Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Self-interacting diffusions Aline Kurtzmann HIM Bonn, Oxford University April 7, 2008 university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Outline Some generalities 1 Self-interacting diffusions on R d 2 Tools: dynamical systems 3 New tools: tightness and uniform estimates 4 General statements 5 university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems Study of some self-interacting diffusions New tools: tightness and uniform estimates General statements What is a self-interacting (or reinforced) diffusion? Solution to d X t = d B t − F ( t , X t , µ t ) d t � t µ t = 1 0 δ X s d s t university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems Study of some self-interacting diffusions New tools: tightness and uniform estimates General statements Brownian polymer Durrett and Rogers (1992) on R d : � t d X t = d B t + f ( X t − X s ) d s d t , 0 where f : R d → R d is measurable and bounded. Applications: physics, biology. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems Study of some self-interacting diffusions New tools: tightness and uniform estimates General statements Cases studied The problem is to find the normalization α ≥ 0 such that X t / t α converges a.s. Three cases have been studied yet: drift on the right in dimension 1 (Cranston & Mountford 1996), self-attracting: ( f ( x ) , x ) ≤ 0 (Cranston & Le Jan 1995, Raimond 1997, Herrmann & Roynette 2003), x self-repelling: f ( x ) = 1 + | x | 1 + β , with 0 < β < 1 (Mountford & Tarrès 2008). university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems Study of some self-interacting diffusions New tools: tightness and uniform estimates General statements A last conjecture (unsolved) Conjecture (Durrett & Rogers, 1992) Suppose f : R → R with compact support, xf ( x ) ≥ 0 and f ( − x ) = − f ( x ) . Then, X t t converges a.s. toward 0. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems Study of some self-interacting diffusions New tools: tightness and uniform estimates General statements Self-interacting diffusions on a compact set Benaïm, Ledoux & Raimond (2002), Benaïm & Raimond (2003, 2005) on a compact manifold: � t d X t = d B t − 1 ∇ x W ( X t , X s ) d s d t . t 0 Heuristic: show that µ t is close to a deterministic flow (stochastic approximation). university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Why is it more difficult? Théorème (Chambeu & K) Let d X t = d B t − ( log t ) 3 W ′ ( X t − µ t ) d t , X 0 = x � t where µ t = 1 0 X s d s and W is strictly convex out of a compact t set. Then The process Y t = X t − µ t converges a.s. to Y ∞ , where Y ∞ 1 belongs to the set of all the local minima of W. Moreover, for each local minimum m, one has P ( Y ∞ = m ) > 0 . On the set { Y ∞ = 0 } , both X t and µ t converge a.s. to 2 � ∞ 0 Y s d s µ ∞ := s . Moreover, on the set { Y ∞ � = 0 } , one has university-logo t →∞ X t / log t = Y ∞ . lim Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Our study: � t � ∇ V ( X t ) + 1 � d X t = d B t − ∇ x W ( X t , X s ) d s d t t 0 δ X t − µ t µ t ˙ = t university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Hypotheses on the potentials (H) V ≥ 1 is C 2 , strictly uniformly convex, W is C 2 and such that ∇ 2 ( V + W ) is bounded by below, and asymptotically ( x , ∇ x W ( x , y )) + ( x , ∇ V ( x )) ≥ M | x | 2 δ with δ > 1 and M > 0, there exists κ > 0 such that W ( x , y ) + |∇ x W ( x , y ) | + |∇ 2 xx W ( x , y ) | ≤ κ ( V ( x ) + V ( y )) . university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Example on R 2 Theorem Suppose V ( x ) = V ( | x | ) and W ( x , y ) = ( x , Ry ) , where R is a rotation matrix (angle θ ). Let I := Z − 1 � ∞ 0 e − 2 V ( ρ ) ρ 2 d ρ . One of the following holds: if I cos θ + 1 > 0 , then a.s. µ t converges to Z − 1 e − 2 V , if I cos θ + 1 ≤ 0 , then: if θ = π , then a.s. µ t converges to a random measure µ ∞ � = Z − 1 e − 2 V , if θ � = π , then µ t does not converge: it circles around and the ω − limit-set ω ( µ t , t ≥ 0 ) is a “circle” of measures. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements If W is symmetric Theorem Suppose (H). If W is symmetric, then the ω − limit set ω ( µ t , t ≥ 0 ) is a.s. a compact connected subspace of the fixed points of Π , with Π( µ )( d x ) = Z ( µ ) − 1 e − 2 ( V + W ∗ µ )( x ) d x. In particular, if Π admits only a finite number of fixed points, then µ t converges a.s. to one of these fixed points. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Markovian system related to the diffusion µ t is asymptotically close to a deterministic dynamical system: µ = Π( µ ) − µ . ˙ university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Asymptotic pseudotrajectory (APT) for a flow Definition ( E , d ) metric. The continuous function ξ : R → E is a APT for the flow Φ if ∀ T > 0 , one has lim t →∞ sup d ( ξ t + s , Φ s ( ξ t )) = 0 . 0 ≤ s ≤ T university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Deterministic example Let the ODE (on R ): ˙ ξ = f ( ξ ) + g ( t ) , (1) where f : R → R is a Lipschitz function and g : R + → R is a continuous function such that lim t →∞ g ( t ) = 0. Consider the solution of ˙ x = f ( x ) (2) ξ is an asymptotic pseudotrajectory of the flow generated by (2). university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements � t + s � t + s ξ t + s − Φ s ( ξ t ) = ( f ( ξ u ) − f (Φ u ( ξ t ))) d u + g ( u ) d u . t t f is Lipschitz ( c ) Gronwall � t + T | ξ t + s − Φ s ( ξ t ) | ≤ e cT sup | g ( u ) | d u . 0 ≤ s ≤ T t g ( t ) converges toward 0 conclusion: lim t →∞ sup | ξ t + s − Φ s ( ξ t ) | = 0 0 ≤ s ≤ T university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Attractor free set Definition A ⊂ E is an attractor for the flow Φ if it is A � = ∅ , compact, invariant and A admits a neighbourhood V ⊂ E such that d (Φ t ( x ) , A ) → 0 uniformly for x ∈ V . A is said to be attractor free if A is the only attractor for the flow restricted to A. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Tightness Using some martingale techniques and the law of large numbers, we get: Lemma There exists a subset P of the set of probability measures on R d , which is compact (for the weak topology) such that a.s. µ t ∈ P for all t large enough. The family ( µ t , t ≥ 0 ) is a.s. tight. university-logo Aline Kurtzmann Self-interacting diffusions
Some generalities Self-interacting diffusions on R d Tools: dynamical systems New tools: tightness and uniform estimates General statements Uniform ultracontractivity Using some techniques of Röckner and Wang, one has: Proposition The semi-group family ( P µ t , t , µ ) is uniformly ultracontractive i.e. � P µ t f � ∞ ≤ exp { ct − δ/ ( δ − 1 ) } , sup � f � 2 f ∈ L 2 (Π( µ )) \{ 0 } with a uniform constant c > 0 . university-logo Aline Kurtzmann Self-interacting diffusions
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