Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 1 / 32

Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space David Applebaum School of Mathematics and Statistics, University of Sheffield, UK Talk at "Workshop on Infinite Dimensional Probability", Kings College London. 23rd February 2017 Talk based on work of Christian Fonseca Mora (Costa Rica) Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space, (Jan 2017) arxiv:1701.06630 Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 1 / 32

Outline of Talk Review Lévy processes on Euclidean, Hilbert and Banach spaces. Define Lévy processes in the dual of a nuclear space. Establish the Lévy-Itô decomposition. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 2 / 32

Outline of Talk Review Lévy processes on Euclidean, Hilbert and Banach spaces. Define Lévy processes in the dual of a nuclear space. Establish the Lévy-Itô decomposition. Describe the correspondence between Lévy processes and infinitely divisible measures on the dual of a nuclear space. Derive the Lévy-Khintchine formula for infinitely divisible measures on duals of nuclear spaces. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 2 / 32

Lévy Processes in Banach Spaces Review Let (Ω, F, P) be a probability space, and (F t, t 0) be a filtration (usual hypotheses assumed). Let E be a real Banach space, and X = (X(t), t 0) be an adapted process where each X(t) : Ω E. We say if it is a Lévy process if (Independent increments) X(t) X(s) independent of F s for all t s. (Stationary increments) X(t) X(s) d = X(t s) X(0) for all t s. X(0) = 0 (a.s.) (Stochastic continuity) lim t 0 P( X(t) a) = 0 for all a E, a 0 X has a.s. càdlàg paths. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 3 / 32

We want to have a Lévy-Itô decomposition which gives a pathwise decomposition: drift + diffusion + small jumps + large jumps. Lévy Khintchine formula: explicit form of Fourier transform of law/ characteristic function. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 4 / 32

The Case E = R d Lévy-Itô decomposition X(t) = bt + σb(t) + x 1 xñ(t, dx) + x >1 xn(t, dx). b R, σ is (d m) matrix, B(t) = (B 1 (t),..., B m (t)) is BM(R m ). Poisson random measure: N(t, A) = #{0 s t; X(s) A}, where jump X(s) = X(s) X(s ). E(N(t, A)) = tν(a), Ñ(t, A) = N(t, A) tν(a). ν is a Lévy measure: ν({0}) = 0, R d (1 x 2 )ν(dx) <. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 5 / 32

Lévy Khintchine formula E(e i u,x(t) ) = e tη(u), η(u) = i b, u 1 Qu, u 2 + (e i u,y 1 i u, y 1 y 1 )ν(dy), R d where Q = σσ T. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 6 / 32

The Case E = H, real separable Hilbert space What changes from R d?, is now Hilbert space inner product. Only other change is σb(t) must be replaced by Q-Brownian motion B Q (t), E( B Q (s), φ B Q (t), ψ ) = s t Qφ, ψ. covariance operator Q is positive, self-adjoint trace class operator. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 7 / 32

Then there exists complete orthonormal basis of eigenfunctions (e n, n N) so that Qe n = λ n e n, B Q (t) = λ n β n (t)e n, n=1 where β n are i.i.d. BM(R). (c.f. Karhunen Loève expansion). Cannot take λ n = 1 for all n, as I not trace class when dim(h) =. This leads to cylindrical Brownian motion. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 8 / 32

The Case E is real separable Banach space What changes from R d and H?, is now dual pairing between E and dual space E. Fourier transform µ of a probability measure µ on E is a mapping E C, so for a E, µ(a) = e i y,a µ(dy). Covariance operator Q is a positive and symmetric linear operator from E to E, so for all φ, ψ E, E Qφ, ψ = Qψ, φ and Qφ, φ 0. Lévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 9 / 32

ν is a symmetric Lévy measure if it is symmetric and satisfies (i) ν({0}) = 0, (ii) The mapping { } a exp [cos( x, a ) 1]ν(dx) E is the characteristic function of a finite measure. ν M(E) is a Lévy measure if ν + ν is a symmetric Lévy measure, where ν(a) = ν( A). For Lévy-Itô, see Riedle and van Gaans, Stoch. Proc. App 119 (2009) Banach and Hilbert spaces enable us to construct processes taking values in function spaces. But what about spaces of distributions? Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 10 / 32

Nuclear Spaces and their Strong Duals A locally convex space Φ is a nuclear space if its topology is generated by a family Π of semi-norms such that: Every p Π is Hilbertian: the completion Φ p of the space (Φ/ker(p), p) where p(φ + ker(p)) = p(φ), is a Hilbert space. For every p Π, q Π, p Cq, for some C > 0, such that i p,q : Φ q Φ p is Hilbert-Schmidt. We denote by Φ p the dual of Φ p and by p its norm. Φ is the usual topological dual of Φ, comprising all continuous linear functions from Φ to R. We will denote the strong dual of Φ by Φ β and φ Φ, f Φ β, f [φ] denotes duality relation. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 11 / 32

The (locally convex, Hausdorff) topology in Φ β is generated by the seminorms {h B, B bounded in E}, where for all f Φ β h B (f ) = sup f [x]. x B Recall that B is bounded in E if given any neighbourhood U of zero, there exists α > 0 so that B αu. This is called the strong topology. We obtain the weak topology, if we replace bounded sets with finite sets, in the above definition. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 12 / 32

Examples of Nuclear Spaces R n and C n. Space of test functions D := C c (R n ) and the Schwartz space of rapidly decreasing functions S(R n ). Their duals are the spaces of distributions D and tempered distributions S (R n ). Space C (M) of smooth functions on a compact manifold M. Space of harmonic functions H(U) on open U R n. Space of analytic functions A(Ω) on open Ω C. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 13 / 32

Lévy processes A Φ β -valued process L = {L t} t 0 is a Lévy process if is starts at zero (a.s.), and has stationary and independent increments as above. The only change is to the definition of stochastic continuity. Here we require: The distribution µ t of L t is a Radon measure, and the mapping t µ t from [0, ) into the space of Radon probability measures on Φ β is continuous at 0 in the weak topology. We don t assume càdlàg paths, but derive the existence of such a version later on. Recall a Borel measure ρ is Radon if for every Borel set B and every ɛ > 0, there exists a compact C so that ρ(b \ C) < ɛ. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 14 / 32

Motivations to study Lévy processes in duals of nuclear spaces Study of SPDEs with Lévy noise: Let Φ, Ψ nuclear spaces. We want to consider stochastic evolution equations of the form: where, dx(t) = (AX(t) + B(t, X(t)))dt + F(t, X(t))dL(t), t [0, T ]. A is the generator of a C 0 -semigroup {S(t)} t 0 on Ψ, B : [0, T ] Ψ Ψ, F : [0, T ] Ψ L(Φ, Ψ ), L = {L(t)} t 0 is a Φ β-valued Lévy process. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 15 / 32

Applications: Modelling dynamics of nerve signal (neurology) ( Kallianpur and Wolper, 1984). Modelling environmental pollution (Kallianpur and Xiong, 1995). Statistical filtering (Üstünel, 1983-4). Chemical kinetics and interacting particle systems ( Gorostiza and Nualart, 1994). SPDE limits of many-server queues (Kaspi and Ramanan, 2013) Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 16 / 32

The Lévy-Itô decomposition Theorem Let L = {L t } t 0 be a Φ β-valued càdlàg Lévy process. Then, for each t 0 it has the following representation L t = tm + W t + f B Ñ(t, df ) + f N(t, df ) ρ (1) c B ρ (1) All the random components of the decomposition are independent. Our result improves previous work by Üstünel [Ann. Prob, 12, No.3, 858-868 (1984)] in two directions: We have a more detailed description of the components (useful for stochastic analysis). Our decomposition works on a larger class of spaces (general nuclear spaces). Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 17 / 32

The Lévy-Itô decomposition Consider a càdlàg Lévy process L = {L t } t 0. Define by L t := L t L t the jump of the process L at the time t ) 0. It is a stationary Poisson point processes on (Φ β \ {0}, B(Φ β \ {0}). For A B(Φ β \ {0}) and t 0 define { } N(t, A)(ω) = # 0 s t = d L s (ω) A = 1 A ( L s (ω)), 0 s t the Poisson random measure associated to L with respect to the ring A of sets bounded below, i.e. A A if 0 / A. Let ν satisfying ν({0}) = 0 and E (N(t, A)) = tν(a), t 0, A B ( Φ β \ {0}). Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 18 / 32

Lévy measure The Borel measure ν on Φ β is a Lévy measure in the following sense: ν({0}) = 0, for each neighborhood of zero U Φ β, ν U c is a bounded Radon measure on Φ β, there exists a continuous Hilbertian semi-norm ρ on Φ such that ρ (f ) 2 ν(df ) <, B ρ (1) and ν Bρ (1) c is a bounded Radon measure on Φ β, where B ρ (1) := {f Φ : ρ (f ) 1} is the unit ball of the Hilbert space (Φ ρ, ρ ) dual to Φ ρ. It is a bounded, closed subset of Φ β. Lévy measures on Φ β are σ-finite Radon measures. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 19 / 32

Discontinuous part: small jumps From B ρ (1) ρ (f ) 2 ν(df ) < we can prove: Theorem There exists a continuous Hilbertian semi-norm q on Φ, ρ q, such that i ρ,q is Hilbert-Schmidt and a Φ q-valued zero-mean, square { } integrable, càdlàg Lévy process Bρ (1) f Ñ(t, df ) : t 0 with ( E q B ρ (1) ) 2 f Ñ(t, df ) = q (f ) 2 ν(df ), t 0. B ρ (1) and characteristic function given φ Φ by ( E e i B ρ (1) f Ñ(t,df )[φ]) { } ( ) = exp t e if [φ] 1 if [φ] ν(df ). B ρ (1) Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 20 / 32

Discontinuous part: large jumps Because ν Bρ is a bounded Radon measure on Φ (1) c β we have: Theorem { } The Φ β -valued process B f N(t, df ) : t 0 defined by ρ (1) c B ρ (1) c f N(t, df )(ω)[φ] = 0 s t L s (ω)[φ]1 Bρ (1) c ( L s(ω)), ω Ω, φ Φ, is a Φ β-valued càdlàg Lévy process. Moreover, φ Φ, ( { }) { } ( ) E exp i f N(t, df )[φ] = exp t e if [φ] 1 ν(df ). B ρ (1) c B ρ (1) c Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 21 / 32

Linear deterministic part Now, define the process Y = {Y t } t 0 by Y t = L t fn(t, df ), t 0. (1.1) B ρ (1) c For every φ Φ, {Y t [φ]} t 0 has moments of all orders. There exists m Φ β such that E (Y t [φ]) = tm[φ]. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 22 / 32

Continuous part Theorem The process W = {W t } t 0 defined t 0 by W t = L t tm f B Ñ(t, df ) f N(t, df ), ρ (1) c B ρ (1) is a Φ η-valued Wiener process with mean-zero and covariance E (W t [φ]w s [ϕ]) = (t s)q(φ, ϕ), φ, ϕ Φ, s, t 0, where Q and η are continuous Hilbertian semi-norms on Φ such that Q K η (for some K > 0) and the map i Q,η is Hilbert-Schmidt. The characteristic function of W is given t 0, φ Φ by ( ) E e iw t [φ] = exp ( t2 ) Q(φ)2. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 23 / 32

Theorem (Lévy-Khintchine theorem for Lévy processes) If L = {L t } t 0 is a Φ β -valued, càdlàg Lévy process, there exist m Φ β, a continuous Hilbertian semi-norm Q on Φ, a Lévy measure ν on Φ β with continuous Hilbertian semi-norm ρ on Φ; such that t 0, φ Φ, (e ) il t [φ] E ( = exp itm[φ] t 2 Q(φ)2 + t Φ β ) ( ) e if [φ] 1 if [φ]1 Bρ (1) (f ) ν(df ). Conversely, let m Φ β, Q be a continuous Hilbertian semi-norm on Φ, and ν be a Lévy measure on Φ β with continuous Hilbertian semi-norm ρ on Φ. There exists a Φ β -valued càdlàg Lévy process L = {L t} t 0 defined on some probability space (Ω, F, P), unique up to equivalence in distribution, whose characteristic function is given as above. In particular, ν is the Lévy measure of L. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 24 / 32

Infinitely Divisible Measures, Convolution Semigroups and cylindrical Lévy A Radon probability measure µ on Φ β is called infinitely divisible if n N a Radon probability measure µ n on Φ β such that µ = µ n µ n (n-times). A family {µ t } t 0 of Radon probability measures on Φ β is a convolution semigroup if: s, t 0, µ s µ t = µ s+t, and µ 0 = δ 0 It is continuous if the mapping t µ t from [0, ) into the space of Radon probability mesures on Φ β is continuous in the weak topology. A family of linear maps L t : Φ L 0 (Ω, F, P), t 0 is a cylindrical Lévy process if for every n N, φ 1,..., φ n Φ, the R n -valued process {(L t (φ 1 ),..., L t (φ n ))} t 0 is a Lévy process. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 25 / 32

Correspondence Lévy processes and Infinitely Divisible Measures A barrelled space is a l.c.s Ψ for which every closed linear map from Ψ into a Banach space is continuous. Theorem If L = {L t } t 0 is a Lévy process in Φ β, the family of probability distributions {µ Lt } t 0 of L is a continuous convolution semigroup on Φ β. Moreover, each µ L t is infinitely divisible for every t 0. Furthermore, if Φ is also a barrelled space, then for each T > 0 the family {µ Lt : t [0, T ]} is uniformly tight. The first part of the theorem is a consequence of the definition of Lévy process. The second part is a consequence of a result due to Siebert (1974) for convolution semigroups. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 26 / 32

Correspondence Lévy processes and Infinitely Divisible Measures Theorem Let Φ be a barrelled nuclear space. If µ is an infinitely divisible measure on Φ β, there exist a Φ β-valued, càdlàg Lévy process } L = { Lt : t 0 such that the probability distribution of L µ L1 1 is µ. The above result is new in our general context. The proof follows in 5 steps: Step 1 If µ is infinitely divisible on Φ β, there exist a continuous convolution semigroup {µ t } t 0 on Φ β such that µ 1 = µ. Also, for every T > 0 the family {µ t : t [0, T ]} is uniformly tight (Siebert 1974). Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 27 / 32

Step 2 {µ t } t 0 defines a cylindrical convolution semigroup : n N, φ 1,..., φ n Φ, the family {µ t π 1 φ 1,φ 2,...,φ n } t 0 is a continuous convolution semigroup of probability measures on R n, where π φ1,...,φ n : Φ R n is the linear and continuous map given by Step 3 Theorem π φ1,...,φ n (f ) = (f [φ 1 ],..., f [φ n ]), f Φ. There exists a cylindrical Lévy process L = {L t } t 0 in Φ defined on a probability space (Ω, F, P), such that t 0, φ 1,..., φ n Φ and Γ B(R n ), P ((L t (φ 1 ), L t (φ 2 ),..., L t (φ n )) Γ) = µ t π 1 φ 1,φ 2,...,φ n (Γ). Moreover, for every T > 0 the family of linear maps L t : Φ L 0 (Ω, F, P) is equicontinuous at zero. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 28 / 32

Step 4 Theorem Let L = {L t } t 0 be a cylindrical Lévy process in Φ such that for every T > 0, the family {L t : t [0, T ]} of linear maps from Φ into L 0 (Ω, F, P) is equicontinuous. Then, there exists a Φ β-valued càdlàg Lévy process L = { L t } t 0, such that for every φ Φ, L[φ] = { L t [φ]} t 0 is a version of L(φ) = {L t (φ)} t 0. Moreover, L is unique up to indistinguishable versions. A key step for the proof is a result called the regularization theorem (Fonseca-Mora, to appear J.Theor. Prob. 2017). Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 29 / 32

Step 5 For every t 0, φ 1,..., φ n Φ and Γ B(R n ), ) π 1 µ Lt φ 1,φ 2,...,φ n (Γ) = P ( Lt π 1 φ 1,...,φ n (Γ) = P ((L t (φ 1 ), L t (φ 2 ),..., L t (φ n )) Γ) = µ t π 1 φ 1,φ 2,...,φ n (Γ). Therefore, µ Lt = µ t. Now, as µ 1 = µ, we then have that µ L1 = µ. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 30 / 32

Theorem (Lévy-Khintchine theorem) Let µ be a Radon probability measure on Φ β. Then: If Φ is also a barrelled space and if µ is infinitely divisible, then there exists m Φ β, a continuous Hilbertian semi-norm Q on Φ, a Lévy measure ν on Φ β with continuous Hilbertian semi-norm ρ on Φ; such that the characteristic function of µ satisfies for every φ Φ: [ µ(φ) = exp im[φ] 1 ] ( ) 2 Q(φ)2 + e if [φ] 1 if [φ]1 Bρ (1) (f ) ν(df ). Φ β Conversely, let m Φ β, Q be a continuous Hilbertian semi-norm on Φ, and ν be a Lévy measure on Φ β with continuous Hilbertian semi-norm ρ on Φ. If µ has characteristic function given as above, then µ is infinitely divisible. Our result improves previous work by Dettweiler, [Zeit. Wahrsch. Verw. Gebiete, 34, 285 311 (1976)]. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 31 / 32

Thank you for listening. Lévy Processes and Infinitely Divisible Measures in the Dual offebruary a Nuclear 2017 Space 32 / 32