Interpretation of generalized Langevin equations David Sabin-Miller Center for the Study of Complex Systems University of Michigan Ann Arbor MI 48109

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Interpretation of generalized Langevin equations

David Sabin-Miller∗

Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109

Daniel M. Abrams†

Department of Engineering Sciences and Applied Mathematics,

Northwestern University, Evanston, IL, USA and

Department of Physics and Astronomy, Northwestern University, Evanston, IL, USA

Many real-world systems exhibit “noisy” evolution in time; interpreting their ﬁnitely-sampled

behavior as arising from continuous-time processes (in the Itˆo or Stratonovich sense) has led to

signiﬁcant success in modeling and analysis in a wide variety of ﬁelds. Yet such interpretation hinges

on a fundamental linear separation of randomness from determinism in the underlying dynamics.

Here we propose some theoretical systems which resist easy and self-consistent interpretation into

this well-deﬁned class of equations, requiring an expansion of the interpretive framework. We

argue that a wider class of stochastic diﬀerential equations, where evolution depends nonlinearly

on a random or eﬀectively-random quantity, may be consistently interpreted and in fact exhibit

ﬁnite-time stochastic behavior in line with an equivalent Itˆo process, at which point many existing

numerical and analytical techniques may be used.We put forward a method for this conversion, and

demonstrate its use on both a toy system and on a system of direct physical relevance: the velocity of

a meso-scale particle suspended in a turbulent ﬂuid. This work enables the theoretical and numerical

examination of a wide class of mathematical models which might otherwise be oversimpliﬁed due

to a lack of appropriate tools.

I. GENERALIZING LANGEVIN

EQUATIONS

Langevin equations are often used to represent

theoretical diﬀerential behavior for systems exhibit-

ing stochastic dynamics (see, e.g., [1, 2]). These

equations have a standard form, which we will aim

to generalize:

dt=f(x, t) + g(x, t)ηt,

where ηtrepresents the “Gaussian white noise”

term, δ-correlated in continuous time. If g(x, t) ex-

hibits xdependence, such Langevin equations are

ill-deﬁned, necessitating a choice of either the Itˆo or

Stratonovich interpretation—these will diﬀer in the

resulting “drift” behavior of the system, but both

are internally consistent and able to be simulated

and analyzed by various techniques [2, 3].

Here, we seek to generalize to systems of the form

dt=R(x, t),(1)

where Ris some random variable with some (possi-

bly non-Gaussian) probability distribution over the

domain. We argue that, with the proper conver-

sion procedure based on the central limit theorem

∗dasami@umich.edu

†dmabrams@northwestern.edu

[4], these Langevin-type systems may be reduced to

equivalent Itˆo behavior, allowing for consistent sim-

ulation and theoretical analysis.

As a motivating example, we start by highlighting

the diﬀerence between two similar-looking Langevin-

type equations:

dt=−x3+ηt,(2)

dt=−(x+ηt)3.(3)

Equation (2) is a classic Langevin equation with

cubic attraction towards zero and diﬀusive noise—

easily interpreted (in either the Itˆo or Stratonovich

sense) as the stochastic diﬀerential equation (SDE)

dx=−x3dt+ dW(where dWrepresents the usual

derivative of a Wiener process), enabling all the an-

alytical and numerical options that entails.

Equation (3), however, is notably diﬀerent in that

the nonlinear cubing operation happens to a funda-

mentally random quantity, linking the deterministic

and random parts of the equation. Na¨ıve numeri-

cal simulation simply converges to deterministic be-

havior as the time-step shrinks, since the ﬂuctua-

tions average out before xchanges considerably. If

timestep-independent stochastic behavior is desired,

we must develop a new consistent and coherent in-

terpretation of this equation.

We note that the notation of Eq, (3) would be

better written as

dt=−X3,where X∼N(x, 1).(4)

arXiv:2210.03781v3 [math-ph] 3 Nov 2024

This is because the “randomness” term ηrequires an

explicit distribution in this case, rather than leaning

on the Central Limit Theorem to abstract that in-

formation away. While Eq. (2) would behave iden-

tically regardless of which distribution ηrepresents,

as long as it has zero mean and standard deviation

1—a broad equivalence class which allows the use

of the small-dtlimit, the normal distribution, with-

out loss of generality—the nonlinear operation being

applied in (3) requires an explicit choice of this un-

derlying “noise,” which we might, for example, pick

to be a normal distribution which is then distorted

by cubing as indicated in Eq. (4).

A. Potential Applications

“Baked-in” stochasticity of this type might arise

in a variety of physical modeling scenarios. For ex-

ample, nonlinear drag forces acting on a macroscopic

object in a turbulent ﬂow would cause velocity to

evolve according to this type of Langevin equation,

with “noise” coming from rapidly ﬂuctuating rela-

tive ﬂuid velocity—including, e.g., viscous drag on a

cylinder in a turbulent wake [5]. We compute results

for this velocity distribution, and its stark diﬀerence

from a na¨ıve approach, at the end of this section.

Physical systems with nonlinear feedback based on

rapidly ﬂuctuating quantities or quantities subject

to random measurement error would also be of this

type. Inasmuch as measurement error acts as inde-

pendent random variation of a quantity, the behav-

ior of simulated or artiﬁcially-forced feedback-based

dynamical systems would also beneﬁt from this anal-

ysis. Our interest was motivated by an earlier model

for individuals reacting to a stochastic political en-

vironment [6]. A variety of other physics-inspired

nonlinear models of complex real-world phenomena

may also share this form.

We note that the systems we are concerned with

diﬀer from other ways in which nonlinearity can arise

in stochastic systems, for example in the determin-

istic part (e.g. [7]) or when x-dependence appears

multiplied by the stochastic quantity (e.g., [8, 9]), or

when functions are applied to a continuous random-

walking quantity (as Itˆo’s lemma would handle [2])

rather than the uncertain/noisy quantity itself. Cer-

tain speciﬁc problems exhibiting nonlinear depen-

dence on stochastic quantities have been examined

[10], but a general theory of this class of stochastic

equations has not been developed.

B. The Proposed Equivalence

We seek to bridge the gap from the theoretical,

possibly non-Gaussian “intrinsic” noise (represented

by the distribution Rin Eq. (1)) to some equiv-

alent emergent system which is well-deﬁned, self-

consistent, and able to be simulated.

Our argument is based on the consideration that

over any ﬁnite time-scale, a theoretical system such

as Eq. (3) will have experienced a large enough

number of nearly-independent increments that the

Generalized Central Limit Theorem should apply

[11]. That is, the net increment over any ﬁnite

time must be drawn from the family of stable distri-

butions, or—if the ”intrinsic” noise represented by

the diﬀerential update distribution itself has ﬁnite

variance—a Gaussian distribution in particular [11].

This intuitively dovetails with the more practically-

motivated necessary condition that, in the numer-

ical simulation of any continuous-time system, its

behavior must not depend sensitively on the simu-

lated timestep; that is, one relatively large step must

result in the same distribution (in an ensemble aver-

age sense) as the commensurate number of arbitrar-

ily small steps.

We proceed henceforth with the assumption of ﬁ-

nite underlying variance. This means that the net

increment over any small but ﬁnite time must be

drawn from a Gaussian distribution with mean equal

to the mean of the underlying process. We may also

choose this Gaussian’s distribution’s variance per

unit time to likewise match the underlying process,

maintaining consistency with the classic Langevin-

Itˆo conversion and agreement in standard cases.

By this reasoning, we argue that every such

stochastic process with ﬁnite variance is in fact

equivalent to an Itˆo SDE over any ﬁnite time-

scale: in particular, the SDE with deterministic part

matching the “true” distribution’s mean behavior

and random part matching its standard deviation.

We note that this is not a one-to-one mapping, but

rather many-to-one: any stochastic process with the

same mean and standard deviation would behave

identically, and thus be represented by the same Itˆo

SDE.

That is, for a general stochastic system of the form

dt=R(x, t)∼P(r|x, t),

where Ris some ﬁnite-variance stochastic quantity

dependent on xand δ-correlated in time, with dis-

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InterpretationofgeneralizedLangevinequationsDavidSabin-Miller∗CenterfortheStudyofComplexSystems,UniversityofMichigan,AnnArbor,MI48109DanielM.Abrams†DepartmentofEngineeringSciencesandAppliedMathematics,NorthwesternUniversity,Evanston,IL,USAandDepartmentofPhysicsandAstronomy,NorthwesternUniversity,Eva...

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Interpretation of generalized Langevin equations David Sabin-Miller Center for the Study of Complex Systems University of Michigan Ann Arbor MI 48109

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