Sharp Estimates of Transition Probability Density for Bessel Process in Half-Line

Bogus, Kamil; Małecki, Jacek

doi:10.1007/s11118-015-9461-x

Cited by 22 publications

(40 citation statements)

References 26 publications

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“…For

μ = 0

we will simply write

p_{a} (t, x, y)

and we will also omit superscripts in the notation of the other objects related to that case. The explicit formula for

p^{(μ)} (t, x, y) : = p_{0}^{(μ)} (t, x, y)

in terms of the modified Bessel functions is well‐known

p^{(μ)} (t, x, y) = \frac{1}{t} {(xy)}^{- μ} exp (- \frac{x^{2} + y^{2}}{2 t}) I_{| μ |} (\frac{xy}{t}), x, y, t > 0 .

For

a > 0

and

μ \neq 0

the sharp two‐sided estimates of

p_{a}^{(μ)} (t, x, y)

of the form

p_{a}^{(μ)} (t, x, y) \overset{μ}{\approx} [1 \land \frac{(x - a) (y - a)}{t}] {(() 1 \land \frac{italicxy}{t})}^{| μ | - \frac{1}{2}} \frac{1}{{(italicxy)}^{μ + 1 / 2}} \frac{1}{\sqrt{t}} exp (- \frac{{(x - y)}^{2}}{2 t}),

for every

x, y > a

and

t > 0

, were obtained recently in …”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the logarithmic behaviour of

p_{a} (t, x, y)

require more delicate methods, i.e. those used in cannot be applied to prove .…”

Section: Introductionmentioning

confidence: 99%

“…the constants appearing in the exponential factors in the upper and lower bounds are different. The presented result given in Theorem , as well as the result of , is free from this defect and here the exponential behaviour is described explicitly. However, this requirement imposes more complicated form of the part describing non‐exponential behaviour of

p_{a} (t, x, y)

.…”

Section: Introductionmentioning

confidence: 99%

“…However, this requirement imposes more complicated form of the part describing non‐exponential behaviour of

p_{a} (t, x, y)

. We refer the reader to the Introduction in , where more detailed discussion on this phenomenon was conducted, as well as to the recent work , where the results of the same kind were obtain in the case

x, y < a

, i.e. for the interval (0, a ).…”

Section: Introductionmentioning

confidence: 99%

“…In the paper we consider the Bessel differential operator

L^{(μ)} = \frac{d^{2}}{{italicdx}^{2}} + \frac{2 μ + 1}{x} \frac{d}{italicdx}

in half‐line

(a, \infty)

a > 0

, and its Dirichlet heat kernel

p_{a}^{(μ)} (t, x, y)

. For

μ = 0

, by combining analytical and probabilistic methods, we provide sharp two‐sided estimates of the heat kernel for the whole range of the space parameters

x, y > a

and every

t > 0

, which complements the recent results given in , where the case

μ \neq 0

was considered.…”

mentioning

confidence: 99%

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Heat kernel estimates for the Bessel differential operator in half‐line

Bogus

Małecki

2016

Mathematische Nachrichten

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Abstract. In the paper we consider the Bessel differential operator L (µ) = d 2 dx 2 + 2µ + 1 x d dx in half-line (a, ∞), a > 0, and its Dirichlet heat kernel p (µ) a (t, x, y). For µ = 0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x, y > a and every t > 0, which complements the recent results given in [1], where the case µ = 0 was considered.

show abstract

“…For

μ = 0

we will simply write

p_{a} (t, x, y)

and we will also omit superscripts in the notation of the other objects related to that case. The explicit formula for

p^{(μ)} (t, x, y) : = p_{0}^{(μ)} (t, x, y)

in terms of the modified Bessel functions is well‐known

p^{(μ)} (t, x, y) = \frac{1}{t} {(xy)}^{- μ} exp (- \frac{x^{2} + y^{2}}{2 t}) I_{| μ |} (\frac{xy}{t}), x, y, t > 0 .

For

a > 0

and

μ \neq 0

the sharp two‐sided estimates of

p_{a}^{(μ)} (t, x, y)

of the form

p_{a}^{(μ)} (t, x, y) \overset{μ}{\approx} [1 \land \frac{(x - a) (y - a)}{t}] {(() 1 \land \frac{italicxy}{t})}^{| μ | - \frac{1}{2}} \frac{1}{{(italicxy)}^{μ + 1 / 2}} \frac{1}{\sqrt{t}} exp (- \frac{{(x - y)}^{2}}{2 t}),

for every

x, y > a

and

t > 0

, were obtained recently in …”

Section: Introductionmentioning

confidence: 99%

“…Moreover, the logarithmic behaviour of

p_{a} (t, x, y)

require more delicate methods, i.e. those used in cannot be applied to prove .…”

Section: Introductionmentioning

confidence: 99%

p_{a} (t, x, y)

.…”

Section: Introductionmentioning

confidence: 99%

“…However, this requirement imposes more complicated form of the part describing non‐exponential behaviour of

p_{a} (t, x, y)

x, y < a

, i.e. for the interval (0, a ).…”

Section: Introductionmentioning

confidence: 99%

“…In the paper we consider the Bessel differential operator

L^{(μ)} = \frac{d^{2}}{{italicdx}^{2}} + \frac{2 μ + 1}{x} \frac{d}{italicdx}

in half‐line

(a, \infty)

a > 0

, and its Dirichlet heat kernel

p_{a}^{(μ)} (t, x, y)

. For

μ = 0

, by combining analytical and probabilistic methods, we provide sharp two‐sided estimates of the heat kernel for the whole range of the space parameters