. Matsumoto,The probability distribution of the first hitting time of Besselprocesses,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.


References 4/5

Heat kernel estimatesE. 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.


References 5/5

Bessel process killed upon leaving a half–line or interval

K. Bogus, J. MałeckiSharp estimates of transition probability density for Besselprocess in half-line,Potential Analysis (2015), DOI 10.1007/s11118-015-9461-x.K. Bogus, J. MałeckiHeat kernel estimates for the Bessel differential operator inhalf-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 acontinuous range of the type parameter,Acta Math. Sin. (Engl. Ser.), vol. 30 p.437–444 (2014).


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Bessel process killed upon leaving a half-linebcc.impan.pl/15PandA/uploads/files/Kamil-Bogus.pdf · Articles Kamil Bogus, Jacek Małecki Sharp estimates of transition probability