A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium
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1 1/ 22 A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium I. Gentil CEREMADE, Université Paris-Dauphine International Conference on stochastic Analysis and Applications Hammamet, Tunisia, November 5-10, 2007 Work written in collaboration with C. Imbert from Paris-Dauphine
2 Outline 2/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
3 Outline 3/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
4 Ornstein-Uhlenbeck and Fokker-Planck equations 4/ 22 { dxt = 2dB t V(X t )dt X 0 = x where B t is a standard Brownian Motion in R n. Ito formula implies : the Semigroup P t f(x) = E x (f(x t )) satisfies the PDE { t P tf(x) = LP t f(x) P 0 f = f, where Lf = f V f is the IG of P t. This is the Ornstein-Uhlenbeck equation. Consider L or Pt, the dual with respect to dx, Lfgdx = fl gdx, or P t fgdx = fp t gdx, then L g = g + div(g. V).
5 5/ 22 The Semigroup Pt f(x) satisfies the PDE { t P t f(x) = L Pt f(x) P 0 f = f, This is the Fokker-Planck equation. Let µ V = e V dx (assume that µ V is a probability measure), (P t ) t 0 or L is self adjoint in L 2 (µ V ) and the by integration by parts Lf gdµ V = f gdµ V. Under smooth assumptions : or equivalently lim P tf(x) = t lim t ev(x) Pt g(x) = The good question is HOW FAST? fdµ V. gdx.
6 Outline 6/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
7 Tools for the asymptotic behaviour 7/ 22 Poincaré inequality : a L 2 convergence. d dt var µ V (P t f) = 2 P t flp t fdµ V 0 = 2 If Poincaré inequality holds var µv (f) C f 2 dµ V P t f 2 dµ V, var µv (P t f) e 2t/C var µv (f). Logarithmic Sobolev inequality a L log L convergence d dt Ent µ V (P t f) := d P t f P t f log dµ V = 4 P t f 2 dµ V, dt Pt fdµ V If Logarithmic Sobolev inequality holds Ent µv (f 2 ) C f 2 dµ V Ent µv (P t f) e 4t/C Ent µv (f).
8 8/ 22 When do we have a Poincaré or a logarithmic Sobolev inequality? * The Gaussian measure, V(x) = x 2 /2 (Inequality proved by Gross). * The Bakry-Emery Γ 2 -criterion implies that if Hess(V) λid, with λ > 0 then logarithmic Sobolev inequality holds with C = 2/λ and Poincaré inequality holds with C = 1/λ. * There are also many technical methods to prove Poincaré or Log-Sobolev : Hardy, transportation...
9 Outline 9/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
10 Definition of Lévy process Lévy process L t = process with stationary & indep increments Fourier transform (L t ) = e tψ(ξ) where ψ is characterized by the Lévy-Khinchine formula. ψ(ξ) = σξ ξ + ib ξ + (e iz ξ 1 iz ξ1 B (ξ))ν(dz) where ν is a singular measure satisfying z 2 ν(dz) < + B R d \B ν(dz) < +, σ is a positive definite matrix and b is a vector. Parameters (σ, b, ν) characterize the Lévy process (or a inifinite divisible law). For all t > 0 the law of L t is an infinite divisible law : µ = µ n µ }{{ n. } n times 10/ 22
11 11/ 22 Associated infinitesimal generators as for the Brownian Motion. I(u) = div (σ u) + b u + (u(x + z) u(x) u(x) z1 B (z))ν(dz) These operators appear everywhere (mathematical finance, mechanics, fluids etc. ) Laws with heavy tails (decrease as power laws) Example : (σ, b,ν) = (0, 0, 1 z α+d dz), the α (0, 2) stable process. In that case ψ(ξ) = ξ α. The case α = 2 is the Brownian motion, I =.
12 The Lévy-Fokker-Planck equation 12/ 22 Replace by I a IG of a Lévy process in the Fokker-Planck equation : { t u = I(u) + div(ux) (LFP) u(0, x) = f(x) The goal is to understand the asymptotic behaviour of the semigroup. Remark : We assume that V = x 2 /2. Questions : Find a steady state as e V as for the classical case. Find the asymptotic behaviour of the Lévy-Fokker-Planck equation (LFP). Find conditions to get an asymptotic behaviour using inequalities as Poincaré or logarithmic Sobolev.
13 Outline 13/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
14 14/ 22 An equilibrium u def = a stationary solution of the LFP u can be seen as an invariant measure µ V in the case of the Laplacian. Proposition (Existence of an equilibrium) Assume that R d \B ln z ν(dz) < +. There then exists an positive equilibrium u : I(u ) + div(u x) = 0. Moreover, u dx is an infinite divisible law whose characteristic exponent A is 1 A(ξ) = ψ(sξ) ds s. 0
15 15/ 22 The condition R d \B ln z ν(dz) < +. (Con1) is satisfied for the α-stable Lévy process. In that case u is the infinite divisible law of the Lévy process, A = ψ/λ. Proof : The Fourier transform û satisfies ψ(ξ)û + ξ û = 0 so that û = exp( A) with A such that : then A(ξ) ξ = ψ(ξ), A(ξ) = 1 0 ψ(sξ) ds s. Con1 prove that A is well defined and is the characteristic exponent of a Lévy process.
16 Outline 16/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
17 17/ 22 For φ : R + R convex and smooth and µ a probability measure, consider the φ-entropy ( ) Eµ φ (f) = φ(f)dµ φ fdµ Examples For φ(x) = 1 2 x 2 (Eµ φ =the variance), D φ (a, b) = 1 2 (a b)2 Fµ(v) φ = 1 (v(x + z) v(x)) 2 ν(dz)µ(dx) 2 For φ(x) = x ln x x 1 (E φ µ =entropy), D φ (a, b) = a ln a b + b a This is a natural interpolation between the variance and the Entropy.
18 Define also a Bregman distance D φ (a, b) = φ(a) φ(b) φ (b)(a b) 0 Theorem Let µ(dx) = u (x)dx, ν the Lévy measure associated to I and consider v(t, x) = u(t,x) u (x), then d dt Eφ µ (v(t, )) = ) D φ (v(x + z), v(x) ν(dz)µ(dx). Fisher information ) Fµ(v) φ = D φ (v(x + z), v(x) ν(dz)µ(dx). Can be seen as a Dirichlet form with respect to the measure u (x)dx 18/ 22
19 19/ 22 The proof is related to : A related equation : the Lévy-Ornstein-Ulenbeck equation (LOU) The function v = u/u satisfies t v = 1 ( ) I(u v) I(u )v u + x v def = Lv. Dual operator of L wrt µ = u (x)dx ( ) w 1 Lw 2 )dµ = (Ǐ(w 1 ) x w 1 w 2 dµ, where Ǐ is I with ˇν(dx) = ν( dx). Recall that in the classical case L is a self-adjoint operator with respect to µ.
20 Outline 20/ 22 1 Introduction Ornstein-Uhlenbeck and Fokker-Planck equations Tools for the asymptotic behaviour 2 The Lévy-Fokker-Planck equation The Lévy-Fokker-Planck equation 3 Results The equilibrium 4 Results Entropies Convergence towards the equilibrium
21 Convergence towards the equilibrium 21/ 22 Theorem We assume that ν I has a density N with respect to dx and satisfies ln z N(z) dz < +. R d \B If N is even and satisfies, z, + 1 N(sz)s d 1 ds CN(z) then for any smooth convex function Φ one gets : ( ) u(t) t 0, Ent Φ u u e t C Ent Φ u ( u0 u ).
22 22/ 22 Proof d dt E φ µ(v(t)) = F φ µ(t) it is enough to compare F φ µ with E φ µ. A functional inequality [Wu 00,Chafaï 04] If µ is an infinite divisible law (without gaussian part) φ satisfies φ > 0 and... Then E φ µ(f) 1 D φ (v(x + z), v(x))ν µ (dz)µ(dx) ν µ is the derivation associated to the probability measure µ. Example : If φ(x) = ln x x 1 and for the Gaussian measure this is exactly the Log-Sobolev inequality Generalization of Log-Sobolev inequality to the infinite divisible law.
23 Proof d dt E φ µ(v(t)) = F φ µ(t) it is enough to compare F φ µ with E φ µ. A functional inequality [Wu 00,Chafaï 04] If µ is an infinite divisible law (without gaussian part) φ satisfies φ > 0 and... Then If E φ µ(f) D φ (v(x + z), v(x))ν µ (dz)µ(dx) ν µ Cν I, This is true for fractional Laplacians and for Lévy process near α-stable process. 22/ 22
24 Proof d dt E φ µ(v(t)) = F φ µ(t) it is enough to compare F φ µ with E φ µ. A functional inequality [Wu 00,Chafaï 04] If µ is an infinite divisible law (without gaussian part) φ satisfies φ > 0 and... Then If E φ µ(f) C D φ (v(x + z), v(x))ν(dz)µ(dx) ν µ Cν I, This is true for fractional Laplacians and for Lévy process near α-stable process. 22/ 22
25 Proof d dt E φ µ(v(t)) = F φ µ(t) it is enough to compare F φ µ with E φ µ. A functional inequality [Wu 00,Chafaï 04] If µ is an infinite divisible law (without gaussian part) φ satisfies φ > 0 and... Then If E φ µ(f) CF φ µ(f) ν µ Cν I, This is true for fractional Laplacians and for Lévy process near α-stable process. 22/ 22
26 Proof d dt E φ µ(v(t)) = F φ µ(t) it is enough to compare F φ µ with E φ µ. A functional inequality [Wu 00,Chafaï 04] If µ is an infinite divisible law (without gaussian part) φ satisfies φ > 0 and... Then If E φ µ(f) CF φ µ(f) ν µ Cν I, then E φ µ(u/u ) E φ µ ( u0 u ) e t C This is true for fractional Laplacians and for Lévy process near α-stable process. 22/ 22
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