[PPT] - Spectral functions of subordinated Brownian motion M.A. Fahrenwaldt PowerPoint Presentation

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Spectral functions of subordinated Brownian motion

M.A. Fahrenwaldt12

1Institut für Mathematische Stochastik

Leibniz Universität Hannover, Germany

2EBZ Business School, Bochum, Germany

Berlin, 23 October 2014

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We explore a correspondence between stochastic processes and analytical objects — simplified idea

Stochastics Analysis Process on Rn Heat kernel BXt e−At  





α = limx→0 1

x ❊(XT(x)−),

where Tx = inf{t ≥ 0|Xt > x}

Lévy’s arcsine law

− − − − → TR(e−At) ∼ ect

∞

k=0

ck(α)t(n−k)/2α +ect

∞

k=0

˜ ck(α)t(n−k)/2α log t

heat trace asymptotics

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Motivation

The heat kernel is of significant intrinsic interest in mathematics

small time asymptotics of the trace of the heat kernel encodes important information about the topology of a manifold M Tr(e−∆t) ∼ (4πt)n/2     a0

vol(M)

+ a1

1

6

κ(M)vol(M)

t + · · ·     , the heat kernel is used for ground state calculations in quantum field theory, it is the transition density of a stochastic process on the manifold and as such significant in stochastics and its applications

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Motivation

Similar investigations have centred on processes living on compact manifolds

Investigations on Rn typically consider estimates of Tr

e−tH

− Tr

e−tH0

where H and H0 are of Schrödinger type with or without potential, for example H = ∆α/2 + V and H0 = ∆α/2. Two schools of thought: stochastic analysis and scattering theory — no explicit trace asymptotics On compact manifolds there are explicit heat trace asymptotics Applebaum (2011), partly extended by Bañuelos & Baudoin (2012):

◮ infinitely divisible central probability measures on compact Lie groups ◮ fully explicit example is generator

√ −∆ on Tn, SU(2), SO(3)

◮ Fourier analysis on Lie groups and global pseudodifferential operators

Bañuelos, Mijena & Nane (2014):

◮ relativistic stable process on a bounded domain in Rn ◮ almost closed form expression for the first two terms in the heat trace

with probabilistic interpretation

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Key result

We consider subordinate Brownian motion on Rn for a wide class of subordinators – fully tractable yet exciting

Bt a canonical Brownian motion on Rn and Xt a subordinator: increasing Lévy process with values in [0, ∞) and X0 = 0 a.s. Distribution of Xt in terms of Bernstein function f (Laplace exponent)

◮ characteristic function ❊

eiξ·BXt

= e−tf (|ξ|2) for ξ ∈ Rn

◮ generating function ❊

e−λXt

= e−tf (λ) for λ > 0

Our class of Laplace exponents is small enough to be analytically tractable yet large enough to be interesting in applications and to show surprising phenomena

◮ assume that f (λ) =

∞

0 (1 − e−λt)m(t)dt with

m(t) ∼ t−1−α p0 + p1t + p2t2 + · · ·

as t → 0 and α ∈ (0, 1), also m of rapid decay as t → ∞

◮ includes relativistic stable Lévy process with f (λ) =

√ 1 + λ − 1

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Key result

We obtain the heat trace asymptotics as t → 0

Theorem

Let −A be the generator of the process BXt. Recall f (λ) = ∞

0 (1 − e−λt)m(t)dt with m(t) ∼ t−1−α

p0 + p1t + p2t2 + · · ·

as t → 0.

Set m(0, ∞) = ∞

0 m(t) − p0t−1−αdt.

(i) α rational: there are constants ck and ˜ cl such that TR

e−tA

∼ e−m(0,∞)t ∞

k=0

ckt−(n−k)/2α −

∞

k=0

˜ ckt−(n−k)/2α log t

(ii) α irrational: TR
e−tA

∼ e−m(0,∞)t ∞

k=0 ckt−(n−k)/2α

Strikingly different behaviour depending on α with the appearance of logarithmic terms

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Key result

One can compute any term explicitly and recover probabilistic information

Recall Lévy’s arcsine law (simplified): Let Xt a subordinator and define the first passage time T(x) = inf{t ≥ 0|Xt > x}. Then TFAE

1 the random variables 1

x XT(x)− converge in distribution to an arcsine

distribution with parameter α ∈ (0, 1) as x → 0

2 α = limx→0 x−1❊(XT(x)−)

In dimension n > 2, the lowest order term of em(0,∞)tTR

e−At

is Ω2

n

n(2π)n 1 2αΓ n 2α

(−p0Γ(−α))−n/2α t−n/2α,

where Ωn = volume of the unit sphere in Rn p0 = − 1

t lim λ→∞ λ−α log ❊

e−λXt

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Derivation of the heat trace expansion

The idea is to use a global calculus of pseudodifferential

perators

1 Bt a Brownian motion and Xt a subordinator with suitable Laplace

exponent f

2 The generator −A of the associated semigroup and the heat operator

e−At itself are classical pseudodifferential operators on Rn

3 The regularized zeta function ζ(z) = TR(A−z) can be

meromorphically continued to C with at most simple poles

4 The heat trace TR(e−At) has an asymptotic expansion given by the

pole structure of Γ(z)ζ(z) The key technical aspect is the use of the regularized trace functional TR

n classical pseudos in Rn to allow the definition of ζ(z) and the heat
trace. It was rigorously defined in Maniccia, Schrohe & Seiler (2014)

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Derivation of the heat trace expansion

1. A class of Bernstein functions

We assume that –with respect to Lebesgue measure– the Lévy measure has a density with certain small-time behaviour

Hypothesis

Let f (λ) = ∞

1 − e−λt

m(t)dt be a Bernstein function with locally integrable density m : (0, ∞) → R such that (i) it has an asymptotic expansion m(t) ∼ ∞

k=0 pkt−1−α+k as t → 0+

with α ∈ (0, 1); (ii) m is of rapid decay at ∞, i.e. m(t)tβ is bounded a.e. for t > 1 for all β ∈ R; and (iii) m(0, ∞) = ∞

0 m(t) − p0t−1−αdt < 0.

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Derivation of the heat trace expansion

1. This class is nonempty and contains interesting examples

Bernstein function f Lévy density m (λ + 1)α − 1

α Γ(1−α)e−tt−α−1

λ/(λ + a)α

sin(απ)Γ(1−α) π

e−attα−2(at + 1 − α) λ

1 − e−2

√ λ+a

/ √ λ + a

e−1/t−at(1+t(e1/t−1)(1+2at) 2√πt5/2

Γ λ+a

2a

/Γ(λ/2a)

a3/2e2at 2√π(e2at−1)3/2

Γ(αλ + 1)/Γ(αλ + 1 − α)

e−t/α Γ(1−α)(1−e−t/α)1+α

in each case a > 0 and 0 < α < 1

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Derivation of the heat trace expansion

2. Such subordinators lead to classical pseudos

Theorem

Let ˜ A = A + m(0, ∞)I. Set αk = −pkΓ(−α + k). (i) The operator ˜ A is a classical selfadjoint elliptic pseudo of order 2α. Its symbol has the asymptotic expansion σ

˜

A

∼

∞

k=−1

αk|ξ|−2(−α+k). (ii) The heat operator e−t ˜

A has symbol expansion

σ

et ˜

A

(ξ) ∼ e−tα0|ξ|2α −

α1|ξ|2α−2 + α2|ξ|2α−4

te−tα0|ξ|2α ± · · · . Proof: The idea is that local properties of m translate into global properties

f σ( ˜

A) by the Mellin transform (trick from number theory & QFT).

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Derivation of the heat trace expansion

3. The regularized zeta function generalizes the Riemann

zeta function

Theorem

The function ζ(z) = TR

˜

A−z is meromorphic on C with at most simple poles at the points zk = (n − k)/2α for k = 0, 1, 2, . . .. The point zn = 0 is a removable singularity. In the lowest orders, this residue becomes resz=z0ζ(z) = 1 2α nΩ2

n

(2π)n α−n/2α resz=z2ζ(z) = − 1 2α nΩ2

n

(2π)n α−z2−1 α1z2 resz=z4ζ(z) = 1 2α nΩ2

n

(2π)n

−α−z4−1

α2z4 + 1

2α−z4−2

α2

1z4(z4 + 1)

,

where Ωn = 2πn/2

Γ(n/2) is the volume of the unit sphere in Rn.

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Derivation of the heat trace expansion

4. The heat trace asymptotics as t → 0+ follow from the

zeta function

Theorem

(i) α rational: there are constants ck and ˜ cl depending on the residues of Γ(z)ζ(z) at the points (n − k)/2α for k = 0, 1, 2, . . . such that TR

e−t ˜

A

∼

∞

k=0

ckt−(n−k)/2α −

∞

k=0

˜ ckt−(n−k)/2α log t. The logarithmic terms correspond to double poles of Γ(z)ζ(z). (ii) α irrational: TR

e−t ˜

A

∼ ∞

k=0 ckt−(n−k)/2α

In the lowest orders (dimension n>2), the expansion becomes

nΩ2

n

2α(2π)n

Γ( n

2α)α − n 2α

t− n

2α − Γ( n−2 2α + 1)α − n−2 2α −1

α1 n−2

2α t− n−2 2α + · · ·

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Selected open questions

1 Is there a probabilistic characterization of this class of Lévy measures

(and hence Bernstein functions)?

2 What is the probabilistic significance of the logarithmic terms in the

heat trace?

3 What is the probabilistic significance of the dichotomy

rational/irrational α in the heat trace expansion?

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Bibliography

Applebaum, D. (2011), ‘Infinitely divisible central probability measures on compact Lie groups – regularity, semigroups and transition kernels’,

Ann. Prob. 39(6), 2474–2496.

Bañuelos, R. & F. Baudoin (2012), ‘Trace estimates for subordinate semigroups’, Tbd X(X), X–X. Bañuelos, R., J.B. Mijena & E. Nane (2014), ‘Two-term trace estimates for relativistic stable processes’, J. Math. Anal. Appl. 410(2), 837–846. Maniccia, L., E. Schrohe & J. Seiler (2014), ‘Determinants of classical SG-pseudodifferential operators’, Math. Nachr. 287(7), 782–802.

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Spectral functions of subordinated Brownian motion

M.A. Fahrenwaldt12

Leibniz Universität Hannover, Germany

Berlin, 23 October 2014

We explore a correspondence between stochastic processes and analytical objects — simplified idea

Stochastics Analysis Process on Rn Heat kernel BXt e−At  



x ❊(XT(x)−),

where Tx = inf{t ≥ 0|Xt > x}

− − − − → TR(e−At) ∼ ect

∞

ck(α)t(n−k)/2α +ect

∞

˜ ck(α)t(n−k)/2α log t

Contents

1

Motivation

2

Key result

3

Derivation of the heat trace expansion

4

Selected open questions

5

Bibliography

The heat kernel is of significant intrinsic interest in mathematics

small time asymptotics of the trace of the heat kernel encodes important information about the topology of a manifold M Tr(e−∆t) ∼ (4πt)n/2     a0

+ a1

6

t + · · ·     , the heat kernel is used for ground state calculations in quantum field theory, it is the transition density of a stochastic process on the manifold and as such significant in stochastics and its applications

Similar investigations have centred on processes living on compact manifolds

Investigations on Rn typically consider estimates of Tr

− Tr

√ −∆ on Tn, SU(2), SO(3)

Bañuelos, Mijena & Nane (2014):

with probabilistic interpretation

Contents

1

Motivation

2

Key result

3

Derivation of the heat trace expansion

4

Selected open questions

5

Bibliography

We consider subordinate Brownian motion on Rn for a wide class of subordinators – fully tractable yet exciting

Bt a canonical Brownian motion on Rn and Xt a subordinator: increasing Lévy process with values in [0, ∞) and X0 = 0 a.s. Distribution of Xt in terms of Bernstein function f (Laplace exponent)

= e−tf (|ξ|2) for ξ ∈ Rn

= e−tf (λ) for λ > 0

Our class of Laplace exponents is small enough to be analytically tractable yet large enough to be interesting in applications and to show surprising phenomena

∞

m(t) ∼ t−1−α p0 + p1t + p2t2 + · · ·

√ 1 + λ − 1

We obtain the heat trace asymptotics as t → 0

Theorem

Let −A be the generator of the process BXt. Recall f (λ) = ∞

0 (1 − e−λt)m(t)dt with m(t) ∼ t−1−α

p0 + p1t + p2t2 + · · ·

Set m(0, ∞) = ∞

0 m(t) − p0t−1−αdt.

(i) α rational: there are constants ck and ˜ cl such that TR

∼ e−m(0,∞)t ∞

ckt−(n−k)/2α −

∞

˜ ckt−(n−k)/2α log t

∼ e−m(0,∞)t ∞

k=0 ckt−(n−k)/2α

Strikingly different behaviour depending on α with the appearance of logarithmic terms

One can compute any term explicitly and recover probabilistic information

Recall Lévy’s arcsine law (simplified): Let Xt a subordinator and define the first passage time T(x) = inf{t ≥ 0|Xt > x}. Then TFAE

x XT(x)− converge in distribution to an arcsine

distribution with parameter α ∈ (0, 1) as x → 0

In dimension n > 2, the lowest order term of em(0,∞)tTR

is Ω2

n

n(2π)n 1 2αΓ n 2α

where Ωn = volume of the unit sphere in Rn p0 = − 1

t lim λ→∞ λ−α log ❊