Cuplength estimates for time-periodic measures of Hamiltonian systems with diffusion

Abstract

We show how methods from Hamiltonian Floer theory can be used to establish lower bounds for the number of different time-periodic measures of time-periodic Hamiltonian systems with diffusion. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/n for every $n\in \mathbb {N}$, we show that, after passing to a subsequence, they converge in probability distribution as $n\rightarrow \infty$. Besides using standard results from Hamiltonian Floer theory and about convergence of tame probability measures, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent $0<\alpha <\frac{1}{2}$.

1 Summary

In classical mechanics one studies the dynamics of a small number of particles which interact with each other and are driven by external forces. In its Hamiltonian formulation, this amounts to finding solutions $u=(q,p):\mathbb {R}\rightarrow T^*Q$ of Hamilton’s equation $\partial _t u=X^H_t(u)$, where Q is a smooth manifold, called configuration space. The cotangent bundle $T^*Q$ is called the phase space and its tautological 1-form $\theta$ equips $T^*Q$ with a canonical symplectic 2-form $\textrm{d}\theta$. For a time-periodic Hamilton function $H_t=H_{t+1}:T^*Q\rightarrow \mathbb {R}$ one obtains a uniquely defined time-periodic Hamiltonian vector field $X^H_t=X^H_{t+1}$ by requiring that $\textrm{d}H_t=\textrm{d}\theta (X^H_t,\cdot )$. It is a natural question to ask for lower bounds for the number of 1-periodic solutions of Hamilton’s equations in terms of the topology of Q. Provided that the time-periodic Hamiltonian $H_t$ is asymptotically quadratic in the cotangent fibre coordinates, in [8] it is shown that the number of 1-periodic solutions is bounded from below by the cuplength of the loop space of Q, which in the case of the d-dimensional torus $Q=\mathbb {T}^d$ agrees with $d+1$, the cuplength of $\mathbb {T}^d$ itself.

On the other hand, if the number of interacting particles is getting very large, it turns out to be more suitable to shift the focus from individual particles to probability densities and to incorporate probabilistic concepts such as diffusion, that is, Brownian motion, which is understood as limit of random walks where the step width converges to zero. By doing so, we move away from classical mechanics to statistical mechanics and thermodynamics. While equilibrium statistical mechanics studies static phenomena, in non-equilibrium statistical mechanics one studies systems undergoing random movements such as diffusion as well as deterministic movements such as drifts given by vector fields. The Fokker–Planck equation (or forward Kolmogorow equation or drift-diffusion equation) modelling the change of probability densities under drift and diffusion effects plays a fundamental role in non-equilibrium statistical mechanics, see [21]. In the case when $Q=\mathbb {T}^d$ and the drift is given by the Hamiltonian vector field $X^H_t(q,p)=(\partial _p H_t,-\partial _q H_t)$ on the phase space $T^*\mathbb {T}^d$, the corresponding Hamiltonian version of the Fokker–Planck equation reads

$$\begin{aligned} \frac{\partial \rho }{\partial t}=-\frac{\partial }{\partial q}\left( \frac{\partial H_t}{\partial p}\cdot \rho \right) +\frac{\partial }{\partial p}\left( \frac{\partial H_t}{\partial q}\cdot \rho \right) +\frac{\sigma ^2}{2}\cdot \frac{\partial ^2\rho }{\partial q^2},\end{aligned}$$

(1)

where $\rho \in \mathbb {R}$ denotes the diffusion constant. Note that if (q(t), p(t)) is a solution of Hamilton’s equation for $H_t$, then it is immediate to check that $\rho (t,q,p)=\delta (q-q(t))\cdot \delta (p-p(t))$ solves the Fokker–Planck equation with $\sigma =0$, that is, the continuity equation. On the other hand, when $H_t=0$, then we get back the classical diffusion or heat equation on $Q=\mathbb {T}^d$.

It is the main goal of this paper to illustrate how methods from Hamiltonian Floer theory in symplectic geometry can be applied to problems in non-equilibrium statistical mechanics. In analogy with the existence proofs of periodic orbits, it is natural to ask for the existence of time-periodic solutions of Fokker–Planck equations, see, e.g., [7, 9, 12]. In this paper we prove the following result, see also Theorem 2.3.

Theorem 0.1

Assume that the time-periodic Hamiltonian $H:\mathbb {T}^1\times T^*\mathbb {T}^d\rightarrow \mathbb {R}$ is of the form $H_t(q,p)=\frac{1}{2}|p|^2+F_t(q,p)$ with a smooth, time-periodic function $F_{t+1}=F_t$ with finite $C^1$-norm, and let $\sigma \in \mathbb {R}$ be arbitrary. Then there exist $d+1$ $=$ cuplength of (the loop space of) $\mathbb {T}^d$ different time-periodic probability measures $\rho$ on $[0,1]\times T^*\mathbb {T}^d$ solving the corresponding Hamiltonian Fokker–Planck equation, which can be distinguished by the expectation value of their symplectic actions.

Here, the expectation value of the symplectic action of the time-periodic probability density $\rho$ on $T^*\mathbb {T}^d$ is defined as

$$\begin{aligned}\int _0^1\int _{T^*\mathbb {T}^d} \left( p\frac{\partial H_t}{\partial p}(q,p)- H_t(q,p)\right) \,\rho (t,q,p) \,\textrm{d}p\,\textrm{d}q \,\textrm{d}t.\end{aligned}$$

For the proof we use that Brownian motion arises as limit of normalised random walks when the step width converges to zero. After proving the existence of closed random periodic solutions and of the corresponding Floer curves for Hamiltonian systems with random walks with step width 1/n for every $n\in \mathbb {N}$, we show that, after passing to a subsequence, they converge in probability distribution as $n\rightarrow \infty$. Besides using standard results from Hamiltonian Floer theory for the case of finite n, we heavily make use of standard results about convergence of tame probability measures from probability theory, which themselves are generalizations of the fact that the space of Borel probability measures on a compact set (equipped with its standard weak topology) is compact itself. Furthermore, we crucially use that sample paths of Brownian motion are almost surely Hölder continuous with Hölder exponent $0<\alpha <\frac{1}{2}$, and we use standard results from interpolation theory for linear operators to be able to work with fractional Sobolev spaces. We would like to emphasize that this paper is written for researchers with a background in Hamiltonian Floer theory, in particular, no prior knowledge about stochastic processes is required.

2 Random walks and the diffusion equation

In this section, we provide the reader in a condensed manner with the necessary definitions and results about random walks, the Wiener measure and the link with the diffusion equation, for being able to follow the rest of the paper; for more details and background we refer to [6, 15].

Let $\Omega _n=(\Omega _n,{\mathcal {P}}(\Omega _n),\nu _n)$ with $\Omega _n=\mathbb {F}_2^n$, $\mathbb {F}_2=\{-1,+1\}$ denote the probability space for the n-fold coin-flip experiment equipped with the canonical counting measure $\nu _n$ as probability measure. Denoting by $\epsilon _i:\Omega _n\rightarrow \mathbb {F}_2$ the projection onto the i-th component, we hence have for every $i=1,\ldots ,n$ that $\nu _n(\{\omega _n\in \Omega _n:\epsilon _i(\omega _n)=\pm 1\})=\frac{1}{2}$ and $\epsilon _i$ is stochastically independent of $\epsilon _j$ for $j\ne i$. We define the discrete stochastic process $W_n:\Omega _n\times [0,1]\rightarrow \mathbb {R}$ as the rescaled linear interpolation of the n-fold random walk with step width $\Delta t=\frac{1}{n}$ given by

$$\begin{aligned}W_n(\omega _n,t)=\frac{1}{\sqrt{n}}\left( \sum _{i=1}^{\lfloor nt\rfloor } \epsilon _i(\omega _n)+\left( t-\frac{\lfloor nt\rfloor }{n}\right) \epsilon _{\lfloor nt\rfloor +1}(\omega _n)\right) \,\,\text {for every}\,\,\omega _n\in \Omega _n.\end{aligned}$$

While the expectation value of $W_n(\omega _n,1)$ is $\mathbb {E}W_n(\omega _n,1)=0$, note that the rescaling factor $\frac{1}{\sqrt{n}}$ is chosen to guarantee that the variance is normalized to $\mathbb {E}\left( W_n(\omega _n,1)-\mathbb {E}W_n(\omega _n,1)\right) ^2=1$.

It follows from the central limit theorem that $W_n(\cdot ,1)$ converges in distribution to ${\mathcal {N}}(0,1)$, the normal distribution with expectation value 0 and variance 1. In the same way one finds that for each $t\in [0,1]$ we have that $W_n(\cdot ,t)$ converges in distribution to ${\mathcal {N}}(0,t)$. By the latter we mean that

$$\begin{aligned}\nu _n(\{\omega _n\in \Omega _n:W_n(\omega _n,t)\le a\})\,\,\text {converges to}\,\, \frac{1}{\sqrt{2\pi t}}\int _{-\infty }^a \exp \left( -\frac{x^2}{2t}\right) \,\textrm{d}x\end{aligned}$$

as $n\rightarrow \infty$. Note that the limiting distribution agrees with the fundamental solution of the diffusion (or heat) equation, so that the stochastic processes $W_n$ model diffusion as $n\rightarrow \infty$.

More precisely, it is known by the functional central limit theorem that the stochastic processes $W_n:\Omega _n\rightarrow C^0([0,1],\mathbb {R})$ converge in distribution in the sense that the pushforward measures $\mu _n:=\nu _n\circ W_n^{-1}$ on $C^0([0,1],\mathbb {R})$ converge as Borel measures to a limit measure $\mu$, called the Wiener measure. Note that, due to the fact that each probability space $\Omega _n$ is just a finite set, the measures $\mu _n$ are indeed (weighted sums of) Dirac measures. Denoting by $\rho _n$, $\rho$ the distribution corresponding to the Borel measure on $[0,1]\times \mathbb {R}$ obtained as pushforward of the product measure $\lambda \otimes \mu _n$, $\lambda \otimes \mu$ ($\lambda$ = Lebesgue measure on [0, 1]) under the canonical continuous map $[0,1]\times C^0([0,1],\mathbb {R})\rightarrow [0,1]\times \mathbb {R}$, $(t,W)\mapsto (t,W(t))$, we find that $\rho _n$ converges in the distributional sense to $\rho$, the fundamental solution of the one-dimensional diffusion equation. For this observe that, for every $t\in [0,1]$, the Borel measure $\rho _n(t,\cdot )$ on $\mathbb {R}$ is obtained as push-forward of the counting measure $\nu _n$ under the map $W_n(\cdot ,t):\Omega _n\rightarrow \mathbb {R}$, by functoriality.

Generalizing from one-dimensional random walks $W_n:\Omega _n\rightarrow C^0([0,1],\mathbb {R})$ to d-dimensional random walks $W_n=(W_n^1,\ldots ,W_n^d):\Omega _n^d\rightarrow C^0([0,1],\mathbb {R}^d)$, $W_n(\omega _n^1,\ldots ,\omega _n^d)=(W_n^1(\omega _n^1),\ldots ,W_n^d(\omega _n^d))$ with one-dimensional random walks $W^1_n,\ldots ,W^d_n$, we find that the limiting Borel measure $\rho$ on $[0,1]\times \mathbb {R}^d$ now solves the diffusion equation on $[0,1]\times \mathbb {R}^d$ given by

$$\begin{aligned}\frac{\partial \rho }{\partial t} = \frac{1}{2}\cdot \frac{\partial ^2\rho }{\partial x^2}\,\,\text {with the Laplace operator}\,\,\frac{\partial ^2}{\partial x^2}=\frac{\partial ^2}{\partial x_1^2}+\cdots +\frac{\partial ^2}{\partial x_d^2}. \end{aligned}$$

Remark 1.1

The functional limit theorem can be rephrased by saying that the n-fold random walks $W_n:\Omega _n\times [0,1]\rightarrow \mathbb {R}$ converge in distribution as $n\rightarrow \infty$ to the Wiener process $W:\Omega \times [0,1]\rightarrow \mathbb {R}$, also called Brownian motion, where $\Omega =C^0([0,1],\mathbb {R})$ is equipped with the Wiener measure $\mu$ and one defines $W(\omega ,t)=\omega (t)$ for all $(\omega ,t)\in \Omega \times [0,1]$. For an intuitive understanding, note that, after replacing $\Omega$ by $\Omega _N=\mathbb {F}_2^N$, W agrees “up to an infinitesimal error” with $W_N$, where N is an arbitrary “unlimited” (hyper)natural number number, i.e., the Wiener process is a random walk with “infinitesimal” step width $\frac{1}{N}$. Indeed this idea was made fully rigorous in the framework of nonstandard analysis using the concept of Loeb measures, see [3].

3 Random Hamiltonian systems and the Fokker–Planck equation

While the above relation between random walks and the diffusion equation can be generalized from $\mathbb {R}^d$ to arbitrary Riemannian manifolds Q, see [19], based on the definition of the Laplace operator for Riemannian manifolds and piecewise geodesic paths, in this paper we restrict our focus to random walks and diffusions on $\mathbb {T}^d$ which are simply obtained by passing to the quotient in the target. While it is hence an interesting project to generalize the results of this paper from $T^*\mathbb {T}^d$ to the cotangent bundle $T^*Q$ of an arbitrary Riemannian manifold, let $H:\mathbb {T}^1\times T^*\mathbb {T}^d\rightarrow \mathbb {R}$ be a time-periodic Hamiltonian function on the cotangent bundle of the d-dimensional torus $\mathbb {T}^d=\mathbb {R}^d/\mathbb {Z}^d$, where we set $H_t:=H(t,\cdot )$, and fix some diffusion constant $\sigma \in \mathbb {R}$.

Definition 2.1

Given H and $\sigma$ as above, we call $u=(u_n)_{n\in \mathbb {N}}$ with $u_n=(q_n,p_n):\Omega _n^d\times [0,1]\rightarrow T^*\mathbb {T}^d$ and $u_n(\omega _n,1)=u_n(\omega _n,0)$ for every $\omega _n\in \Omega _n^d$ a sequence of closed random walk Hamiltonian orbits if for every $n\in \mathbb {N}$ and for every $\omega _n\in \Omega _n^d$ we have

$$\begin{aligned} q_n(\omega _n,t)-q_n(\omega _n,0)= & {} \int _0^t \frac{\partial H_t}{\partial p}(q_n(\omega _n,\tau ),p_n(\omega _n,\tau ))\,\textrm{d}\tau + \sigma \cdot W_n(\omega _n,t),\\ \frac{\partial p_n}{\partial t}(\omega _n,t)= & {} -\frac{\partial H_t}{\partial q}(q_n(\omega _n,t),p_n(\omega _n,t)). \end{aligned}$$

Analogous to Sect. 1, every sequence of (closed) random walk Hamiltonian orbits $u=(u_n)_{n\in \mathbb {N}}$ defines a sequence of Dirac measures $\mu _n^u:=\nu _n\circ u_n^{-1}$, $n\in \mathbb {N}$ on $C^0([0,1],T^*\mathbb {T}^d)$. Assume for the moment that $(u_n)_{n\in \mathbb {N}}$ converges in distribution in the sense that the corresponding sequence of Dirac measures $(\mu _n^u)_{n\in \mathbb {N}}$ converges as Borel measures to a Borel measure $\mu ^u$ on $C^0([0,1],T^*\mathbb {T}^d)$. Then the distribution $\rho ^u$ on $[0,1]\times T^*\mathbb {T}^d$, corresponding to the Borel measure obtained again as pushforward of $\lambda \otimes \mu ^u$ under the canonical evaluation map $[0,1]\times C^0([0,1],T^*\mathbb {T}^d)\rightarrow [0,1]\times T^*\mathbb {T}^d$, satisfies the periodicity condition $\rho ^u(0,\cdot )=\rho ^u(1,\cdot )$ and is a solution of the following Hamiltonian version of the Fokker–Planck equation (or forward Kolmogorov equation or drift-diffusion equation)

$$\begin{aligned} \frac{\partial \rho ^u}{\partial t}=-\frac{\partial }{\partial q}\left( \frac{\partial H_t}{\partial p}\cdot \rho ^u\right) +\frac{\partial }{\partial p}\left( \frac{\partial H_t}{\partial q}\cdot \rho ^u\right) +\frac{\sigma ^2}{2}\cdot \frac{\partial ^2\rho ^u}{\partial q^2}.\end{aligned}$$

(2)

Note that the latter equation combines the Hamiltonian version of the continuity equation modelling the change of $\rho ^u$ under drift with the heat equation from Sect. 1 modelling the change of $\rho ^u$ under diffusion. To see that $\rho ^u$ is indeed a solution of the Hamiltonian Fokker–Planck equation, observe that the equation is equivalent to

$$\begin{aligned}D_t\rho ^u:=\frac{\partial \rho ^u}{\partial t}+\frac{\partial H_t}{\partial p}\cdot \frac{\partial \rho ^u}{\partial q}-\frac{\partial H_t}{\partial q}\cdot \frac{\partial \rho ^u}{\partial p}=\frac{\sigma ^2}{2}\cdot \frac{\partial ^2\rho ^u}{\partial q^2},\end{aligned}$$

where $D_t\rho ^u$ denotes the material derivative describing the change of the $\rho ^u$ under the influence of the drift given by the Hamiltonian vector field.

Remark 2.2

When $\sigma =0$ and when $u_n(\omega _n,\cdot )\equiv u=(q,p)$ for all $n\in \mathbb {N}$, $\omega _n\in \Omega _n^d$, then $\mu _n^u=\mu ^u=\delta _u$ is the Dirac measure localized at $u\in C^0([0,1],T^*\mathbb {T}^d)$ and $\rho ^u(t,q,p)=\delta (q-q(t))\cdot \delta (p-p(t))$, and it is immediate to check that $\rho ^u$ solves Eq. (2) with $\sigma =0$, that is, the continuity equation.

Since for every $n\in \mathbb {N}$ the stochastic process $W_n$ extends from [0, 1] to the entire real line in such a way that $W_n(\cdot ,t_2+1)-W_n(\cdot ,t_1+1)$ and $W_n(\cdot ,t_2)-W_n(\cdot ,t_1)$ both agree in distribution for all $t_1\le t_2$, $\rho ^u$ indeed extends to a measure on $\mathbb {R}\times T^*\mathbb {T}^d$ satisfying the periodicity condition $\rho ^u(t+1,\cdot )=\rho ^u(t,\cdot )$ for all $t\in \mathbb {R}$, see also [7, Proposition 4.2]. By generalizing methods from Hamiltonian Floer theory we show the following main result of this paper.

Theorem 2.3

Assume that the time-periodic Hamiltonian $H:\mathbb {T}^1\times T^*\mathbb {T}^d\rightarrow \mathbb {R}$ is of the form $H_t(q,p)=\frac{1}{2}|p|^2+F_t(q,p)$ with a smooth, time-periodic function $F_{t+1}=F_t$ with finite $C^1$-norm, and let $\sigma \in \mathbb {R}$ be arbitrary. Then there exist $d+1$ $=$ cuplength of (the loop space of) $\mathbb {T}^d$ different sequences $u=(u_n)_{n\in \mathbb {N}}$ of contractible closed random walk Hamiltonian orbits in the sense of Definition 2.1. After passing to a suitable subsequence, they converge in distribution to $d+1$ different limiting time-periodic probability measures $\rho ^u$ on $[0,1]\times T^*\mathbb {T}^d$. In particular, we obtain at least $d+1$ different solutions of the corresponding Hamiltonian Fokker–Planck equation.

Apart from the fact that we expect that this statement can be proven for a larger class of Hamiltonians as long as they fulfill a suitable quadratic growth condition in the cotangent fibre, we think that there are many generalizations and elaborations which present the opportunity for further work:

• It is an interesting open question whether the results of this paper can be elaborated to establish the existence of $d+1$ different solutions of the Hamiltonian stochastic differential equation

  $$\begin{aligned} \textrm{d}q(\omega ,t)= & {} \frac{\partial H_t}{\partial p}(q(\omega ,t),p(\omega ,t))\,\textrm{d}t + \sigma \cdot \textrm{d}W_t(\omega ),\\ \frac{\partial p}{\partial t}(\omega ,t)= & {} -\frac{\partial H_t}{\partial q}(q(\omega ,t),p(\omega ,t)) \end{aligned}$$

  on the probability space $\Omega ^d=C^0([0,1],\mathbb {R}^d)$ which are strong in the sense of [17, Section 5.3]. Since a precise formulation of the statement as well as of the proof would require some substantial extra theoretical background, we decided to focus on the convergence of probability distributions. Following up on Remark 1.1, as a first step it seems very promising to combine the convergence result in Theorem 2.3 with nonstandard analysis methods from [3] to start with establishing weak solutions in the sense of [17, Section 5.3], where the original probability space $\Omega ^d$ is replaced by the nonstandard probability space $\Omega ^d_N=\mathbb {F}_2^{N\cdot d}$ for a suitable “unlimited” (hyper)natural number N.

• Going even further, it seems to be a very natural but also challenging project to develop a generalization of Hamiltonian Floer theory which can directly be used to prove the existence of periodic solutions of the Hamiltonian Fokker–Planck equation or even of the above Hamiltonian stochastic differential equations, without using the approximation via random walks. If successful, it might be possible to define an extension of the full Hamiltonian Floer homology and to prove a nondegenerate version of our statement using Betti numbers instead of cuplength estimates as lower bounds. To the author’s knowledge, this would require to incorporate Malliavin calculus, see [16], into Floer theory to make sure that periodic solutions as well as the Floer curves depend smoothly on the elements in the underlying probability space.

• More on the side of possible further applications, it is noteworthy to mention that the incorporation of Brownian motion methods into classical mechanics and field theories is the cornerstone of all stochastic approaches to quantum mechanics and quantum field theories, see the Euclidean quantum mechanics of Zambrini et al. [20], the stochastic quantization of Parisi–Wu and other Euclidean path integral quantization methods, see [13].

• Finally, more on the geometric side, it is an interesting project to generalize the above theorem from time-periodic random walk Hamiltonian systems on $T^*\mathbb {T}^d$ to those on the cotangent bundle $T^*Q$ of other Riemannian manifolds Q. While it seems promising to first consider the case when a global orthonormal frame exists as in the case of Lie groups, the latter might be combined with a symplectic group action to generalize our result appropriately to more general classes of symplectic manifolds.

In view of these broader questions about the interplay between Hamiltonian systems with random walks and solutions of Fokker–Planck equations on more general symplectic manifolds, this paper focuses on a proof using methods from Hamiltonian Floer theory, where our main aim is to illustrate how the weak notion of convergence in distribution and the limiting Brownian motion with its almost surely non-differentiable sample paths can still be incorporated in the analytical framework of Hamiltonian Floer theory. Let $n\in \mathbb {N}$ be arbitrary. Instead of looking for closed random walk Hamiltonian orbits $u_n=(q_n,p_n):\Omega _n^d\times [0,1]\rightarrow T^*\mathbb {T}^d$ in the sense of Definition 2.1, we make use of the fact that we can equally well look for random Hamiltonian orbits ${\bar{u}}_n=({\bar{q}}_n,p_n):\Omega _n^d\times [0,1]\rightarrow T^*\mathbb {T}^d$ with boundary condition $({\bar{q}}_n,p_n)(\omega _n,1)=\phi ^{\omega _n}_1( ({\bar{q}}_n,p_n)(\omega _n,0))$ for the $\omega _n$-dependent symplectic flow

$$\begin{aligned}\phi ^{\omega _n}_t: T^*\mathbb {T}^d\rightarrow T^*\mathbb {T}^d,\,\,\phi ^{\omega _n}_t(q,p)=(q-\sigma \cdot W_n(\omega _n,t),p),\end{aligned}$$

solving the Hamiltonian equation

$$\begin{aligned} \frac{\partial {\bar{q}}_n}{\partial t}(\omega _n,t)=\frac{\partial K^{\omega _n}_t}{\partial p}({\bar{u}}_n(\omega _n,t)),\,\,\frac{\partial p_n}{\partial t}(\omega _n,t)=-\frac{\partial K^{\omega _n}_t}{\partial q}({\bar{u}}_n(\omega _n,t))\end{aligned}$$

(3)

for the $\omega _n$-dependent time-dependent Hamiltonian

$$\begin{aligned}K^{\omega _n}_t=H_t\circ (\phi ^{\omega _n}_t)^{-1}:T^*\mathbb {T}^d\rightarrow \mathbb {R},\,\, K^{\omega _n}_t({\bar{q}},p)=H_t({\bar{q}}+\sigma \cdot W_n(\omega _n,t),p)\end{aligned}$$

with $K^{\omega _n}_{t+1}=K^{\omega _n}_t\circ (\phi ^{\omega _n}_1)^{-1}$. Here the relation between $q_n$ and ${\bar{q}}_n$ is given by

$$\begin{aligned}{\bar{q}}_n(\omega _n,t)=q_n(\omega _n,t)-\sigma \cdot W_n(\omega _n,t).\end{aligned}$$

The motivation for this step is the following: While $W_n(\omega _n,\cdot )$ is only continuous and we hence need to assume the same for $u_n(\omega _n,\cdot )$, each of the Hamiltonian orbits ${\bar{u}}_n(\omega _n,\cdot )$ can be assumed to be differentiable for each $\omega _n\in \Omega _n^d$ as their partial derivatives solve Eq. (3).

For fixed $\omega _n\in \Omega _n$, recall from classical Hamilton dynamics that the Hamiltonian orbits ${\bar{u}}_n(\omega _n,\cdot )$ are critical points of the symplectic actions

$$\begin{aligned}\int _0^1 \left( p_n(\omega _n,t)\partial _t{\bar{q}}_n(\omega _n,t)-K^{\omega _n}_t({\bar{q}}_n(\omega _n,t),p_n(\omega _n,t))\right) \,\textrm{d}t. \end{aligned}$$

Since $\Omega _n$ is a finite set for every $n\in \mathbb {N}$, it immediately follows that the random Hamiltonian orbits ${\bar{u}}_n:\Omega _n^d\times [0,1]\rightarrow T^*\mathbb {T}^d$ are precisely the critical points of the corresponding expectation values

$$\begin{aligned}\mathbb {E}\int _0^1 \left( p_n(\omega _n,t)\partial _t{\bar{q}}_n(\omega _n,t)-K^{\omega _n}_t({\bar{q}}_n(\omega _n,t),p_n(\omega _n,t))\right) \,\textrm{d}t \end{aligned}$$

on the space of paths $({\bar{q}}_n,p_n):\Omega _n^d\times [0,1]\rightarrow T^*\mathbb {T}^d$ satisfying the $\omega _n$-dependent boundary condition $({\bar{q}}_n,p_n)(\omega _n,1)=\phi ^{\omega _n}_1(({\bar{q}}_n,p_n)(\omega _n,0))$. Here $\mathbb {E}$ denotes the expectation value with respect to the counting measure $\nu _n$ on $\Omega _n^d=\mathbb {F}_2^{n\cdot d}$, that is, a weighted finite sum over all elements of $\Omega _n$. As in the non-random setting, the $d+1$ different random Hamiltonian orbits as claimed in Theorem 2.3 are distinguished by their random symplectic action, by studying $L^2$-gradient flow lines of this symplectic action, also called Floer curves, see below.

Since the Hamiltonian orbits ${\bar{u}}_n(\omega _n,\cdot )$ can be assumed to be differentiable for each $\omega _n\in \Omega _n^d$, every sequence of Hamiltonian orbits ${\bar{u}}=({\bar{u}}_n)_{n\in \mathbb {N}}$ defines a sequence of Dirac measures $({\bar{\mu }}_n^{{\bar{u}}})_{n\in \mathbb {N}}$ by setting ${\bar{\mu }}_n^{{\bar{u}}}:=\nu _n\circ {\bar{u}}_n^{-1}$ on $C^1([0,1],T^*\mathbb {T}^d)$ for every $n\in \mathbb {N}$. Apart from the existence result for sequences of random Hamiltonian orbits ${\bar{u}}=({\bar{u}}_n)_{n\in \mathbb {N}}$, the other main finding is that there is a subsequence that converges in distribution, that is, after passing to a suitable subsequence, the sequence of Borel measures $({\bar{\mu }}_n^{{\bar{u}}})_{n\in \mathbb {N}}$ converges to a limiting Borel measure ${\bar{\mu }}^{{\bar{u}}}$ on $C^1([0,1],T^*\mathbb {T}^d)$. Using the continuous map $C^1([0,1],T^*\mathbb {T}^d)\times C^0([0,1],\mathbb {T}^d)\rightarrow C^0([0,1],T^*\mathbb {T}^d)$, $({\bar{u}},W)\mapsto {\bar{u}}+(\sigma \cdot W,0)$, the Borel measure $\mu ^u$, obtained via pushforward of the Borel measure ${\bar{\mu }}^{{\bar{u}}}\otimes \mu$ ($\mu$ = Wiener measure), is the limit of the Dirac measures $\mu _n^u=\nu _n\circ u_n^{-1}$ given by $u=(u_n)_{n\in \mathbb {N}}$. Since the random symplectic action

$$\begin{aligned}{} & {} \mathbb {E}\int _0^1 \left( p_n(\omega _n,t)\partial _t{\bar{q}}_n(\omega _n,t)-K^{\omega _n}_t({\bar{q}}_n(\omega _n,t),p_n(\omega _n,t))\right) \,\textrm{d}t \\{} & {} \quad = \mathbb {E}\int _0^1 \left( p_n(\omega _n,t)\partial _t{\bar{q}}_n(\omega _n,t)-H_t(q_n(\omega _n,t),p_n(\omega _n,t))\right) \,\textrm{d}t\end{aligned}$$

can be written as

$$\begin{aligned}{} & {} \int _{C^1}\int _0^1 p(t)\partial _t{\bar{q}}(t)\,\textrm{d}t\,\textrm{d}{\bar{\mu }}_n^{{\bar{u}}} - \int _{C^0} \int _0^1 H_t(q(t),p(t))\,\textrm{d}t\,\textrm{d}\mu _n^u \\{} & {} \quad =\int _{C^0}\int _0^1 \left( p(t)\frac{\partial H_t}{\partial p}(q(t),p(t))- H_t(q(t),p(t))\right) \,\textrm{d}t\,\textrm{d}\mu _n^u\\{} & {} \quad =\int _0^1\int _{T^*\mathbb {T}^d} \left( p\frac{\partial H_t}{\partial p}(q,p)- H_t(q,p)\right) \,\rho ^u_n(t,q,p) \,\textrm{d}p\,\textrm{d}q \,\textrm{d}t, \end{aligned}$$

using the integral over all paths $({\bar{q}},p)$ in $C^1([0,1],T^*\mathbb {T}^d)$ equipped with the Dirac measure ${\bar{\mu }}^{{\bar{u}}}_n$ and over all paths (q, p) in $C^0([0,1],T^*\mathbb {T}^d)$ equipped with the Dirac measure $\mu ^u_n$, respectively, after passing to the subsequence as above, the random symplectic actions converge to

$$\begin{aligned}{} & {} \int _{C^1}\int _0^1p(t)\partial _t{\bar{q}}(t)\,\textrm{d}t\,\textrm{d}{\bar{\mu }}^{{\bar{u}}}-\int _{C^0}\int _0^1 H_t(q(t),p(t))\,\textrm{d}t\,\textrm{d}\mu ^u \\{} & {} \quad =\int _{C^0}\int _0^1 \left( p(t)\frac{\partial H_t}{\partial p}(q(t),p(t))- H_t(q(t),p(t))\right) \,\textrm{d}t\,\textrm{d}\mu ^u \\{} & {} \quad =\int _0^1\int _{T^*\mathbb {T}^d} \left( p\frac{\partial H_t}{\partial p}(q,p)- H_t(q,p)\right) \,\rho ^u(t,q,p) \,\textrm{d}p\,\textrm{d}q \,\textrm{d}t, \end{aligned}$$

where the Dirac measures ${\bar{\mu }}^{{\bar{u}}}_n$, $\mu ^u_n$, $\rho ^u_n$ are replaced by the limiting Borel measure ${\bar{\mu }}^{{\bar{u}}}$, $\mu ^u$, $\rho ^u$ respectively. To show that the limiting Borel measures obtained from the $d+1$ different sequences of time-periodic random walk Hamiltonian orbits still can be distinguished using their symplectic actions, we show that the Floer curves used to distinguish the $d+1$ sequences of random walk Hamiltonian orbits converge as well, possibly after passing to a further subsequence, in the distributional Gromov–Floer sense of Lemma 3.4.

Apart from the use of fractional Sobolev spaces, the main technical input that we use is the following well-known result about the regularity of sample paths of Brownian motion, see [15, Corollary 1.20].

Proposition 2.4

With respect to the Wiener measure the following holds true: For every $0<\alpha <\frac{1}{2}$, a path $\omega \in \Omega ^d=C^0(\mathbb {R},\mathbb {R}^d)$ is almost surely Hölder continuous with Hölder exponent $\alpha$, i.e., there exists $c_\alpha >0$ such that $|\omega (t_2)-\omega (t_1)|\le c_\alpha \cdot |t_2-t_1|^\alpha$ for every $t_1,t_2\in \mathbb {R}$. In other words, the corresponding subspace of Hölder continuous functions has full Wiener measure.

4 Hamiltonian Floer theory with diffusion

The proof consists of the following steps, where for each $\omega _n\in \Omega _n^d$, $n\in \mathbb {N}$ we set $q_n^{\omega _n}(t):=q_n(\omega _n,t)$ and $p_n^{\omega _n}(t):=p_n(\omega _n,t)$, where we refer to [2, 4, 18] for more details on the underlying Hamiltonian Floer theory and [11] for the necessary modifications in the case of boundary conditions twisted by a symplectomorphism.

The case $F_t\equiv 0$: In this case the diffusive Hamiltonian equations simplify to

$$\begin{aligned}q_n^{\omega _n}(t)-q_n^{\omega _n}(0)=\int _0^t p_n^{\omega _n}(\tau )\,\textrm{d}\tau +\sigma \cdot W_n(\omega _n,t),\,\,\frac{\partial p_n^{\omega _n}}{\partial t}(t)=0,\end{aligned}$$

which is equivalent to

$$\begin{aligned}p_n^{\omega _n}(t)=p_n^{\omega _n}(0),\,\, q_n^{\omega _n}(t)=q_n^{\omega _n}(0)+t\cdot p_n^{\omega _n}(0)+\sigma \cdot W_n(\omega _n,t),\,\,t\in [0,1].\end{aligned}$$

Furthermore $(q_n^{\omega _n},p_n^{\omega _n}):[0,1]\rightarrow T^*\mathbb {T}^d$ satisfies the boundary condition $(q_n^{\omega _n},p_n^{\omega _n})(1)=(q_n^{\omega _n},p_n^{\omega _n})(0)$ precisely when

$$\begin{aligned}p_n^{\omega _n}(0)-\sigma \cdot W_n(\omega _n,1)\in \mathbb {Z}^d,\,\,q_n^{\omega _n}(0)\in \mathbb {T}^d\,\,\text {arbitrary.}\end{aligned}$$

Also taking into account that we are interested in contractible solutions, we arrive at

$$\begin{aligned}p_n^{\omega _n}(0)=\sigma \cdot W_n(\omega _n,1),\,\,q_n^{\omega _n}(0)\in \mathbb {T}^d\,\,\text {arbitrary.}\end{aligned}$$

In particular, we note that there is an entire manifold of stochastic time-dependent solutions satisfying the boundary condition, and the cuplength estimates holds in the Morse–Bott sense, i.e., after adding a sufficiently small Morse function on $\mathbb {T}^d$.

Moduli spaces of Floer curves: To prove the existence of $d+1$ (= cuplength of $\mathbb {T}^d$) different solutions $({\bar{q}}_n^{\omega _n},p_n^{\omega _n}):[0,1]\rightarrow T^*\mathbb {T}^d$ using Hamiltonian Floer theory, we now follow the standard strategy, see e.g. [2, 18]; since the details are standard as well, we only outline the key steps.

Since the Hamiltonian $K_t^{\omega _n}=K^0+G_t^{\omega _n}$ with $K^0(q,p)=\frac{1}{2}|p|^2$, $G_t^{\omega _n}=F_t\circ (\phi ^{\omega _n}_t)^{-1}$ as well as the boundary condition $({\bar{q}}_n^{\omega _n},p_n^{\omega _n})(1)=\phi ^{\omega _n}_1(({\bar{q}}_n^{\omega _n},p_n^{\omega _n})(0))$ are depending on the paths in Wiener space, we introduce for every $\omega _n\in \Omega _n^d$, $n\in \mathbb {N}$ a corresponding moduli space ${\mathcal {M}}_n^{\omega _n}={\mathcal {M}}_n^{\omega _n}(F,\sigma )$ of Floer curves, for its definition see below. To be able to employ a maximum principle for proving compactness for moduli spaces of Floer curves, we start with the following standard auxiliary result.

Lemma 3.1

There exists $R>0$ depending on $|W_n(\omega _n,\cdot )|$ such that $K_t^{\omega _n}(q,p)=\frac{1}{2}|p|^2+G_t^{\omega _n}(q,p)$ and ${\bar{K}}_t^{\omega _n}(q,p)=\frac{1}{2}|p|^2+{\bar{G}}_t^{\omega _n}(q,p)$, ${\bar{G}}_t^{\omega _n}(q,p)=\chi _R(|p|)\cdot G_t^{\omega _n}(q,p)$ with the cut-off function $\chi _R:[0,\infty )\rightarrow \mathbb {R}$, $\chi _R(s)=1$ for $s\le R$, $\chi _R(s)=0$ for $s\ge R+1$, have the same Hamiltonian orbits with symplectic action $\le \frac{1}{2}(\sigma \cdot W_n(\omega _n,1))^2+4\Vert F\Vert _{C^1}$.

Proof

We start by noting that the dependence on $|W_n(\omega _n,1)|$ is directly related to the given bound on the symplectic action. Since the symplectic action of a Hamiltonian orbit ${\bar{u}}=({\bar{q}},p)$ of $K_t^{\omega _n}$ is given by

$$\begin{aligned}\int _0^1 \left( \frac{1}{2}p(t)^2+p(t)\frac{\partial G_t^{\omega _n}}{\partial p}({\bar{q}}(t),p(t))-G_t^{\omega _n}({\bar{q}}(t),p(t))\right) \,\textrm{d}t, \end{aligned}$$

it follows from the fact that $F_t$ and hence $G_t^{\omega _n}$ has bounded $C^1$-norm that the symplectic action grows quadratically with the $L^2$-norm of the p-component of the Hamiltonian orbit $({\bar{q}}(t),p(t))$. With the bound on the symplectic action in place, it follows that we get a bound on this $L^2$-norm. Using the Hamiltonian equation we get a bound on the Sobolev $W^{1,2}$-norm which in turn leads to a bound of the p-component in the supremum norm. $\square$

For every $1\le j\le d$ consider the submanifold $C_j=\mathbb {T}^{j-1}\times \{0\}\times \mathbb {T}^{d-j+1}\subset \mathbb {T}^d$. Using the intersection product of homology classes, note that $[C_1]\cdot \cdots \cdot [C_d]=[\text {point}]$, or equivalently, $\text {PD}[C_1]\cup \cdots \cup \text {PD}[C_d]$ is equal to the canonical volume form on $\mathbb {T}^d$. Further we introduce, smoothly depending on $\tau >0$, the smooth cut-off function $\varphi _{\tau }:\mathbb {R}\rightarrow [0,1]$ with $\varphi _{\tau }=0$ for $\tau =0$ and $\varphi _{\tau }(s)=1$ for $0<s<\tau$ and $\varphi _{\tau }(s)=0$ for $s<-1$ and $s>\tau +1$ for $\tau >0$ large. Then for every $n\in \mathbb {N}$ the $\omega _n$-dependent moduli space ${\mathcal {M}}_n^{\omega _n}$ is defined as the set

$$\begin{aligned}{\mathcal {M}}_n^{\omega _n}=\bigcup _{\tau >0}{\mathcal {M}}_{n,\tau }^{\omega _n},\,\,{\mathcal {M}}_{n,\tau }^{\omega _n}=\{{\widetilde{u}}:\mathbb {R}\times [0,1]\rightarrow T^*\mathbb {T}^d: (*1),(*2),(*3),(*4)\}\end{aligned}$$

of Floer curves satisfying the $\omega _n$-dependent Floer equation

$$\begin{aligned}(*1):\,\, {\overline{\partial }}^{\omega _n,\tau }_{J,{\bar{K}}}({\tilde{u}}):=\,\partial _s{\tilde{u}}+J_t({\tilde{u}})\partial _t{\tilde{u}}+\nabla K^0({\tilde{u}})+\varphi _{\tau }(s)\cdot \nabla {\bar{G}}^{\omega _n}_t({\tilde{u}})=0,\end{aligned}$$

the boundary condition

$$\begin{aligned}(*2):\,\,{\widetilde{u}}(s,1)=\phi ^{\omega _n}_1({\widetilde{u}}(s,0))\,\,\text {for every}\,\,s\in \mathbb {R},\end{aligned}$$

and, for ${\widetilde{u}}=({\widetilde{q}},{\widetilde{p}})$, the asymptotic condition

$$\begin{aligned}(*3):\,\,{\widetilde{p}}(s,t)\rightarrow \sigma \cdot W_n(\omega _n,1), \,\,{\widetilde{q}}(s,t)\rightarrow q+ t\cdot \sigma \cdot W_n(\omega _n,1)\,\text {for some}\,q\in \mathbb {T}^d, \end{aligned}$$

that is, the Floer curve converges to a solution for the case $F_t=0$ as $s\rightarrow \pm \infty$. Here $J_t$ denotes a family of almost complex structure on $T^*\mathbb {T}^d$ satisfying the periodicity condition $(\phi ^{\omega _n}_1)^*J_{t+1}=J_t$. Finally we demand the intersection property

$$\begin{aligned}(*4): {\tilde{q}}\left( \frac{j}{d}\cdot \tau ,0\right) \in C_j\,\,\text {for every}\,\,j=1,\ldots ,d.\end{aligned}$$

To prove that ${\mathcal {M}}^{\omega _n}_n$ carries the structure of a one-dimensional manifold one uses that for every $\tau >0$ the submoduli space ${\mathcal {M}}^{\omega _n}_{n,\tau }$ is the zero set of the nonlinear Floer operator ${\overline{\partial }}^{\omega _n,\tau }_{J,{\bar{K}}}$, viewed as a section in the Banach space bundle ${\mathcal {E}}^{k,p}_{\omega _n}$ over the Banach manifold ${\mathcal {B}}^{k+1,p}_{\omega _n}$ with $k=0,1,2,\ldots$ and $p>2$. Here ${\mathcal {B}}^{k+1,p}_{\omega _n}$ consists of $W^{k+1,p}_{\text {loc}}$-maps ${\widetilde{u}}:\mathbb {R}\times [0,1]\rightarrow T^*\mathbb {T}^d$ satisfying $(*2)$, $(*3)$, $(*4)$, while the fibre ${\mathcal {E}}^{k,p}_{\omega _n,{\widetilde{u}}}$ over ${\widetilde{u}}\in {\mathcal {B}}^{k+1,p}_{\omega _n}$ is the linear Banach space $W^{k,p}_{\phi ^{\omega _n}_1}(\mathbb {R}\times [0,1],\mathbb {R}^{2d})$ of $W^{k,p}$-maps satisfying $(*2)$. To prove that ${\overline{\partial }}^{\omega _n,\tau }_{{\bar{K}}}$ defines a nonlinear Fredholm operator, one shows that the linearization $D_{{\widetilde{u}}}: T_{{\widetilde{u}}}{\mathcal {B}}^{k+1,p}_{\omega _n}\rightarrow {\mathcal {E}}^{k,p}_{\omega _n,{\widetilde{u}}}$ is a linear Fredholm operator for every ${\widetilde{u}}\in {\mathcal {M}}^{\omega _n}_{n,\tau }$.

One of the main ingredients is to show that the gradient ${\widetilde{u}}\mapsto \nabla {{\bar{K}}}^{\omega }_t({\widetilde{u}})$ defines a bounded linear map from $T_{{\widetilde{u}}}{\mathcal {B}}^{k,p}_{\omega _n}$ into ${\mathcal {E}}^{k,p}_{\omega _n,{\widetilde{u}}}$, that is, from some $W^{k,p}$-space into another $W^{k,p}$-space. Since $W_n(\omega _n,\cdot )$ is Lipschitz continuous and hence an element of $W^{1,\infty }([0,1],\mathbb {R}^d)$, the space of Hölder continuous functions with Hölder exponent 1, using $K^{\omega _n}_t({\bar{q}},p)=H_t({\bar{q}}+\sigma \cdot W_n(\omega _n,t),p)$ and the embedding of $W^{1,\infty }$ into $W^{1,p}$ it follows that $\nabla K^{\omega _n}_t$ and hence $\nabla {\bar{K}}^{\omega _n}_t$ defines a bounded linear map $T_{{\widetilde{u}}}{\mathcal {B}}^{k,p}_{\omega _n}$ into ${\mathcal {E}}^{k,p}_{\omega _n,{\widetilde{u}}}$ for $k=0,1$. Summarizing we find that ${\mathcal {M}}^{\omega _n}_n$ is a subset of the Banach manifold ${\mathcal {B}}^{k+1,p}_{\omega _n}$ with $k=0,1$ and $p>2$, in particular, we get that each ${\widetilde{u}}$ is an $W^{2,p}$-map and hence at least $C^1$, i.e., differentiable in the classical sense.

Now a standard transversality argument shows, possibly after slightly perturbing the family of almost complex structures $J_t$, that ${\mathcal {M}}_n^{\omega _n}$ is a one-dimensional manifold which is non-empty, since for $\tau =0$ the moduli subspace ${\mathcal {M}}_n^{\omega _n,0}$ contains precisely one element. Since we employ the cut-off Hamiltonian ${\bar{G}}_t^{\omega _n}(q,p)=\chi _R(|p|)\cdot G_t^{\omega _n}(q,p)$ instead of the original Hamiltonian $G_t^{\omega _n}$, it follows that the standard $C^0$-bounds for Floer curves in cotangent bundles are available, see e.g. [8], which in turn implies that also the standard Gromov–Floer compactness is in place. Note that for the latter we use that the energy is uniformly bounded by twice the Hofer norm of F, see [14, Proposition 9.1.4], which in turn is bounded by twice the $C^1$-norm of F. Hence it follows that ${\mathcal {M}}_n^{\omega _n,\tau }$ is non-empty for every $\tau >0$ as ${\mathcal {M}}_n^{\omega _n}$ is compact up to breaking of cylinders. Compactifying the moduli space ${\mathcal {M}}_n^{\omega _n}$ in the Gromov–Floer sense, we find in the limit $\tau \rightarrow \infty$ for every $j=1,\ldots ,d$ a Floer map ${\widetilde{u}}^{\omega _n,j}_n=({\widetilde{q}}^{\omega _n,j}_n,{\widetilde{p}}^{\omega _n,j}_n):\mathbb {R}\times [0,1]\rightarrow T^*\mathbb {T}^d$ satisfying the boundary condition $(*2)$ and the other conditions $(*1)$, $(*3)$, and $(*4)$ being replaced by

$$\begin{aligned}&(*1'):\,\, {\overline{\partial }}^{\omega _n}_{J,{\bar{K}}}({\widetilde{u}}^{\omega _n,j}_n):=\,\partial _s{\widetilde{u}}^{\omega _n,j}_n+J_t({\widetilde{u}}^{\omega _n,j}_n)\partial _t{\widetilde{u}}^{\omega _n,j}_n+\nabla {\bar{K}}^{\omega _n}_t({\widetilde{u}}^{\omega _n,j}_n)=0,&\\&(*3'):\,\,{\widetilde{u}}^{\omega _n,j}_n(s_{k,\pm },\cdot )\rightarrow ({\bar{q}}^{\omega _n,j}_{n,\pm },p^{\omega _n,j}_{n,\pm })\,\,\text {for sequences}\,\,s_{k,\pm }\rightarrow \pm \infty \,\,\text {as}\,\,k\rightarrow \infty ,&\\&(*4'): {\widetilde{q}}^{\omega _n,j}_n(0,0)\in C_j.&\end{aligned}$$

Here $({\bar{q}}^{\omega _n,j}_{n,\pm },p^{\omega _n,j}_{n,\pm }):[0,1]\rightarrow T^*\mathbb {T}^d$ is a solution of the Hamiltonian equation for the Hamiltonian ${\bar{K}}^{\omega _n}_t$ with boundary condition $({\bar{q}}^{\omega _n,j}_{n,\pm }(1),p^{\omega _n,j}_{n,\pm }(1))=\phi ^{\omega _n}_1({\bar{q}}^{\omega _n,j}_{n,\pm }(0),p^{\omega _n,j}_{n,\pm }(0))$. From $(*4')$ it follows that $({\bar{q}}^{\omega _n,j}_{n,-},p^{\omega _n,j}_{n,-})$ and $({\bar{q}}^{\omega _n,j}_{n,+},p^{\omega _n,j}_{n,+})$ can be distinguished by their symplectic action given by

$$\begin{aligned} {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,\pm },p^{\omega _n,j}_{n,\pm })=\int _0^1 \left( d{\bar{K}}^{\omega _n}_t\cdot p\frac{\partial }{\partial p}-{\bar{K}}^{\omega _n}_t\right) ({\bar{q}}^{\omega _n,j}_{n,\pm }(t),p^{\omega _n,j}_{n,\pm }(t))\,\textrm{d}t \end{aligned}$$

Due to the asymptotic condition $(*3)$ in the definition of the moduli spaces ${\mathcal {M}}_n^{\omega _n}$, it follows from [14, Proposition 9.1.4] that the above actions differ from the symplectic action $\frac{1}{2}(\sigma \cdot W_n(\omega _n,1))^2$ of the asymptotic orbit for $F_t\equiv 0$ by at most twice the Hofer norm |||F|||. Since the latter is bounded by twice the $C^1$-norm of F, it follows from our choice of auxiliary Hamiltonian in Lemma 3.1 that $({\bar{q}}^{\omega _n,j}_{n,-},p^{\omega _n,j}_{-,n})$ and $({\bar{q}}^{\omega _n,j}_{+,n},p^{\omega _n,j}_{+,n})$ are indeed Hamiltonian orbits for the original Hamiltonian $K_t^{\omega _n}$ with $K_t^{\omega _n}(q,p)=\frac{1}{2}|p|^2+G_t^{\omega _n}(q,p)$ and we have

$$\begin{aligned}{\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,\pm },p^{\omega _n,j}_{n,\pm })=\int _0^1 \left( dK^{\omega _n}_t\cdot p\frac{\partial }{\partial p}-K^{\omega _n}_t\right) ({\bar{q}}^{\omega _n,j}_{n,\pm }(t),p^{\omega _n,j}_{n,\pm }(t))\,\textrm{d}t\end{aligned}$$

More precisely we have

$$\begin{aligned} {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,1}_{n,-},p^{\omega _n,1}_{n,-})< & {} {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,1}_{n,+},p^{\omega _n,1}_{n,+})\\\le & {} {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,2}_{n,-},p^{\omega _n,2}_{n,-})<\cdots <{\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,d}_{n,+},p^{\omega _n,d}_{n,+})\end{aligned}$$

which in turn implies that there at least $d+1$ different contractible solutions.

Tight family of measures and Gromov–Floer compactness: It remains to show that the random Hamiltonian orbits ${\bar{u}}^j_{\pm }=({\bar{u}}^j_{n,\pm })_{n\in \mathbb {N}}$, with ${\bar{u}}^j_{n,\pm }(\omega _n,t)={\bar{u}}^{\omega _n,j}_{n,\pm }(t)=({\bar{q}}^{\omega _n,j}_{n,\pm }(t),p^{\omega _n,j}_{n,\pm }(t))$ for $(\omega _n,t)\in \Omega _n^d\times [0,1]$, converge in distribution as $n\rightarrow \infty$ in the sense that the corresponding Dirac measures ${\bar{\mu }}^{{\bar{u}}^j_\pm }_n$ converge, possibly after passing to a subsequence. Furthermore, to show that the resulting limiting Borel measures ${\bar{\mu }}^{{\bar{u}}^j_\pm }$ are different, we further show a corresponding statement for the families of Floer curves ${\widetilde{u}}^{\omega _n,j}_n$ connecting ${\bar{u}}^{\omega _n,j}_{n,-}$ and ${\bar{u}}^{\omega _n,j}_{n,+}$ for each $j=1,\ldots ,d$.

Lemma 3.2

For every $\epsilon >0$ there exists a compact subset $C^1_{\epsilon }$ of $C^1([0,1], T^*\mathbb {T}^d)$ such that for each $j=1,\ldots ,d$ we have ${\bar{\mu }}^{{\bar{u}}^j_\pm }_n(C^1_{\epsilon })\ge 1-\epsilon$ for n sufficiently large. In particular, after passing to a subsequence, ${\bar{\mu }}^{{\bar{u}}^j_\pm }_n$ converges to some Borel measure ${\bar{\mu }}^{{\bar{u}}^j_\pm }$ on $C^1([0,1],T^*\mathbb {T}^d)$ as $n\rightarrow \infty$.

Note that the first half of the statement can be rephrased as (asymptotical) tightness of the family of probability measures. Since tight families of probability measures are well-known to be compact, see e.g. [6, Theorem 25.10], the second half of the statement indeed follows from the first.

Proof

Let $\epsilon >0$ be arbitrary. By Lemma 3.3 we know that the space $W^{\frac{1}{4},\infty }([0,1],\mathbb {T}^d)$ of Hölder continuous functions with Hölder exponent $\frac{1}{4}<\frac{1}{2}$ has full Wiener measure $\mu$. Denoting by $W^{\frac{1}{4},\infty }_B([0,1],\mathbb {T}^d)$ the subspace of functions with $W^{\frac{1}{4},\infty }$-norm less than or equal to B, and using that $\mu$ is the limit of the Dirac measures $\mu _n=\nu _n\circ W_n^{-1}$, we find $B>0$ and $n_0\in \mathbb {N}$ such that $\mu _n(W^{\frac{1}{4},\infty }_B([0,1],\mathbb {T}^d))\ge 1-\epsilon$ for $n\ge n_0$. Now all that remains to be shown is that there exists ${\bar{B}}>0$ with the following property for all $n\in \mathbb {N}$, $\omega _n\in \Omega _n^d$: If $W_n(\omega _n,\cdot )\in W^{\frac{1}{4},\infty }_B([0,1],\mathbb {T}^d)$, then ${\bar{u}}^{\omega _n,j}_{n,\pm }\in C^1_{\epsilon }$ for $j=1,\ldots ,d$ with the compact subset $C^1_{\epsilon }:=W^{1\frac{1}{4},\infty }_{{\bar{B}}}([0,1],T^*\mathbb {T}^d)$ of all maps in $C^1([0,1],T^*\mathbb {T}^d)$ with $W^{1\frac{1}{4},\infty }$-norm less than or equal to ${\bar{B}}$. As a first step we observe that, since the $W^{\frac{1}{4},\infty }$-norm of $W_n(\omega _n,\cdot )$ dominates its $C^0$-norm, it follows from Lemma 3.1 that there is a bound for the $C^0$-norm of ${\bar{u}}^{\omega _n,j}_{n,\pm }$ which just depends on the chosen $B>0$. On the other hand, using $J_t({\bar{u}}^{\omega _n,j}_{n,\pm })\partial _t{\bar{u}}^{\omega _n,j}_{n,\pm }+\nabla H_t({\bar{u}}^{\omega _n,j}_{n,\pm }+(\sigma W_n(\omega _n,\cdot ),0))=0$ it follows that the $W^{\frac{1}{4},\infty }$-norm of $\partial _t{\bar{u}}^{\omega _n,j}_{n,\pm }$ is uniformly bounded as well, again depending on $B>0$. $\square$

Recall that we have shown above for every $n\in \mathbb {N}$, $\omega _n\in \Omega _n^d$ that the orbits ${\bar{u}}^{\omega _n,j}_{n,\pm }$ are pairwise different as they can be ordered via their symplectic action. Here the crucial strict inequality is that for each $j=1,\ldots ,d$ we have ${\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,-},p^{\omega _n,j}_{n,-})<{\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,+},p^{\omega _n,j}_{n,+})$, which follows from the existence of the Floer map ${\widetilde{u}}^{\omega _n,j}_n:\mathbb {R}^2\rightarrow T^*\mathbb {T}^d$ connecting ${\bar{u}}^{\omega _n,j}_{n,-}=({\bar{q}}^{\omega _n,j}_{n,-},p^{\omega _n,j}_{n,-})$ and ${\bar{u}}^{\omega _n,j}_{n,+}=({\bar{q}}^{\omega _n,j}_{n,+},p^{\omega _n,j}_{n,+})$ in the sense of $(*3')$. To establish that the symplectic actions for ${\bar{u}}^{\omega _n,j}_{n,-}$ and ${\bar{u}}^{\omega _n,j}_{n,+}$ are different from each other, we crucially use that the Floer curve must be nontrivial due to $(*4')$, i.e., it must intersect a given homology cycle. To see that this argument carries through to the limit as $n\rightarrow \infty$, we show that the Floer maps ${\widetilde{u}}^{\omega _n,j}_n$ themselves converge in distribution.

But before we can state the corresponding statement and prove it, we first need the following technical result about the Cauchy–Riemann operator ${\overline{\partial }}_J({\widetilde{u}})=\partial _s{\widetilde{u}}+J_t({\widetilde{u}})\partial _t{\widetilde{u}}$.

Lemma 3.3

Fix some real number $0\le \alpha \le 1$ and $p>2$. For every $S>0$ there exists $c>0$ such that for every $W^{1,p}$-map ${\widetilde{u}}:[-S,+S]\times [0,1]\rightarrow T^*\mathbb {T}^d$ we have

$$\begin{aligned} \Vert {\widetilde{u}}\Vert _{\alpha +1,p}\le c\left( \Vert {\overline{\partial }}_J({\widetilde{u}})\Vert _{\alpha ,p}+\Vert {\widetilde{u}}\Vert _{0,p}\right) , \end{aligned}$$

where $\Vert \cdot \Vert _{\alpha ,p}$ denotes the $W^{\alpha ,p}$-norm.

Proof

In [14, B.3.4] it is shown that the result holds for $\alpha =0$ and $\alpha =1$. Now it follows from the classical interpolation theory, see [1, 7.22] or [5, Definition 2.4.1], that the same holds true for every $0\le \alpha \le 1$, i.e., when ${\overline{\partial }}_J$ is viewed as a map from $W^{\alpha +1,p}$ to $W^{\alpha ,p}$. Here we would like to remark that fractional Sobolev spaces $W^{\alpha ,p}$ and $W^{\alpha +1,p}$ with a non-integer number of weak derivatives can be either defined using Fourier transform on the space of tempered distributions or as interpolation space in the sense of [1, 7.5.7] or [5, Definition 2.4.1] between $L^p$ and $W^{1,p}$ or $W^{1,p}$ and $W^{2,p}$, respectively, see [5, Theorem 6.4.5]. $\square$

Now, let $S>0$ be chosen arbitrary and fixed. Observe that for each $n\in \mathbb {N}$ the restricted Floer maps ${\widetilde{u}}^{\omega _n,j}_n: [-S,+S]\times [0,1]\rightarrow T^*\mathbb {T}^d$, $\omega _n\in \Omega _n^d$ now define Dirac measures ${\widetilde{\mu }}^{{\widetilde{u}}^j}_n$ on $C^1([-S,+S]\times [0,1],T^*\mathbb {T}^d)$. We can prove the following Gromov–Floer analogue of Lemma 3.2.

Lemma 3.4

For every $\epsilon >0$ there exists a compact subset ${\widetilde{C}}^1_{\epsilon }$ of $C^1([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ such that for each $j=1,\ldots ,d$ we have ${\widetilde{\mu }}^{{\widetilde{u}}^j}_n({\widetilde{C}}^1_{\epsilon })\ge 1-\epsilon$ for n sufficiently large. In particular, after passing to a subsequence, ${\widetilde{\mu }}^{{\widetilde{u}}^j}_n$ converges to some Borel measure ${\widetilde{\mu }}^{{\widetilde{u}}^j}$ on $C^1([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ as $n\rightarrow \infty$.

Proof

As in the proof of Lemma 3.2 it suffices to show is that there exists ${\bar{B}}>0$ with the following property for all $n\in \mathbb {N}$, $\omega _n\in \Omega _n^d$: If $W_n(\omega _n,\cdot )\in W^{\frac{1}{4},\infty }_B([0,1],\mathbb {T}^d)$ with $B>0$ from the proof of Lemma 3.2, then ${\widetilde{u}}^{\omega _n,j}_n\in {\widetilde{C}}^1_{\epsilon }$, $j=1,\ldots ,d$ with the compact subset ${\widetilde{C}}^1_{\epsilon }:=W^{1\frac{1}{4},p}_{{\bar{B}}}([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ of all maps in $C^1([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ with $W^{1\frac{1}{4},p}$-norm less than or equal to ${\bar{B}}$. Note that here $p>2$ is chosen large enough such that $W^{1\frac{1}{4},p}$ embeds compactly into $C^1$. As a first step we observe that, since we employ the cut-off Hamiltonian ${\bar{K}}_t^{\omega _n}(q,p)=\frac{1}{2}|p|^2+{\bar{G}}_t^{\omega _n}(q,p)$, ${\bar{G}}_t^{\omega _n}(q,p)=\chi _R(|p|)\cdot G_t^{\omega _n}(q,p)$ instead of the original Hamiltonian $K_t^{\omega _n}(q,p)=\frac{1}{2}|p|^2+G_t^{\omega _n}(q,p)$, it follows that the standard $C^0$-bounds for Floer curves in cotangent bundles from [8] are available. In particular, there is a bound for the $C^0$-norm of ${\widetilde{u}}^{\omega _n,j}_n$ which just depends on the chosen $B>0$ in view of the choice of $R>0$ in Lemma 3.1. While the uniform energy bound given by twice the Hofer norm of F is sufficient to establish uniform $L^2$-bounds for the first derivatives $T{\widetilde{u}}^{\omega _n,j}_n$, due to the fact that $W^{1,2}([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ does not embed into $C^0([-S,+S]\times [0,1],T^*\mathbb {T}^d)$, the latter is not sufficient. However, together with the $C^0$-bounds mentioned above, we now again make use of the fact that the standard Gromov–Floer compactness is in place, which in turn shows that a uniform $C^0$-bound for the first derivatives $T{\widetilde{u}}^{\omega _n,j}_n$ can be established. For the proof, note that, if the latter bound would not exist, then within the set of all restricted Floer maps ${\widetilde{u}}^{\omega _n,j}_n$ one would find a sequence which would develop a holomorphic sphere in some point in $[-S,+S]\times [0,1]$, which in turn is excluded by the exactness of the symplectic form on $T^*\mathbb {T}^d$. Since ${\widetilde{u}}^{\omega _n,j}_n$ is hence uniformly bounded with respect to the $C^1$-norm and $W_n(\omega _n,\cdot )$ is uniformly bounded with respect to the Hölder $W^{\frac{1}{4},\infty }$-norm, their sum ${\widetilde{u}}^{\omega _n,j}_n+(\sigma W_n(\omega _n,\cdot ),0)$ is uniformly bounded with respect to the $W^{\frac{1}{4},p}$-norm for every $p>2$. Since each Floer map ${\widetilde{u}}^{\omega _n,j}_n$ satisfies the Floer equation ${\overline{\partial }}_J({\widetilde{u}}^{\omega _n,j}_n)+\nabla H_t({\widetilde{u}}^{\omega _n,j}_n+(\sigma W_n(\omega _n,\cdot ),0))=0$, it follows from Lemma 3.3 that ${\widetilde{u}}^{\omega _n,j}_n$ is indeed uniformly bounded with respect to the $W^{1\frac{1}{4},p}$-norm. $\square$

To finish the proof of Theorem 2.3, we use the standard (in)equality relating the energy of the restricted Floer maps ${\widetilde{u}}^{\omega _n,j}_n$ with the symplectic action of their asymptotic orbits ${\bar{u}}^{\omega _n,j}_{n,\pm }$,

$$\begin{aligned}\int _{-S}^{+S}\int _0^1 \left| \partial _s{\widetilde{u}}^{\omega _n,j}_n(s,t)\right| ^2\,\textrm{d}t\,\textrm{d}s \,\le \, {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,+},p^{\omega _n,j}_{n,+}) - {\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,-},p^{\omega _n,j}_{n,-}),\end{aligned}$$

together with

$$\begin{aligned} \mathbb {E}{\mathcal {L}}_n^{\omega _n}({\bar{q}}^{\omega _n,j}_{n,\pm },p^{\omega _n,j}_{n,\pm })= & {} \mathbb {E}\int _0^1 \left( dK^{\omega _n}_t\cdot p\frac{\partial }{\partial p}-K^{\omega _n}_t\right) ({\bar{q}}^{\omega _n,j}_{n,\pm }(t),p^{\omega _n,j}_{n,\pm }(t))\,\textrm{d}t\\= & {} \mathbb {E}\int _0^1 \left( \textrm{d}H_t\cdot p\frac{\partial }{\partial p}-H_t\right) (q^{\omega _n,j}_{n,\pm }(t),p^{\omega _n,j}_{n,\pm }(t))\,\textrm{d}t\\= & {} \int _{C^0}\int _0^1 \left( p(t)\frac{\partial H_t}{\partial p}(q(t),p(t))- H_t(q(t),p(t))\right) \,\textrm{d}t\,\textrm{d}\mu _n^{u^{j,\pm }}\\= & {} \int _0^1\int _{T^*\mathbb {T}^d} \left( p\frac{\partial H_t}{\partial p}(q,p)- H_t(q,p)\right) \,\rho ^{u^{j,\pm }}_n(t,q,p) \,\textrm{d}p\,\textrm{d}q \,\textrm{d}t \end{aligned}$$

to deduce that the energy of ${\widetilde{\mu }}^{{\widetilde{u}}^j}_n$, that is, the expectation value of the energy of the Floer curves ${\widetilde{u}}^{\omega _n,j}_n$,

$$\begin{aligned}\mathbb {E}\int _{-S}^{+S}\int _0^1 \left| \partial _s{\widetilde{u}}^{\omega _n,j}_n(s,t)\right| ^2\,\textrm{d}t\,\textrm{d}s=\int _{C^1}\int _{-S}^{+S}\int _0^1 \left| \partial _s u(s,t)\right| ^2\,\textrm{d}t\,\textrm{d}s\,\textrm{d}{\widetilde{\mu }}^j_n\end{aligned}$$

is bounded from above by the difference ${\mathcal {L}}(\rho ^{u^{j,+}}_n)-{\mathcal {L}}(\rho ^{u^{j,-}}_n)$ of the symplectic actions of $\rho ^{u^{j,\pm }}_n$,

$$\begin{aligned}{\mathcal {L}}(\rho ^{u^{j,\pm }}_n)=\int _0^1\int _{T^*\mathbb {T}^d} \left( p\frac{\partial H_t}{\partial p}(q,p)- H_t(q,p)\right) \,\rho ^{u^{j,\pm }}_n(t,q,p) \,\textrm{d}p\,\textrm{d}q \,\textrm{d}t.\end{aligned}$$

Taking limits as $n\rightarrow \infty$, it follows that the energy of ${\widetilde{\mu }}^{{\widetilde{u}}^j}$,

$$\begin{aligned}\int _{C^1}\int _{-S}^{+S}\int _0^1 \left| \partial _s u(s,t)\right| ^2\,\textrm{d}t\,\textrm{d}s\,\textrm{d}{\widetilde{\mu }}^j\end{aligned}$$

is bounded from above by the difference ${\mathcal {L}}(\rho ^{u^{j,+}})-{\mathcal {L}}(\rho ^{u^{j,-}})$ of the symplectic actions of $\rho ^{u^{j,\pm }}$. With this we can deduce the strict inequality ${\mathcal {L}}(\rho ^{u^{j,-}})<{\mathcal {L}}(\rho ^{u^{j,+}})$: Assuming to the contrary that ${\mathcal {L}}(\rho ^{u^{j,-}})\ge {\mathcal {L}}(\rho ^{u^{j,+}})$, that is, ${\mathcal {L}}(\rho ^{u^{j,-}})={\mathcal {L}}(\rho ^{u^{j,+}})$, it would follow that

$$\begin{aligned}\int _{C^1}\int _{-S}^{+S}\int _0^1 \left| \partial _s u(s,t)\right| ^2\,\textrm{d}t\,\textrm{d}s\,\textrm{d}{\widetilde{\mu }}^j=0,\end{aligned}$$

that is, the limiting Borel measure ${\widetilde{\mu }}^j$ would have full measure on maps in $C^1([-S,+S]\times [0,1],T^*\mathbb {T}^d)$ which are independent of $s\in [-S,+S]$. But the latter is impossible in view of the intersection condition $(*4')$ imposed on all Floer curves.

Summarizing, we find that

$$\begin{aligned}{\mathcal {L}}(\rho ^{u^{1,-}})<{\mathcal {L}}(\rho ^{u^{1,+}})\le {\mathcal {L}}(\rho ^{u^{2,-}})<\cdots<{\mathcal {L}}(\rho ^{u^{d-1,+}})\le {\mathcal {L}}(\rho ^{u^{d,-}})<{\mathcal {L}}(\rho ^{u^{d,+}})\end{aligned}$$

which in turn implies that there at least $d+1$ different solutions of the Hamiltonian Fokker–Planck equation.