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

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 $n\to\infty$. 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 H\"older continuous with H\"older exponent $0<\alpha<\frac{1}{2}$.

It follows from the central limit theorem that W n (·, 1) converges in distribution to N (0, 1), the normal distribution with expectation value 0 and variance 1. In the same way one finds that for each t ∈ [0, 1] we have that W n (·, t) converges in distribution to N (0, t). By the latter we mean that ν n ({ω n ∈ Ω n : W n (ω n , t) ≤ a}) converges to as n → ∞. 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 → ∞. More precisely, it is known by the functional central limit theorem that the stochastic processes W n : Ω n → C 0 ([0, 1], R) converge in distribution in the sense that the pushforward measures µ n := ν n • W −1 n on C 0 ([0, 1], R) converge as Borel measures to a limit measure µ, called the Wiener measure. Denoting by ρ n , ρ the distribution corresponding to the Borel measure on [0, 1] × R obtained as pushforward of the product measure λ ⊗ µ n , λ ⊗ µ (λ = Lebesgue measure on [0, 1]) under the canonical continuous map [0, 1] × C 0 ([0, 1], R) → [0, 1] × R, (t, W ) → (t, W (t)), we find that ρ n converges in the distributional sense to ρ, the fundamental solution of the one-dimensional diffusion equation. For this observe that, for every t ∈ [0, 1], the Borel measure ρ n (t, ·) on R is obtained as push-forward of the counting measure ν n under the map W n (·, t) : Ω n → R, by functoriality.

Random Hamiltonian systems and the Fokker-Planck equation
While the above relation between random walks and the diffusion equation can be generalized from R d to arbitrary Riemannian manifolds Q, based on the definition of the Laplace operator for Riemannian manifolds and using piecewise geodesic paths, in this paper we restrict our focus to random walks and diffusions on T d which are simply obtained by passing to the quotient in the target. Although the definition below can hence be generalized from T * T d to the cotangent bundle T * Q of an arbitrary Riemannian manifold, let H : T 1 × T * T d → R be a time-periodic Hamiltonian function on the cotangent bundle of the d-dimensional torus T d = R d /Z d , where we set H t := H(t, ·), and fix some diffusion constant σ ∈ R.
Definition 2.1. Given H and σ as above, we call u = (u n ) n∈N with u n = (q n , p n ) : Ω d n × [0, 1] → T * T d and u n (ω n , 1) = u n (ω n , 0) for every ω n ∈ Ω d n a sequence of closed random walk Hamiltonian orbits if for every n ∈ N and for every ω n ∈ Ω d n we have Analogous to Section 1, every sequence of (closed) random walk Hamiltonian orbits u = (u n ) n∈N defines a sequence of Dirac measures µ u n := ν n • u −1 n , n ∈ N on C 0 ([0, 1], T * T d ). Assume for the moment that (u n ) n∈N converges in distribution in the sense that the corresponding sequence of Dirac measures (µ u n ) n∈N converges as Borel measures to a Borel measure µ u on C 0 ([0, 1], T * T d ). Then the distribution ρ u on [0, 1] × T * T d , corresponding to the Borel measure obtained again as pushforward of λ ⊗ µ u under the canonical evaluation and is a solution of the following Hamiltonian version of the Fokker-Planck equation (or forward Kolmogorov equation or drift-diffusion equation) Note that the latter equation combines the Hamiltonian version of the continuity equation modelling the change of ρ u under drift with the heat equation from Section 1 modelling the change of ρ u under diffusion. In order to see that ρ u is indeed a solution of the Hamiltonian Fokker-Planck equation, observe that the equation is equivalent to where D t ρ u denotes the material derivative describing the change of the ρ u under the influence of the drift given by the Hamiltonian vector field.
Since for every n ∈ N the stochastic process W n extends from [0, 1] to the entire real line in such a way that W n (·, t 2 + 1)− W n (·, t 1 + 1) and W n (·, t 2 )− W n (·, t 1 ) both agree in distribution for all t 1 ≤ t 2 , ρ u indeed extends to a measure on R×T * T d satisfying the periodicity condition ρ u (t+1, ·) = ρ u (t, ·) for all t ∈ R. 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 : with a smooth, time-periodic function F t+1 = F t with finite C 1 -norm, and let σ ∈ R be arbitrary. Then there exist d+1 = cuplength of (the loop space of ) T d different sequences u = (u n ) n∈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 ρ u on [0, 1] × T * 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 clearly 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, following the comment at the beginning of this section we also expect that the above theorem can be suitably generalized from time-periodic random walk Hamiltonian systems on T * T d to those on the cotangent bundle T * Q of other Riemannian manifolds Q, possibly under additional restrictions such as the existence of a global orthonormal frame. 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. 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.
Remark 2.4. Following up on Remark 1.1, our result can be used to establish the existence of d + 1 different solutions of the Hamiltonian stochastic differential equation .
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. Using the convergence result in Theorem 2.3 combined again with nonstandard analysis methods, the aforementioned solutions however can again be obtained by replacing the natural numbers n in the sequences of closed random walk Hamiltonian orbits by a suitable "unlimited" (hyper)natural number H as in Remark 1.1.
Let n ∈ N be arbitrary. Instead of looking for closed random walk Hamiltonian orbits u n = (q n , p n ) : Ω d n ×[0, 1] → T * T d in the sense of Definition 2.1, we make use of the fact that we can equally well look for random Hamiltonian orbitsū n = (q n , p n ) : Ω d n × [0, 1] → T * T d with boundary condition (q n , p n )(ω n , 1) = φ ωn 1 ((q n , p n )(ω n , 0)) for the ω n -dependent symplectic flow solving the Hamiltonian equation for the ω n -dependent time-dependent Hamiltonian Here the relation between q n andq n is given byq Note that, as in classical Hamilton theory, the random Hamiltonian orbits u n : Ω d n × [0, 1] → T * T d are precisely the critical points of the random symplectic action on the space of paths (q n , p n ) : Ω d n × [0, 1] → T * T d satisfying the ω ndependent boundary condition (q n , p n )(ω n , 1) = φ ωn 1 ((q n , p n )(ω n , 0)), where E denotes the expectation value with respect to the counting measure ν n on Ω d n = F n·d 2 . 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.
While W n (ω n , ·) is only continuous, each of the Hamiltonian orbits u n (ω n , ·) can be assumed to be differentiable for each ω n ∈ Ω d n . Hence every sequence of Hamiltonian orbitsū = (ū n ) n∈N defines a sequence of Dirac measures (μū n ) n∈N by settingμū n := ν n •ū −1 n on C 1 ([0, 1], T * T d ) for every n ∈ N. Apart from the existence result for sequences of random Hamiltonian orbitsū = (ū n ) n∈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 (μū n ) n∈N converges to a limiting Borel measureμū on C 1 ([0, 1], T * T d ). Using the continuous map the Borel measure µ u , obtained via pushforward of the Borel measureμū ⊗ µ (µ = Wiener measure), is the limit of the Dirac measures µ u n = ν n •u −1 n given by u = (u n ) n∈N . Since the random symplectic action E 1 0 (p n (ω n , t)∂ tqn (ω n , t) − K ωn t (q n (ω n , t), p n (ω n , t))) dt = E 1 0 (p n (ω n , t)∂ tqn (ω n , t) − H t (q n (ω n , t), p n (ω n , t))) dt can be written as using the integral over all paths (q, p) in C 1 ([0, 1], T * T d ) equipped with the Dirac measureμū n and over all paths (q, p) in C 0 ([0, 1], T * T d ) equipped with the Dirac measure µ u n , respectively, after passing to the subsequence as above, the random symplectic actions converge to where the Dirac measuresμū n , µ u n , ρ u n are replaced by the limiting Borel measureμū, µ u , ρ u respectively. In order 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 a distributional Gromov-Floer sense.
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 ( [10], corollary 1.20).

Hamiltonian Floer theory with diffusion
The proof consists of the following steps, where for each ω n ∈ Ω d n , n ∈ N we set q ωn n (t) := q n (ω n , t) and p ωn n (t) := p n (ω n , t), where we refer to [4], [2], [12] for more details on the underlying Hamiltonian Floer theory and [9] for the necessary modifications in the case of boundary conditions twisted by a symplectomorphism.
Moduli spaces of Floer curves: In order to prove the existence of d + 1 (= cuplength of T d ) different solutions (q ωn n , p ωn n ) : [0, 1] → T * T d using Hamiltonian Floer theory, we now follow the standard strategy, see e.g. [2], [12]; since the details are standard as well, we only outline the key steps.
Since the Hamiltonian K ωn t = K 0 +G ωn t with K 0 (q, p) = 1 2 |p| 2 , G ωn t = F t • (φ ωn t ) −1 as well as the boundary condition (q ωn n , p ωn n )(1) = φ ωn 1 ((q ωn n , p ωn n )(0)) are depending on the paths in Wiener space, we introduce for every ω n ∈ Ω d n , n ∈ N a corresponding moduli space M ωn n = M ωn n (F, σ) of Floer curves. In order 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 (ω n , ·)| such that K ωn t (q, p) = 1 2 |p| 2 + G ωn t (q, p) andK ωn t (q, p) = 1 2 |p| 2 +Ḡ ωn t (q, p),Ḡ ωn t (q, p) = χ R (|p|) · G ωn t (q, p) with the cut-off function χ R : [0, ∞) → R, χ R (s) = 1 for s ≤ R, χ R (s) = 0 for s ≥ R + 1, have the same Hamiltonian orbits with symplectic action ≤ 1 2 (σ · W n (ω n , 1)) 2 + 4 F C 1 . Proof. We start by noting that the dependence on |W n (ω n , 1)| is directly related to the given bound on the symplectic action. Since the symplectic action of a Hamiltonian orbitū = (q, p) of K ωn t is given by it follows from the fact that F t and hence G ωn t has bounded C 1 -norm that the symplectic action grows quadratically with the L 2 -norm of the p-component of the Hamiltonian orbit (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.  u(s, 0)) for every s ∈ R, and, for u = ( q, p), the asymptotic condition ( * 3) : p(s, t) → σ · W n (ω n , 1), q(s, t) → q + t · σ · W n (ω n , 1) for some q ∈ T d , that is, the Floer curve converges to a solution for the case F t = 0 as s → ±∞. Here J t denotes a family of almost complex structure on T * T d satisfying the periodicity condition (φ ωn 1 ) * J t+1 = J t . Finally we demand the intersection property ( * 4) :q j d · τ, 0 ∈ C j for every j = 1, . . . , d.
In order to prove that M ωn n carries the structure of a one-dimensional manifold one uses that for every τ > 0 the submoduli space M ωn n,τ is the zero set of the nonlinear Floer operator ∂ (R × [0, 1], R 2d ) of W k,p -maps satisfying ( * 2). In order to prove that ∂ ωn,τ K defines a nonlinear Fredholm operator, one shows that the linearization D u : T u B k+1,p ωn → E k,p ωn, u is a linear Fredholm operator for every u ∈ M ωn n,τ . One of the main ingredients is to show that the gradient u → ∇K ω t ( u) defines a bounded linear map from T u B k,p ωn into E k,p ωn, u , that is, from some W k,p -space into another W k,p -space. Since W n (ω n , ·) is Lipschitz continuous and hence an element of W 1,∞ ([0, 1], R d ), the space of Hölder continuous functions with Hölder exponent 1, using K ωn t (q, p) = H t (q + σ · W n (ω n , t), p) and the embedding of W 1,∞ into W 1,p it follows that ∇K ωn t and hence ∇K ωn t defines a bounded linear map T u B k,p ωn into E k,p ωn, u for k = 0, 1. Summarizing we find that M ωn n is a subset of the Banach manifold B k+1,p ωn with k = 0, 1 and p > 2, in particular, we get that each u is an W 2,p -map and hence at least C 1 , i.e., differentiable in the classical sense.
Tight family of measures and Gromov-Floer compactness: It remains to show that the random Hamiltonian orbitsū j ± = (ū j n,± ) n∈N , with u j n,± (ω n , t) =ū ωn,j n,± (t) = (q ωn,j n,± (t), p ωn,j n,± (t)) for (ω n , t) ∈ Ω d n × [0, 1], converge in distribution as n → ∞ in the sense that the corresponding Dirac measures µū j ± n converge, possibly after passing to a subsequence. Furthermore, in order to show that the resulting limiting Borel measuresμū j ± are different, we further show a corresponding statement for the families of Floer curves u ωn,j n connectingū ωn,j n,− andū ωn,j n,+ for each j = 1, . . . , d.

Lemma 3.2.
For every ǫ > 0 there exists a compact subset C 1 ǫ of C 1 ([0, 1], T * T d ) such that for each j = 1, . . . , d we haveμū j ± n (C 1 ǫ ) ≥ 1 − ǫ for n sufficiently large. In particular, after passing to a subsequence,μū j ± n converges to some Borel measureμū 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 ǫ > 0 be arbitrary. By Lemma 3.3 we know that the space W ,∞ -norm less than or equal to B, and using that µ is the limit of the Dirac measures µ n = ν n • W −1 n , we find B > 0 and n 0 ∈ N such that µ n (W Recall that we have shown above for every n ∈ N, ω n ∈ Ω d n that the orbitsū ωn,j n,± are pairwise different as they can be ordered via their symplectic action. Here the crucial strict inequality is that for each j = 1, . . . , d we have L ωn n (q ωn,j n,− , p ωn,j n,− ) < L ωn n (q ωn,j n,+ , p ωn,j n,+ ), which follows from the existence of the Floer map u ωn,j n : R 2 → T * T d connectingū ωn,j n,− = (q ωn,j n,− , p ωn,j n,− ) and u ωn,j n,+ = (q ωn,j n,+ , p ωn,j n,+ ) in the sense of ( * 3 ′ ). In order to establish that the symplectic actions forū ωn,j n,− andū ωn,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. In order to see that this argument carries through to the limit as n → ∞, we show that the Floer maps u ωn,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   Proof. As in the proof of Lemma 3.2 it suffices to show is that there exists B > 0 with the following property for all n ∈ N, ω n ∈ Ω d n : If W n (ω n , ·) ∈ W 4 ,p -norm less than or equal toB. Note that here p > 2 is chosen large enough such that W 1 1 4 ,p embeds compactly into C 1 . As a first step we observe that, since we employ the cutoff HamiltonianK ωn t (q, p) = 1 2 |p| 2 +Ḡ ωn t (q, p),Ḡ ωn t (q, p) = χ R (|p|)·G ωn t (q, p) instead of the original Hamiltonian K ωn t (q, p) = 1 2 |p| 2 + G ωn t (q, p), it follows that the standard C 0 -bounds for Floer curves in cotangent bundles from [7] are available. In particular, there is a bound for the C 0 -norm of u ωn,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 u ωn,j n , due to the fact that , 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 u ωn,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 u ωn,j n one would find a sequence which would develop a holomorphic sphere in some point in [−S, +S] × [0, 1], which in turn is excluded by the exactness of the symplectic form on T * T d . Since u ωn,j n is hence uniformly bounded with respect to the C 1 -norm and W n (ω n , ·) is uniformly bounded with respect to the Hölder W 1 4 ,∞ -norm, their sum u ωn,j n + (σW n (ω n , ·), 0) is uniformly bounded with respect to the W 1 4 ,p -norm for every p > 2. Since each Floer map u ωn,j n satisfies the Floer equation ∂ J ( u ωn,j n ) + ∇H t ( u ωn,j n + (σW n (ω n , ·), 0)) = 0, it follows from Lemma 3.3 that u ωn,j n is indeed uniformly bounded with respect to the W 1 1 4 ,p -norm.
Taking limits as n → ∞, it follows that the energy of µ u j , |∂ s u(s, t)| 2 dt ds d µ j is bounded from above by the difference L(ρ u j,+ ) − L(ρ u j,− ) of the symplectic actions of ρ u j,± . With this we can deduce the strict inequality L(ρ u j,− ) < L(ρ u j,+ ): Assuming to the contrary that L(ρ u j,− ) ≥ L(ρ u j,+ ), that is, L(ρ u j,− ) = L(ρ u j,+ ), it would follow that