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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 dqt = vtdt, mdvt = −γvtdt + σdWt. (1)

◮ γ is the dissipation (or drag) matrix ◮ σ is diffusion (or noise) matrix ◮ Wt 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 ˙

x = V(x, ˙ x), V(x, ˙ x) = 1 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, γs = 1

2(γ + γT), has

eigenvalues bounded below by a constant λ1 > 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, Wt, to a dynamical system

• n 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 (FO(M), π) is a known method that facilitates coupling dynamical systems on manifolds to noise. Canonical horizontal vectors fields on FO(M), Hv (one for each v ∈ Rn), characterize geodesic motion on the frame bundle:

Lemma

Let u ∈ FO(M) and v ∈ Rn. Let τ be the integral curve of Hv starting at u. Then x ≡ π ◦ τ is the geodesic starting at π(u) with initial velocity u(v) and for any w ∈ Rn, 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 FO(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 FO(M) is given by bh(u) = Hu−1b(π(u))(u). (3) Dynamics on M and on FO(M) are related by:

Lemma

If τ is an integral curve of bh starting at u then x ≡ π ◦ τ is an integral curve of b starting at π(u) and for any v ∈ Rn, 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 Hv’s can be used to reformulate forced geodesic motion as a dynamical system on FO(M) × Rn via the vector field: X(u,v) = (Hv(u), u−1V(π(u), u(v))). (4) X characterizes the deterministic dynamics:

Lemma

If (u(t), v(t)) is an integral curve of X starting at (u0, v0) then x(t) = π ◦ u(t) is a solution to ∇˙

x ˙

x = V(x, ˙ x), x(0) = π(u0), (5) Uα(t) ≡ u(t)eα 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 ∈ TxM, x = π(w) ∈ M (6)

◮ m is the particle mass ◮ F is a smooth vector field on M ◮ γ is a smooth

1

1

• tensor field on M

◮ γs, has eigenvalues bounded below by a constant λ1 > 0

F and γ can be lifted to the frame bundle (scalarization):

◮ F(u) = u−1F(π(u)) ◮ γ(u) = u−1γ(π(u))u by γ(u)

These are Rn and Rn×n-valued functions on FO(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 : FO(M) → Rn×k, one can couple noise to the velocity component of the deterministic dynamical system on FO(M) × Rn to obtain the SDE: u(t) =u0 + t Hv(s)(u(s))ds, (7) v(t) =v0 + 1 m t F(u(s)) − γ(u(s))v(s)ds + 1 m t σ(u(s))dWs. (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 (um

t , vm t ) be the solution to the SDE with mass m and initial

condition (u0, v0). A key to proving convergence of um

t in the limit m → 0 is a

sequence of bounds on the momentum process pm

t = mvm t .

The most difficult one is:

Lemma

For any p > 0, T > 0, and 0 < β < 1/2 we have E[ sup

t∈[0,T]

pm

t p]1/p = O(mβ) as m → 0.

(9)

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 dpm

t =

• F(um

t ) − 1

mγ(um

t )pm t

• dt + σ(um

t )dWt.

(10) This is a linear SDE on Rn, and so its unique solution can be written in terms of um

t

pm

t = Φ(t)

• pm

0 +

t Φ−1(s)F(um

s )ds +

t Φ−1(s)σ(um

s )dWs

• ,

(11) where Φ(t) is the fundamental solution to the linear part: d dt Φ(t) = − 1 mγ(um

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. sup

t∈[0,T]

pm

t p ≤O(mp) + C sup t∈[0,T]

Φ(t) t Φ−1(s)σ(um

s )dWsp.

(13) By assumption, the symmetric part of − 1

mγ(u) has eigenvalues

bounded above by −λ1/m < 0 with the bound uniform in u. Therefore, for s ≤ t, Φ(t)Φ−1(s) ≤ e−λ1(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

Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit

Proof Outline

Lemma

Let Wt be an Rk-valued Brownian motion, σ ∈ L2

loc(W) be an

Rn×k-valued process, and let B(t) be a continuous Rn×n-valued adapted process. Let Φ(t) be the adapted C1 process that pathwise solves the initial value problem (IVP) d dt Φ(t) = B(t)Φ(t), Φ(0) = I. (15) Then we have the P-a.s. equality for all t: Φ(t) t Φ−1(s)σsdWs =Φ(t) t σsdWs (16) − Φ(t) t Φ−1(s)B(s) t

s

σrdWr

• ds

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

Integrating by parts gives Φ(t) t Φ−1(s)σ(s)dWs = t σ(s)dWs (17) + t B(s)Φ(s) s Φ−1(r)σ(r)dWrds. Fix an ω ∈ Ω for which the above equality holds and consider the resulting continuous functions:

◮ r(t) =

t

0 σsdWs, ◮ y(t) = Φ(t)

t

0 Φ−1(s)σsdWs.

• Eq. (17) implies that these satisfy the integral equation

y(t) = r(t) + t B(s)y(s)ds, y(0) = 0. (18)

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

• Eq. (17) implies that these satisfy the integral equation

y(t) = r(t) + t B(s)y(s)ds, y(0) = 0. (19) The unique solution to this equation is y(t) = r(t) + t Φ(t)Φ−1(s)B(s)r(s)ds. (20) From here, one can reorganize terms using d dsΦ−1(s) = −Φ−1(s) ˙ Φ(s)Φ−1(s) (21) to get the desired result.

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

Φ(t) t Φ−1(s)σsdWs =Φ(t) t σsdWs (22) − Φ(t) t Φ−1(s)B(s) t

s

σrdWr

• ds

This decomposition allows us to use Φ(t)Φ−1(s) ≤ e−λ1(t−s)/m for s ≤ t (23) to eventually prove E[ sup

t∈[0,T]

pm

t p]1/p = O(mβ) as m → 0.

(24)

Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold

Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit

The Limiting SDE

Recall: um

t =u0 +

t Hvm

s (um

s )ds,

(25) vm

t

=v0 + 1 m t F(um

s ) − γ(um s )vm s ds + 1

m t σ(um

s )dWs. (26)

In the SDE for um

t , the terms not involving pm t (which survive

when m → 0) can be separated from those involving pm

t (which

go to zero) in order to derive a candidate for the limiting SDE. This part of the derivation closely follows the analogous version derived by S. Hottovy, A. McDaniel, G. Volpe, J. Wehr (2014) in Euclidean space.

Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold

Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit

Limiting Equation

Eventually one arrives at the limiting SDE for ut on FO(M): dut = (γ−1F)h(ut)dt + H(γ−1σ)(ut)(ut) ◦ dWt + S(ut)dt. (27) S is the noise induced drift. More precisely:

Theorem

Let ut be the solution to Eq. (27) with initial condition u0. Fix T > 0 and a Riemannian metric tensor field on FO(M) with associated metric d. Then for any q > 0 and any 0 < κ < 1/2 we have E[ sup

t∈[0,T]

d(um

t , ut)q]1/q = O(mκ) as m → 0.

(28) In particular, we obtain a type of uniform Lq convergence for any q with an explicit convergence rate bound.

Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold

Background Deterministic Dynamics Stochastic Dynamics The Small Mass Limit

Limiting Equation

We now discussing the limiting equation: dut = (γ−1F)h(ut)dt + H(γ−1σ)(ut)(ut) ◦ dWt + S(ut)dt. (29)

◮ (γ−1F)h(ut) is the horizontal lift of γ−1F. This term is

expect from the massless limit in the non-random case.

◮ The noise term H(γ−1σ)(ut)(ut) ◦ dWt is also expected from

knowledge of Brownian motion on a manifold.

◮ The third term, however, is an additional drift that arises

when σ is not constant and γ is nonzero-we call it the noise induced drift.

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 Example

We give one example of the noise induced drift: The case where γ and σ are both scalars on M.

Corollary

If γµ

ν (x) = γ(x)δµ ν and σµ ν (x) = σ(x)δµ ν for γ, σ ∈ C∞(M) then

S = − 1 2(γ−2σ∇σ)h. (30) In particular, if σ is a constant then the noise induced drift

• vanishes. If we also set F = 0 and γ = σ then

dut =H(ut) ◦ dWt, (31) whose solution is the lift of a Brownian motion on M to the

• rthogonal frame bundle.

Jeremiah Birrell University of Arizona Small Mass Limit of a Langevin Equation on a Manifold