Intertwining for Interacting Particle Systems in the Continuum I

Example: Symmetric inclusion process & Meixner polynomialsConsistency: labelled particles Markov dynamics for finitely many particles in E = Rd. [...] Symmetric inclusion process α(dx) finite measure on Rd Generator Lf (η) = f (η−δx + δy) − f (η) (α + η)(dy)η(dx) Particles jump. Join another particle or jump to new location. [...] Intertwining with Lenard’s K-transform K-transform of function f : Nf → R Kf δx1 + · · · + δxn = I⊂{1,...,n} f i∈I δxi . [...] Time evolution of k-point correlation functions ↔ time-evolution for k particles.Intertwining with orthogonal polynomials ρ probability measure on Nf. [...] Pn = closure in L2(Nf, ρ) of linear combinations of maps η → η(A1) · · · η(Ak), k ≤ n, A1, . . . , Ak ⊂ Rd. [...] Proof does not need explicit formulas or recurrence relations for polynomials.Orthogonal polynomials of particular interest when ρ is law of L´evy point process η E exp − f dη = exp k∈N e−kf (x) − 1 m(k)α(dx) . [...] Poisson law on N0 with parameter α. Eη0 Pn(ηt; fn) = Pn η0; p⊗n t fn . [...] Application to symmetric inclusion process Lf (η) = f (η−δx + δy) − f (η) (α + η)(dy)η(dx) Pascal point process (negative binomial point process) p ∈ (0, 1), α(dx) ▶ B1, . . . [...] For every p ∈ (0, 1), law ρp,α of negative binomial point process is reversible for the symmetric inclusion process.Negative binomial process and Ewens measure Expectation of functions of negative binomial process f dρ = 1 − p)α(E) f (0) + ∞ n=1 pn n! f (δx1 + · · · + δxn)λn(dx) Measures λ1 = α, λ2(A) = 1lA(x1, x2)α(dx1)α(dx2) + 1lA(x, x)α(dx) more generally, λn is a sum over set partition [...] , xn) Theorem (ηt)t≥0 symmetric inclusion process, p ∈ (0, 1) fixed, orthogonalization in L2(ρp,α) Eη0 Pn ηt; fn = Pn(η0; p(n) t fn) for ρp,α almost all η0.