THE INVARIANT MEASURE OF A WALKING DROPLET IN HYDRODYNAMIC PILOTWAVE THEORY HUNG D. NGUYEN1AND ANAND U. OZA2

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THE INVARIANT MEASURE OF A WALKING DROPLET IN

HYDRODYNAMIC PILOT–WAVE THEORY

HUNG D. NGUYEN1AND ANAND U. OZA2

Abstract. We study the long time statistics of a walker in a hydrodynamic pilot-wave system,

which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior

experiments and numerical simulations have indicated that the system may reach a statistically

steady state, its long-time behavior has not been studied rigorously. For a broad class of external

potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path

spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we

demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting

an example of such an invariant measure, as obtained from a numerical simulation of a walker in a

harmonic potential.

1. Introduction

In 2005, Yves Couder and Emmanuel Fort discovered that an oil droplet may self-propel (or

“walk”) while bouncing on the surface of a vertically vibrating bath of the same ﬂuid [20,19].

The so-called “walker” is comprised of the droplet and its guiding or “pilot” wave. The pilot

wave is created by the droplet’s impacts on the bath surface, and in turn the droplet receives a

propulsive force from the pilot wave during its impact. The coupling between the droplet and its

associated wave ﬁeld leads to behavior reminiscent of that observed in the microscopic quantum

realm. Speciﬁcally, experiments have demonstrated analogs of tunneling [28,69], the quantum

corral [39,67,21], the quantum mirage [67], Landau levels [32,38], Friedel oscillations [68], Zeeman

splitting [29], and the quantum harmonic oscillator [63,62,61]. Recent review articles [8,9,10]

have discussed the potential and limitations of the walker system as a hydrodynamic analog of

quantum systems.

A number of theoretical models for this hydrodynamic pilot-wave system have been developed,

with varying degrees of complexity [74,10]. This paper will be concerned with the so-called “strobo-

scopic model” [58], which describes the horizontal dynamics of a droplet with position and velocity

(x(t), v(t)) ∈R2in the presence of an external potential U. In its dimensionless form, the trajectory

equation reads

dx(t) = vdt,

κdv(t) = −vdt−U′(x(t)) dt+αZt

−∞

H(x(t)−x(s))K(t−s) dsdt+σdW(t),(1.1)

where H(x)=J1(x) is the Bessel function of the ﬁrst kind of order one, K(t)=e−t,κ > 0

the dimensionless droplet mass, σ≥0 the noise strength and Wa standard Brownian motion.

Equation (1.1) posits that the droplet moves in response to four forces: a drag force proportional

to its velocity v, an external force −U′, a pilot-wave force proportional to α, and a stochastic

force proportional to σ. The pilot-wave force is proportional to −h′, the slope of the interface at

the droplet’s position. The interface height his a sum of the standing waves generated by the

droplet during each impact. These waves are subthreshold Faraday waves [31] and oscillate at the

same frequency that the drop bounces, as is typically the case in experiments [64,51,78]. In the

high-frequency limit relevant to the experiments, in which the bouncing period is small relative to

arXiv:2210.11767v2 [math.PR] 31 Jan 2024

the timescale of the walker’s horizontal motion, the aforementioned sum may be replaced by the

integral shown in Eq. (1.1). The kernel is comprised of the functions Hand K, whose functional

forms were originally derived by Mol´aˇcek & Bush [51]. More sophisticated models for Hthat

incorporate the observed far-ﬁeld decay of the wave ﬁeld have since been developed [18,71,72].

We assume for the sake of simplicity that the noise strength σis independent of both the droplet

position xand velocity v.

The stroboscopic model (1.1) without stochastic forcing (σ= 0) has been used to model free

walkers [58,25]; walkers in a rotating frame [57,59]; walkers in linear [75], quadratic [47,46,61,7],

quartic [52] and Bessel [70] potentials; and pairs [76], rings [16,71,72,73], chains [2] and lattices [17]

of droplets. The walker’s “path memory” [32] is a key feature of Eq. (1.1): the pilot-wave force on

the walker at a given time is inﬂuenced by the walker’s entire past, with the near past having a

larger inﬂuence than the far owing to the exponential decay of K.

A recent review article [66] has discussed the mechanisms by which a walker’s dynamics may

become chaotic, and studies have characterized the long-time statistical properties of a walker in

the chaotic regime. Speciﬁcally, an experimental study [38] of a walker in a rotating frame showed

that its trajectory is characterized by chaotic jumps between unstable circular orbits. The proba-

bility distribution of the walker’s radius of curvature thus converges to a peaked multimodal form

in the long-time limit, a ﬁnding that was corroborated by a numerical study using the stroboscopic

model [59]. Experimental [63] and numerical [45,22] studies of a walker in a harmonic potential

similarly revealed that, in the long-time limit, the trajectories exhibit a quantization in their radius

and angular momentum. Studies of a walker in circular [39,34,21,65] and elliptical [67] “corral”

geometries have shown that the long-time statistical behavior of the walker’s position is related

to the eigenmodes of the domain. Prior studies have also established a link between the walker’s

position probability density and the time-averaged pilot-wave ﬁeld [23,70], and have shown that

persistent oscillations in the walker’s speed lead to multimodal probability distributions with dis-

tinct peaks [68,25]. While chaotic dynamics is the mechanism that generates coherent multimodal

statistics in all of the aforementioned studies, there has not yet been an investigation into the role

of stochastic forcing (σ̸= 0 in Eq. (1.1)) on the walker dynamics. Moreover, it has been an open

question as to whether Eq. (1.1) admits an invariant measure. Therefore, the main goal of this

paper is to rigorously study the long time behavior of (1.1). More speciﬁcally, we prove that (1.1)

admits a unique stationary distribution, assuming a general set of conditions on the functions U,

Hand K.

We note that without the memory term (α= 0), Eq. (1.1) is reduced to the Langevin equation

dx(t) = vdt,

κdv(t) = −vdt−U′(x(t)) dt+σdW(t),(1.2)

whose asymptotic behavior is well-understood. In particular, (1.2) naturally possesses a Markovian

structure on R2, which is amenable to analysis. Furthermore, for a broad class of potentials U,

it can be shown that (1.2) admits a unique invariant probability measure and that the system is

exponentially attracted toward equilibrium [40,49,50,60]. On the other hand, due to the presence

of past information, the dynamics (x(t), v(t)) of (1.1) itself is not really a Markov process. To

circumvent this diﬃculty, we will draw upon the framework of [1,41,43,50], which dealt with the

same issue, to construct the dynamics of (1.1) on suitable path spaces. More speciﬁcally, given an

initial trajectory (x0, v0)∈C((−∞,0]; R2), we ﬁrst evolve (1.1) on the time interval [0, t] to obtain

a path x(·), v(·)on (−∞, t]. Then, letting θt:C((−∞, t]; R2)→C((−∞,0]; R2) be the shift map

deﬁned as

θtf(r) = θtf(r) = f(t+r), r ≤0,

we observe that θtx(·), v(·)again lives in C((−∞,0]; R2). Consequently, this induces a Markov

semi-ﬂow on C((−∞,0]; R2), which allows for the use of asymptotic analysis to investigate statis-

tically steady states. In Section 2, we will discuss the construction of the solution to (1.1) in more

detail.

Historically, stochastic diﬀerential equations (SDEs) with inﬁnite past were studied as early as

in the seminal work of Itˆo and Nisio, 1964 [43]. Motivated by the approach developed in [43],

stochastic dissipative PDEs such as the Navier-Stokes equation and Ginzburg-Landau equation

were considered in the context of memory [26,27]. Making use of a strategy similar to that in [43],

the existence and uniqueness of stationary solutions of these speciﬁc equations were established.

Later on, a more general method to study invariant structures of SDEs with memory was developed

in [1]. For the analysis of equilibrium of other stochastic dynamics in inﬁnite-dimensional spaces,

we refer the reader to [4,5,6,12,13,14,54]. Related to (1.1) in ﬁnite-dimensional settings, the

generalized Langevin equation (GLE) was introduced in [53] and popularized in [79]. In particular,

statistically steady states of the GLE were explored in many papers [36,41,56]. The signiﬁcant

diﬀerences from (1.1) are that the GLE assumes the so called ﬂuctuation-dissipation relationship

[53,79], and that it involves an integral convolution with the velocity instead of the displacement.

Turning back to (1.1), our ﬁrst main result is the existence of an invariant probability measure

µ, cf. Theorem 2.10. The approach that we employ is drawn from the argument in [43] via the

classical Krylov-Bogoliubov Theorem. More speciﬁcally, under the assumptions that the kernel K

has exponential decay and that the potential Udominates the pilot-wave force H, cf. Remark

2.7, we are able to establish suitable moment bounds of the solution. Then, by a compactness

argument, the solution is shown to converge to at least one steady state. It is worth mentioning

that the existence proof relies heavily on the exponential decay of the memory kernel [30,51].

Furthermore, as a consequence of the energy estimates obtained in the proof, a typical stationary

path must have moderate growth. This property is used to prove the second main result of the paper

concerning the uniqueness of µ, cf. Theorem 2.12. The proof of uniqueness employs a coupling

argument asserting that, starting from two distinct initial paths, the solutions always converge to

the same point, thereby establishing that µis unique [1,35,37,48]. While the main ingredient

for the existence proof is the well-known Krylov-Bogoliubov Theorem, the uniqueness proof makes

use of Girsanov’s Theorem, which ensures that any coupling can be accomplished under suitable

changes of variables.

We note that, while the existence and uniqueness of an invariant measure in a SDE with memory

were established in [1], the existence of a Lyapunov function was assumed. In this work we eﬀectively

construct a Lyapunov function and thereby obtain ergodicity results. For the sake of simplicity,

we restrict our attention to one-dimensional dynamics, x(t)∈Rand v(t)∈R, but expect that

analogous results should hold in higher dimensions. Finally, we note that in this work, we adopt

the Itˆo approach, which was previously employed in [1,26,27,41], so as to allow for the convenience

of both using Itˆo’s formula and performing moment estimates on Martingale processes.

The rest of the paper is organized as follows. In Section 2, we introduce the relevant func-

tion spaces as well as the assumptions. We also state the main results of the paper, including

Theorem 2.10 and Theorem 2.12, giving the existence and uniqueness of an invariant probability

measure. In Section 3, we collect useful moment bounds on the solutions. In Section 4, we address

the asymptotic behavior of (1.1) and use the energy estimates collected in Section 3to prove the

main results. In the Appendix, following the classical theory of SDEs, we explicitly construct the

solutions of (1.1).

2. Assumptions and main results

We start by discussing the well-posedness of (1.1). Since the parameters κ, α, σ do not aﬀect

the analysis, we set κ=α=σ= 1 for the sake of simplicity and reduce (1.1) to

dx(t) = v(t)dt,

dv(t) = −v(t)dt−U′(x(t))dt−Zt

−∞

H(x(t)−x(s))K(t−s)dsdt+ dW(t).(2.1)

To construct a phase space for (2.1), we denote by C(−∞,0] the set of past trajectories in R2, i.e.,

C(−∞,0] = C((−∞,0]; R2).

The topology in C(−∞,0] is induced by the Prokhorov metric [1,43]

d(f1, f2) = X

n≥1

2−n∥f1−f2∥n

1 + ∥f1−f2∥n

where

∥f∥n= max

s∈[−n,0] ∥f(s)∥

and ∥ · ∥ denotes the Euclidean norm in R2. More generally, for −∞ ≤ t1< t2≤ ∞, we denote by

C(t1, t2), the set of trajectories in (t1, t2) given by

C(t1, t2) = C((t1, t2); R2).

In particular, the set of future paths is given by

C[0,∞) = C([0,∞); R2).

Given a path ξ= (x, v)∈ C(−∞,∞), we denote by πxand πvrespectively the projections onto the

marginal path spaces, namely,

πxξ(·) = x(·) and πvξ(·) = v(·).

Moreover, given a set A⊂R, the projection of ξonto C(A;R2) is given by

πAξ(s) = ξ(s), s ∈A.

As mentioned in Section 1, we will also make use of θt, the shift map on the spaces of trajectories,

deﬁned as

θtξ(s) = ξ(t+s), s ∈R.

Throughout, we will ﬁx a stochastic basis S= (Ω,F,P,{Ft}t≥0, W ) satisfying the usual condi-

tions [44], i.e., the set Ω is endowed with a probability measure Pand a ﬁltration of sigma-algebras

{Ft:t∈R}generated by W.

Having introduced the needed spaces, we are now in a position to deﬁne a strong solution of (2.1)

[1,43,41].

Deﬁnition 2.1. Given an initial condition ξ0∈ C(−∞,0], a process (x, v) = ξ(−∞,T ]∈ C(−∞, T ]

is called a solution of (2.1) if the following holds:

1. For all s≤0, (x(s), v(s)) = ξ0(s);

2. The process (x(t), v(t)) is adapted to the ﬁltration {Ft}; and

3. P-almost surely (a.s.), for all 0 ≤t1≤t2≤T

x(t2)−x(t1) = Zt2

v(r)dr,

v(t2)−v(t1) = Zt2

t1−v(r)−U′(x(r)) −Zr

−∞

H(x(r)−x(s))K(r−s)dsdr+W(t2)−W(t1).

Next, we introduce the main assumptions that will be employed throughout the analysis. Con-

cerning the memory kernel K, we impose the following assumption, which is characteristic of some

prior theoretical models of walking droplets [30,51]:

Assumption 2.2. K: [0,∞)→R+satisﬁes

K′(t)≤ −δK(t), t ≥0,

for some δ > 0.

Remark 2.3. We note that using Gronwall’s inequality, Assumption 2.2 implies that the kernel

decays exponentially fast, i.e.,

K(t)≤K(0)e−δt, t ≥0.

However, the diﬀerential inequality in Assumption 2.2 is slightly more general than the above

exponential decay, and will be employed later in Section 3, see e.g., estimate (3.6).

With regards to the pilot-wave force H, we assume Hhas polynomial growth as follows:

Assumption 2.4. H∈C1(R)satisﬁes

max{|H′(x)|,|H(x)|} ≤ aH(|x|p1+ 1), x ∈R,(2.2)

for some constants aH>0and p1≥0.

Remark 2.5. 1. One particular example of the pilot-wave nonlinear term His J1, the Bessel

function of the ﬁrst kind of order one, which has been used in theoretical models of walking

droplets [51]. In this case, condition (2.2) is satisﬁed with p1= 0 since J1and J′

1are actually

bounded. In general, the functions Hand H′as in Assumption 2.4 need not be.

2. The bound on H′as in (2.2) is only employed to establish the well-posedness of (2.1).

In order to establish moment bounds on the solutions, the potential Uis required to dominate

the pilot-wave force H. We thus make the following assumption about U.

Assumption 2.6. The potential U∈C1(R; [1,∞)) satisﬁes

|U′(x)| ≤ a0(U(x)n0+ 1), x ∈R,(2.3)

for positive constants a0, n0. Furthermore,

xU′(x)≥a1U(x)−a2, x ∈R,(2.4)

and

U(x)≥a3|x|2 max{1,p1+ε1}, x ∈R,(2.5)

for some positive constants ε1and ai,i= 1,2,3, where p1is as in Assumption 2.4.

Remark 2.7. We note that conditions (2.3)-(2.4) are standard and can be found in the litera-

ture [36,49,50]. In view of condition (2.2) on the growth rate of H, condition (2.5) implies that

the potential U(x) dominates both x2and H(x)2, i.e.,

c(U(x) + 1) ≥max{x2, H(x)2}, x ∈R,

for some positive constant cindependent of x. In particular, when H= J1, cf. Remark 2.5, the

potential Ucan be chosen as a quadratic function, e.g., U(x) = x2+ 1, a situation that has been

considered in experiments [63,62].

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THEINVARIANTMEASUREOFAWALKINGDROPLETINHYDRODYNAMICPILOT–WAVETHEORYHUNGD.NGUYEN1ANDANANDU.OZA2Abstract.Westudythelongtimestatisticsofawalkerinahydrodynamicpilot-wavesystem,whichisastochasticLangevindynamicswithanexternalpotentialandmemorykernel.Whilepriorexperimentsandnumericalsimulationshaveindicate...

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THE INVARIANT MEASURE OF A WALKING DROPLET IN HYDRODYNAMIC PILOTWAVE THEORY HUNG D. NGUYEN1AND ANAND U. OZA2

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