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

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 2014 1 / 20

We explore a correspondence between stochastic processes and analytical objects — simplified idea Stochastics Analysis Process on R n Heat kernel e − At B X t     � � ∞ α = lim x → 0 1 x ❊ ( X T ( x ) − ) , � c k ( α ) t ( n − k ) / 2 α e ct → TR ( e − At ) ∼ k = 0 − − − − where ∞ � c k ( α ) t ( n − k ) / 2 α log t + e ct ˜ T x = inf { t ≥ 0 | X t > x } k = 0 � �� � � �� � heat trace asymptotics Lévy’s arcsine law 2 / 20

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

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 ) ∼ ( 4 π t ) n / 2 + t + · · ·  ,  a 0 a 1   ���� ���� vol ( M ) 1 � κ ( M ) vol ( M ) 6 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 4 / 20

Motivation Similar investigations have centred on processes living on compact manifolds � e − tH � � e − tH 0 � Investigations on R n typically consider estimates of Tr − Tr 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

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

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) � e i ξ · B Xt � = e − tf ( | ξ | 2 ) for ξ ∈ R n ◮ characteristic function ❊ � e − λ X t � = e − tf ( λ ) for λ > 0 ◮ generating function ❊ 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 − λ t ) 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

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

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 ❊ ( X T ( x ) − ) � e − At � In dimension n > 2, the lowest order term of e m ( 0 , ∞ ) t TR is � n Ω 2 1 � ( − p 0 Γ( − α )) − n / 2 α t − n / 2 α , n 2 α Γ n ( 2 π ) n 2 α where Ω n = volume of the unit sphere in R n � e − λ X t � λ →∞ λ − α log ❊ p 0 = − 1 t lim 9 / 20

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

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

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 � ∞ � 1 − e − λ t � Let f ( λ ) = m ( t ) dt be a Bernstein function with locally 0 integrable density m : ( 0 , ∞ ) → R such that (i) it has an asymptotic expansion m ( t ) ∼ � ∞ k = 0 p k t − 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 � ∞ 0 m ( t ) − p 0 t − 1 − α dt < 0 . (iii) m ( 0 , ∞ ) = 12 / 20

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 e − at t α − 2 ( at + 1 − α ) π √ √ � λ + a � e − 1 / t − at ( 1 + t ( e 1 / t − 1 )( 1 + 2 at ) 1 − e − 2 λ / λ + a 2 √ π t 5 / 2 � λ + a � a 3 / 2 e 2 at Γ / Γ( λ/ 2 a ) 2 √ π ( e 2 at − 1 ) 3 / 2 2 a e − t /α Γ( αλ + 1 ) / Γ( αλ + 1 − α ) Γ( 1 − α )( 1 − e − t /α ) 1 + α in each case a > 0 and 0 < α < 1 13 / 20

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

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

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 ) / 2 α for k = 0 , 1 , 2 , . . . such that ∞ ∞ � A � � � e − t ˜ c k t − ( n − k ) / 2 α − c k t − ( n − k ) / 2 α log t . ∼ ˜ TR k = 0 k = 0 The logarithmic terms correspond to double poles of Γ( z ) ζ ( z ) . � A � e − t ˜ ∼ � ∞ k = 0 c k t − ( n − k ) / 2 α (ii) α irrational: TR In the lowest orders (dimension n>2), the expansion becomes � � − n − 2 − n t − n 2 α t − n − 2 n Ω 2 2 α − 1 Γ( n 2 α 2 α − Γ( n − 2 α 1 n − 2 2 α + · · · n 2 α ) α 2 α + 1 ) α 2 α ( 2 π ) n 0 0 16 / 20