Spectral functions of subordinated Brownian motion

Transcription

1 Spectral functions of subordinated Brownian motion M.A. Fahrenwaldt 12 1 Institut für Mathematische Stochastik Leibniz Universität Hannover, Germany 2 EBZ Business School, Bochum, Germany Berlin, 23 October / 20

2 We explore a correspondence between stochastic processes and analytical objects simplified idea Stochastics Process on R n Analysis Heat kernel B Xt α = lim x 0 1 x E(X T (x) ), where T x = inf{t 0 X t > x} }{{} Lévy s arcsine law e At e ct c k (α)t (n k)/2α TR(e At ) k=0 +e ct c k (α)t (n k)/2α log t k=0 } {{ } heat trace asymptotics 2 / 20

3 Motivation Contents 1 Motivation 2 Key result 3 Derivation of the heat trace expansion 4 Selected open questions 5 Bibliography 3 / 20

4 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 a 0 }{{} vol(m) + a }{{} 1 t +, κ(m)vol(m) 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 / 20

5 Motivation Similar investigations have centred on processes living on compact manifolds Investigations on R n typically consider estimates of Tr ( e th) Tr ( e th ) 0 where H and H 0 are of Schrödinger type with or without potential, for example H = α/2 + V and H 0 = 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 T n, 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 R n almost closed form expression for the first two terms in the heat trace with probabilistic interpretation 5 / 20

6 Key result Contents 1 Motivation 2 Key result 3 Derivation of the heat trace expansion 4 Selected open questions 5 Bibliography 6 / 20

7 Key result We consider subordinate Brownian motion on R n for a wide class of subordinators fully tractable yet exciting B t a canonical Brownian motion on R n and X t a subordinator: increasing Lévy process with values in [0, ) and X 0 = 0 a.s. Distribution of X t in terms of Bernstein function f (Laplace exponent) characteristic function E ( e iξ B X t ) = e tf ( ξ 2) for ξ R n ( ) generating function E e λx t = 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 ( p 0 + p 1 t + p 2 t 2 + ) 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 7 / 20

8 Key result We obtain the heat trace asymptotics as t 0 Theorem Let A be the generator of the process B Xt. Recall f (λ) = 0 (1 e λt )m(t)dt with m(t) t 1 α ( p 0 + p 1 t + p 2 t 2 + ) as t 0. Set m(0, ) = 0 m(t) p 0 t 1 α dt. (i) α rational: there are constants c k and c l such that [ ] TR (e ta) e m(0, )t c k t (n k)/2α c k t (n k)/2α log t k=0 k=0 (ii) α irrational: TR ( e ta) e m(0, )t [ k=0 c kt (n k)/2α] Strikingly different behaviour depending on α with the appearance of logarithmic terms 8 / 20

9 Key result One can compute any term explicitly and recover probabilistic information Recall Lévy s arcsine law (simplified): Let X t a subordinator and define the first passage time T (x) = inf{t 0 X t > x}. Then TFAE 1 the random variables 1 x X T (x) converge in distribution to an arcsine distribution with parameter α (0, 1) as x 0 2 α = lim x 0 x 1 E(X T (x) ) In dimension n > 2, the lowest order term of e m(0, )t TR ( e At) is where Ω 2 n 1 ( n ) n(2π) n 2α Γ ( p 0 Γ( α)) n/2α t n/2α, 2α Ω n = volume of the unit sphere in R n p 0 = 1 t lim λ λ α log E ( ) e λxt 9 / 20

10 Derivation of the heat trace expansion Contents 1 Motivation 2 Key result 3 Derivation of the heat trace expansion 4 Selected open questions 5 Bibliography 10 / 20

11 Derivation of the heat trace expansion The idea is to use a global calculus of pseudodifferential operators 1 B t a Brownian motion and X t 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 R n 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 on classical pseudos in R n to allow the definition of ζ(z) and the heat trace. It was rigorously defined in Maniccia, Schrohe & Seiler (2014) 11 / 20

12 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) = ( 0 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 p kt 1 α+k as t 0 + with α (0, 1); (ii) m is of rapid decay at, i.e. m(t)t β is bounded a.e. for t > 1 for all β R; and (iii) m(0, ) = 0 m(t) p 0 t 1 α dt < / 20

13 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 t t α 1 sin(απ)γ(1 α) λ/(λ + a) α π e at t α 2 (at + 1 α) ( ) λ 1 e 2 λ+a / e λ + a 1/t at (1+t(e 1/t 1)(1+2at) 2 πt 5/2 Γ ( ) λ+a 2a /Γ(λ/2a) a 3/2 e 2at Γ(αλ + 1)/Γ(αλ + 1 α) in each case a > 0 and 0 < α < 1 2 π(e 2at 1) 3/2 e t/α Γ(1 α)(1 e t/α ) 1+α 13 / 20

14 Derivation of the heat trace expansion 2. Such subordinators lead to classical pseudos Theorem Let à = A + m(0, )I. Set α k = p k Γ( α + k). (i) The operator à is a classical selfadjoint elliptic pseudo of order 2α. Its symbol has the asymptotic expansion ) σ (à k= 1 α k ξ 2( α+k). (ii) The heat operator e tã has symbol expansion ( σ e tã) (ξ) e tα 0 ξ 2α [ α 1 ξ 2α 2 + α 2 ξ 2α 4] te tα 0 ξ 2α ±. Proof: The idea is that local properties of m translate into global properties of σ(ã) by the Mellin transform (trick from number theory & QFT). 14 / 20

15 Derivation of the heat trace expansion 3. The regularized zeta function generalizes the Riemann zeta function Theorem (Ã z ) The function ζ(z) = TR is meromorphic on C with at most simple poles at the points z k = (n k)/2α for k = 0, 1, 2,.... The point z n = 0 is a removable singularity. In the lowest orders, this residue becomes res z=z0 ζ(z) = 1 2α nω 2 n (2π) n α n/2α 0 res z=z2 ζ(z) = 1 2α res z=z4 ζ(z) = 1 nω 2 ( n 2α (2π) n nω 2 n (2π) n α z α 1 z 2 ) 0 α 2 z α z α1z 2 4 (z 4 + 1), α z 4 1 where Ω n = 2πn/2 Γ(n/2) is the volume of the unit sphere in Rn. 15 / 20

16 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 c k and c l depending on the residues of Γ(z)ζ(z) at the points (n k)/2α for k = 0, 1, 2,... such that TR ( e tã) c k t (n k)/2α c k t (n k)/2α log t. k=0 k=0 The logarithmic terms correspond to double poles of Γ(z)ζ(z). ( ) (ii) α irrational: TR e tã k=0 c kt (n k)/2α In the lowest orders (dimension n>2), the expansion becomes [ nω 2 n 2α(2π) Γ( n n n 2α )α 2α 0 t n 2α Γ( n 2 n 2 ] 2α + 1)α 2α 1 n 2 n 2 0 α 1 2α t 2α + 16 / 20

17 Selected open questions Contents 1 Motivation 2 Key result 3 Derivation of the heat trace expansion 4 Selected open questions 5 Bibliography 17 / 20

18 Selected open questions 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? 18 / 20

19 Bibliography Contents 1 Motivation 2 Key result 3 Derivation of the heat trace expansion 4 Selected open questions 5 Bibliography 19 / 20

20 Bibliography Bibliography Applebaum, D. (2011), Infinitely divisible central probability measures on compact Lie groups regularity, semigroups and transition kernels, Ann. Prob. 39(6), 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), Maniccia, L., E. Schrohe & J. Seiler (2014), Determinants of classical SG-pseudodifferential operators, Math. Nachr. 287(7), / 20