Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Small Mass Limit of a Langevin Equation on a Manifold Jeremiah Birrell Department of Mathematics The University of Arizona Joint work with S. Hottovy, G. Volpe, J. Wehr Ann. Henri Poincar´ e , (2017) 18: 707 Preprint arXiv:1604.04819 35th Western States meeting of Mathematical Physics February 12-13, 2017 Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Background In the simplest case, the motion of a diffusing particle of non-zero mass, m , is governed by a stochastic differential equation (SDE), of the form dq t = v t dt , mdv t = − γ v t dt + σ dW t . (1) ◮ γ is the dissipation (or drag) matrix ◮ σ is diffusion (or noise) matrix ◮ W t is a Wiener process Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Earlier Work Pioneered by by Smoluchowski (1916) and Kramers (1940). The field has grown to explore a large array of models and phenomena: ◮ Coupled fluid-particle systems: Pavliotis and Suart (2003) ◮ Relativistic diffusion: Chevalier and Debbasch (2008), Bailleul (2010) ◮ A variety of models on manifolds: Pinsky (1976,1981), Jørgensen (1978), Dowell (1980), Bismut (2005,2015), Li (2014) Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Noise Induced Drift Noise induced drift can arise in the small mass limit: An additional drift term not present in the original system. Originates from state dependence of the drag/noise matrices. First derived formally by H¨ anggi (1982). Proven rigorously: ◮ In 1-dim. and fluctuation dissipation case: Sancho, Miguel, D¨ urr (1982) ◮ General N-dim. case: S. Hottovy, A. McDaniel, G. Volpe, J. Wehr (2014) Effects observed experimentally by Volpe et. al. (2010). We generalize to N-dim. compact Riemannian manifolds. Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Forced Geodesic Motion on the Tangent Bundle ( M , g ) is an n -dimensional smooth, connected, compact, Riemannian manifold without boundary. Equation for forced geodesic motion: x ) = 1 x ˙ x = V ( x , ˙ x ) , V ( x , ˙ m ( F ( x ) − γ ( x ) ˙ ∇ ˙ x ) , (2) ◮ ∇ is the Levi-Civita connection. ◮ Forcing, V , depends on the position and velocity. ◮ Linear drag term with drag tensor γ . Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Forced Geodesic Motion on the Tangent Bundle Goal: Couple the system to noise and study the small mass limit. Main (Physical) Assumption Assume the symmetric part of γ , γ s = 1 2 ( γ + γ T ) , has eigenvalues bounded below by a constant λ 1 > 0 on all of M . This is needed to ensure that the momentum degrees of freedom are sufficiently damped and are negligible in the limit. Problem: To couple n -dim. noise, W t , to a dynamical system on a n -manifold you need n vector fields, but a general n -manifold doesn’t have n -canonical vector fields. Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Frame Bundle and Horizontal Vector Fields Formulating equations on the orthogonal frame bundle ( F O ( M ) , π ) is a known method that facilitates coupling dynamical systems on manifolds to noise. Canonical horizontal vectors fields on F O ( M ) , H v (one for each v ∈ R n ), characterize geodesic motion on the frame bundle: Lemma Let u ∈ F O ( M ) and v ∈ R n . Let τ be the integral curve of H v starting at u. Then x ≡ π ◦ τ is the geodesic starting at π ( u ) with initial velocity u ( v ) and for any w ∈ R n , τ ( t ) w is parallel transported along x ( t ) . Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Frame Bundle and Horizontal Vector Fields The H ’s also let one lift vector fields from M to F O ( M ) : Lemma Let ( M , g ) be a smooth Riemannian manifold and b be a smooth vector field on M. The horizontal lift of b to F O ( M ) is given by b h ( u ) = H u − 1 b ( π ( u )) ( u ) . (3) Dynamics on M and on F O ( M ) are related by: Lemma If τ is an integral curve of b h starting at u then x ≡ π ◦ τ is an integral curve of b starting at π ( u ) and for any v ∈ R n , τ ( t ) v is the parallel translate of u ( v ) along x ( t ) . Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Forced Geodesic Motion on the Frame Bundle The H v ’s can be used to reformulate forced geodesic motion as a dynamical system on F O ( M ) × R n via the vector field: X ( u , v ) = ( H v ( u ) , u − 1 V ( π ( u ) , u ( v ))) . (4) X characterizes the deterministic dynamics: Lemma If ( u ( t ) , v ( t )) is an integral curve of X starting at ( u 0 , v 0 ) then x ( t ) = π ◦ u ( t ) is a solution to x ˙ x = V ( x , ˙ ∇ ˙ x ) , x ( 0 ) = π ( u 0 ) , (5) U α ( t ) ≡ u ( t ) e α is an orthonormal basis that is parallel transported along x ( t ) , and ˙ x ( t ) = u ( t ) v ( t ) . Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Forced Geodesic Motion with Noise Recall the forcing: V ( x , w ) = 1 m ( F ( x ) − γ ( x ) w ) , w ∈ T x M , x = π ( w ) ∈ M (6) ◮ m is the particle mass ◮ F is a smooth vector field on M � 1 ◮ γ is a smooth � tensor field on M 1 ◮ γ s , has eigenvalues bounded below by a constant λ 1 > 0 F and γ can be lifted to the frame bundle (scalarization): ◮ F ( u ) = u − 1 F ( π ( u )) ◮ γ ( u ) = u − 1 γ ( π ( u )) u by γ ( u ) These are R n and R n × n -valued functions on F O ( M ) , respectively. Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Forced Geodesic Motion with Noise Given noise coefficients σ : F O ( M ) → R n × k , one can couple noise to the velocity component of the deterministic dynamical system on F O ( M ) × R n to obtain the SDE: � t u ( t ) = u 0 + H v ( s ) ( u ( s )) ds , (7) 0 � t � t v ( t ) = v 0 + 1 F ( u ( s )) − γ ( u ( s )) v ( s ) ds + 1 σ ( u ( s )) dW s . m m 0 0 (8) Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Rate of Decay of the Momentum Let ( u m t , v m t ) be the solution to the SDE with mass m and initial condition ( u 0 , v 0 ) . A key to proving convergence of u m t in the limit m → 0 is a sequence of bounds on the momentum process p m t = mv m t . The most difficult one is: Lemma For any p > 0 , T > 0 , and 0 < β < 1 / 2 we have t � p ] 1 / p = O ( m β ) as m → 0 . � p m E [ sup (9) t ∈ [ 0 , T ] Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Proof Outline The momentum solves the SDE � t ) − 1 � dp m F ( u m m γ ( u m t ) p m dt + σ ( u m t = t ) dW t . (10) t This is a linear SDE on R n , and so its unique solution can be written in terms of u m t � t � t � � p m p m Φ − 1 ( s ) F ( u m Φ − 1 ( s ) σ ( u m t = Φ( t ) 0 + s ) ds + s ) dW s , 0 0 (11) where Φ( t ) is the fundamental solution to the linear part: dt Φ( t ) = − 1 d m γ ( u m t )Φ( t ) , Φ( 0 ) = I . (12) Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit Proof Outline The following bound is straightforward. � t t � p ≤ O ( m p ) + C sup � p m Φ − 1 ( s ) σ ( u m s ) dW s � p . � Φ( t ) sup t ∈ [ 0 , T ] t ∈ [ 0 , T ] 0 (13) By assumption, the symmetric part of − 1 m γ ( u ) has eigenvalues bounded above by − λ 1 / m < 0 with the bound uniform in u . Therefore, for s ≤ t , � Φ( t )Φ − 1 ( s ) � ≤ e − λ 1 ( t − s ) / m . (14) One would like to use this to bound the last term in Eq. (13), but the fact that Φ − 1 ( s ) is under the stochastic integral makes this difficult in its current form. Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold
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