Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums
Abstract
This paper introduces a geometric method for proving ergodicity of degenerate noise driven stochastic processes. The driving noise is assumed to be an arbitrary Levy process with nondegenerate diffusion component (but that may be applied to a single degree of freedom of the system). The geometric conditions are the approximate controllability of the process the fact that there exists a point in the phase space where the interior of the image of a point via a secondarily randomized version of the driving noise is non void.
The paper applies the method to prove ergodicity of a sliding disk governed by Langevintype equations (a simple stochastic rigid body system). The paper shows that a key feature of this Langevin process is that even though the diffusion and drift matrices associated to the momentums are degenerate, the system is still at uniform temperature.
1 Introduction
This paper is concerned with proving ergodicity of mechanical systems governed by Langevintype equations driven by Levy processes and with a singular diffusion matrix applied on the momentums.
Such systems arise, for instance, when one models stochastically forced mechanical systems composed of rigid bodies. In such systems one would like to introduce a certain structure to the noise and observe its effect on the dynamics of the system. For instance, one would like to apply stochastic forcing to a single degree of freedom and characterize the ergodicity of the system. The stochastic process associated to the dynamics of these systems is in general only weak Feller and not strong Feller.
The paper provides a concrete weak Feller (but not strong Feller) stochastic process to illustrate this lack of regularity. The example is a simple mechanical system that is randomly forced and torqued and that preserves the Gibbs measure. In this case one would like to determine if this Gibbs measure is the unique, invariant measure of the system.
A new strategy based on the introduction of the asymptotically strong Feller property has been introduced in [6]. This paper proposes an alternative method based on two conditions: weak irreducibility and closure under second randomization of the stochastic forcing (see theorem 3.1). Our strategy is in substance similar to the one proposed by Meyn and Tweedie for discrete Markov Chains in chapter 7 of [9].
Although the Hörmander condition ([11] 38.16) can also be used to obtain local regularity properties of the semigroup, hence a local strong Feller condition and ergodic properties. The alternative approach proposed here doesn’t require smooth vector fields or manifolds, it can directly be applied to Levy processes and (this is our main motivation) it allows for an explicit geometric understanding of the mechanisms supporting ergodicity.
2 General set up.
Let be a Markov stochastic process on a (separable) manifold with model space .
Let be dimensional Levy process, i.e. a stochastic process on that has has independent increments, is stationary, is stochastically continuous and such that (almost surely) trajectories are continuous from the left and with limits from the right.
We assume that there exists a family deterministic mappings (indexed by ) such that
(2.1) 
Recall that the first three condition defining a Levy process mean that is independent of , the law of depends only on and .
Recall also [11, 12] that since is a Levy process, there exists a , a constant matrix , a standard dimensional Brownian Motion and an independent Poisson process of jumps with intensity of measure on (such that ) such that
(2.2) 
Where is a compound Poisson point process (of jumps of norm larger than one) and
(2.3) 
is a martingale (of small jumps compensated by a linear drift). Recall also that any process that can be represented as (2.2) is a dimensional Levy process, in particular a dimensional Brownian Motion is a Levy process. In this paper, the only assumption on the stochastic forcing will be the following one:
Condition 2.1.
is non degenerate (has a non null determinant).
We will then prove the ergodicity of based on the following geometric conditions on .
Condition 2.2.
is approximately controllable, i.e., for all and there exists and so that .
This condition is illustrated in Fig. 1.
Condition 2.3.
For all , the mapping is continuous with respect to the norm where .
Let be deterministic continuous mappings from onto equal to at time . For , write
(2.4) 
Condition 2.4.
There exists and , such that in a neighborhood of :

is differentiable in .

is invertible and uniformly bounded.

is uniformly bounded.
3 Main theorem
Theorem 3.1.
Proof.
We will need the following two lemmas on the Levy process .
Lemma 3.1.
Assume that satisfies condition 2.1. Let and be arbitrary. The laws of and are absolutely continuous with respect to each other.
Proof.
Lemma 3.1 follows by applying Girsanov’s theorem to the diffusive component () of . ∎
Lemma 3.2.
Assume that satisfies condition 2.1. Let . For all , the inequality holds almost surely.
Proof.
Let . Let be the LevyKhintchine characteristics of . Let such that
(3.1) 
Observe that [12] can be written as
(3.2) 
where
(3.3) 
and
(3.4) 
is a compound Poisson point process (of jumps of norm larger than one) and
(3.5) 
is a martingale (of small jumps compensated by a linear drift). Observe that with strictly positive probability , is uniformly equal to 0 over . Furthermore by the Martingale maximal inequality
(3.6) 
and using ([12])
(3.7) 
and Chebyshev’s inequality we obtain that
(3.8) 
hence
(3.9) 
We conclude the proof of lemma 3.2 by applying Schilder’s theorem to and using the fact that is not degenerate. ∎
Let us now prove that is ergodic. Let be an invariant set of positive measure, i.e.,
(3.10) 
and . We will prove that . Assume . Then , which is also an invariant set, has strictly positive measure, i.e. . Now let us prove the following lemma
Lemma 3.3.
If then

For all and , .

For all and , .
Proof.
We will restrict the proof to . Since there exists such that for all , (otherwise one would get by covering the separable manifold with a countable number of balls such that ). Assume that there exists and such that . Since is weakly controllable (condition 2.2) there exists and so that . From the continuity condition 2.3 on and the Schilder type lemma 3.2 imply that there exists such that
(3.11) 
Write the semigroup associated with . Equation (3.11) leads to a contradiction with the fact that
(3.12) 
since and for all , .
∎
From condition 2.4 there exists and and such that for , and , is differentiable in , and . It follows from the condition 2.4 and the continuity condition 2.3 that can be chosen small enough so that there exists , , such that for we have for all ,
(3.13) 
Let . From the previous lemma there exists and such that and . Set () to be the process started from the point () and set to be the measure of probability associated to . We obtain from the Markov property that
(3.14) 
Write
(3.15) 
The Girsanov type lemma 3.1 implies that the laws of and are absolutely continuous with respect to each other. Hence for all ,
(3.16) 
Which leads to
(3.17) 
Let be the event . Observe that from the Schilder type lemma 3.2 the measure of probability of is strictly positive. It follows from (3.17) and (3.13) that
(3.18) 
Using the change of variable we obtain from (3.13) that
(3.19) 
Hence
(3.20) 
We deduce from equation (3.13) and the fact that is bounded from below by that
(3.21) 
However a similar computation leads from (3.17) and (3.13) to
(3.22) 
Hence a contradiction. Thus must be ergodic. Let us now prove that is the unique invariant measure. Assume that is also invariant with respect to the semigroup . By the argument presented above is ergodic and it follows from Proposition 3.2.5 of [10] that and are singular and it is easy to check from the argument presented above that this can’t be the case (the proof is similar to the one given in theorem 4.2.1 of [10]). Hence is the unique invariant distribution. The proof of the fact that is weakly mixing follows from theorem 3.4.1 of [10] and is similar to the one given at page 44 of [10] (theorem 4.2.1). ∎
4 Sliding Disk at Uniform Temperature.
Consider a disk on a surface as shown in Fig. 3 [1]. The disk is free to slide and rotate. We assume that one rescales position its radius and time by some characteristic frequency of rotation or other timescale. The dimensionless Lagrangian is given by
(4.1) 
where stands for the velocity of the center of mass, the angular velocity of the disk and is a strictly positive dimensionless constant given by (where is the radius of the disk, is its mass and its moment of inertia). is an arbitrary periodic potential which is assumed to be smooth, and of period one.
The contact with the surface is modeled using a sliding friction law. For this purpose we introduce a symmetric matrix defined as,
Observe that is degenerate since the frictional force is actually applied to only a single degree of freedom, and hence, one of its eigenvalues is zero. In addition to friction a white noise is applied to the same degree of freedom to which friction is applied. The governing stochastic differential equations are
(4.2) 
where is the matrix square root of . The matrix square root is easily computed by diagonalizing and computing square roots of the diagonal entries (eigenvalues of ) as shown:
Write . It easy to check that the Gibbs distribution
(4.3) 
is invariant for (4.2), where , , and is the energy of the mechanical system and is given by
Define
(4.4) 
The system (4.2) can be written
(4.5) 
where , and is a one dimensional Brownian Motion. Observe that condition 2.1 is satisfied with , and .
Observe also that if is a constant then the quantity is conserved and the system (4.2) can’t be ergodic. Let us assume that is not constant, our purpose is to prove that the Gibbs distribution is ergodic with respect to the stochastic process .
Remark 4.1.
Observe that when is not constant over a non void open subset of (say ), needs to travel a distance that is uniformly (in ) bounded from below by a strictly positive amount to get from to the domain . It follows that in that situation that the process and hence is not strong Feller and theorems requiring this property can’t be applied.
Remark 4.2.
Observe also that the condition in a neighborhood of doesn’t guarantee that is strongly Feller in that neighborhood. For instance observe that and imply that the drift on is uniformly bounded by a strictly positive constant on a neighborhood of it follows that is discontinuous in the neighborhood of () close to the line .
We believe that the system is asymptotically strong Feller so one could in principle obtain the ergodicity of by controlling the semigroup associated to as it is suggested in [6]. We propose an alternative method based on the controllability of the ODE associated to and theorem 3.1. We believe that it is much simpler to control the geometric properties of the ODE associated to rather than the gradient of its semigroup.
One can also check that the generator of satisfies a local Hörmander condition ([11] 38.16) at a point such that so an alternative method to prove ergodicity would be to use that condition to obtain a local regularity of the semi group associated to . Here we propose an alternative method which doesn’t require to be smooth and which can be applied with Levy processes.
Theorem 4.1.
Assume that is not constant. Then the Gibbs measure is ergodic and strongly mixing with respect to the stochastic process (4.2). Furthermore, it is the unique invariant distribution of .
First let us prove that codition 2.2 is satisfied by .
Lemma 4.1.
Assume is not constant. Then is approximately controllable.
Proof.
Since is not constant, there exists such that for there exists a smooth path such that , , , , , and
(4.6) 
Take and interpolate smoothly between (obtained from the control problem (4.6)) and
(4.7) 
Observe that the extension of to as a solution of (4.6) satisfies
(4.8) 
Taking be the smooth curve defined by and
(4.9) 
completes the proof. ∎
Proof.
The proof that satisfies condition 2.3 is a standard application of Gronwall’s lemma. Observe that the semigroup associated to is not strongly irreducible and never equivalent to because . Let us now show that condition 2.4 is satisfied.
Write the stochastic process defined by
(4.10) 
To prove that satisfies condition 2.4 it is sufficient to show that satisfies condition 2.4.
Since is smooth and not constant, there exists a point , such that for , . Let be a point of the phase space such that and . Let and .
Let be continuous mappings from onto , equal to zero at time zero. For we write the solution of
(4.11) 
It follows that
(4.12) 
Writing the solution of
(4.13) 
we obtain that up to the first order in , and at the order in and , can be approximated by . It follows that can be written as where is continuous in and in the neighborhood of . Moreover, can be chosen so that , and are uniformly bounded in that neighborhood. Choosing and implies condition 2.4. By invoking theorem 3.1 one obtains that the process is ergodic and weakly mixing.
It follows from theorem 3.4.1 of [10] that for there exists a set of relative measure such that
(4.14) 
Furthermore since is continuous when is continuous and bounded we deduce that when is continuous and bounded then
(4.15) 
The fact that the process is strongly mixing then follows from corollary 3.4.3 of [10]. ∎
In [1], using thoerem 4.1 we prove that if is nonconstant then the displacement of the sliding disk is a.s. not ballistic (see Proposition 4.1). However, the meansquared displacement with respect to the invariant law is ballistic (see theorem 4.2). More precisely, we show that the squared standard deviation of the displacement with respect to its noiseaverage grows like . This implies that the process exhibits not only ballistic transport but also ballistic diffusion. If is constant then the squared standard deviation of the displacement is diffusive (grows like ). See below for theoretical results and numerical experiments using efficient stochastic variational integrators.
Proposition 4.1.
Provided that is nonconstant, then a.s.
Proposition 4.2.
The squared standard deviation of the degree of freedom is diffusive, i.e.,
(4.16) 
Proposition 4.3.
Assume that is non constant, then
(4.17) 
and
(4.18) 
Theorem 4.2.
We have ([1])

If is constant then
(4.19) 
If is non constant then
(4.20) and
(4.21)
Classical homogenization techniques can’t be applied to obtain theorem 4.2 (since the stochastic forcing is degenerate on momentums). We refer to [1] for a proof of that theorem. The ballistic diffusion is caused by long time memory effects created by the degeneracy of the noise and the coupling between the two degrees of freedom through . Figure 5 gives an illustration of the meansquared displacement of the rolling disk versus time started from rest. In [1] we have used that phenomenon to propose a fluctuation driven magnetic motor characterized by ballistic diffusion at uniform. A plot of the angular displacement of that magnetic motor versus time for a single realization started from rest is given in figure 6.
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