Spectral Analysis of a Discrete Metastable System
Driven by Lévy Flights
In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation
where is a multi-well potential with local minima and is a symmetric -stable Lévy process (Lévy flights process). We investigate the spectrum of the generator of this Markov chain in the limit and localize the top eigenvalues . These eigenvalues turn out to be of the same algebraic order and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits , , and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of .
Keywords: metastability; Lévy flights; -stable Lévy process; eigenvalues; spectral gap; Markov chain; transition times; fractional Laplacian; semiclassical limit.
AMS Subject Classification: 60J10, 60J75, 47A75, 47G20, 15B51, 15A18
1 Introduction. Spectral properties of Gaussian small noise diffusions
Let be a one-dimensional multi-well potential (see, e.g. Fig. 1) such that as for some . Assume that has exactly local minima , as well as local maxima , , enumerated in increasing order
Moreover, we suppose that all these extrema are non-degenerate, i.e.
The points , , are exponentially stable attractors of a deterministic dynamical system generated by the ordinary differential equation . For any initial position , the solution does not leave the domain of attraction and as .
Although deterministic gradient systems with multiple attractors appear naturally in various application problems (e.g. climate modelling [2, 17, 38], physics , robotics , economics ), their evident drawback often consists in an impossibility of transitions between the domains of attraction. To overcome this limitation, a small noise is often added to the model. Intuitively the perturbed random system stays in the vicinity of one of the attractors most of the time, making sporadic transitions to the other domains of attraction. This type of behaviour is often referred to as metastability. The essential characteristics of the small noise system as a metastable hopping process are the life times in the domains of attraction, transition probabilities between the wells, the most probable state etc. Naturally, these characteristics depend heavily on the type of the noisy perturbation. The case of Brownian perturbations is the most well studied.
For a standard Brownian motion , consider the SDE
Under the conditions on formulated above, for any , (1.4) has a unique strong solution (see e.g. ). Since the classical results by Kramers , it is known that the mean life times of in the potential wells are exponentially long with respect to the parameter , more precisely of the order } for the -th well. Considering a generic perturbed gradient system on exponentially long time scales of the order , , one can recover different behaviours. In particular on the slowly growing (short) scales, when is less than the life times in all the wells (, ), the trajectory does not succeed in leaving any of the domains of attraction. On longer time scales, the trajectory may leave some of the domains of attraction thus making transitions and eventually forming a sublimit distribution. On the longest scale, converges to its stationary law which is concentrated in the vicinity of the potential’s global minimum. For a generic potential, there are such critical values which are different and can be ordered in decreasing order . These values determine the time scales which distinguish the sublimit distributions. The values , , can be calculated probabilistically in terms of the heights of the potential barriers of with the help of the large deviations theory by Freidlin and Wentzell (see [21, §VI.6] or a recent exposition ).
Recall that is a strong Markov processes with the infinitesimal generator
The existence of multiple sublimit distributions is reflected in the spectral properties of the operator . The generator is a negative definite essentially self-adjoint operator in the weighted space . It has a discrete non-positive spectrum and a corresponding sequence of eigenfunctions which form an orthonormal basis (see [24, Chapter 3] for the detailed exposition). Whereas its top eigenvalue is identically zero, , with the corresponding eigenfunction being a constant, it turns out that the next eigenvalues are of the order where the rates are obtained above. The principle non-zero eigenvalue corresponding to the longest time scale determines the speed of convergence of the law of to the stationary distribution as ; further eigenvalues recover the speeds of convergence to sublimit distributions. With the help of the Fourier method of separation of variables, the functional which is the solution of the Cauchy problem , , can be represented as
The remainder term can be shown to be negligible compared to the leading terms of the expansion due to the spectral gap property, namely, that there is a constant not depending on such that for all . Moreover, the eigenfunctions , , are almost constant over the potential wells.
A rich literature is devoted to the analysis of metastability of the system of the type (1.4) on different levels of rigour. A probabilistic characterization of the principle eigenvalue of in a bounded domain with the Dirichlet boundary conditions was obtained by Khasminski  (see also Friedman [22, Lemma 1.1]). Schuss et al. [43, 35, 42] determined the shape of the eigenfunction and the asymptotics of the eigenvalue , especially its subexponential prefactor in terms of the values and with the help of formal asymptotic expansions of solutions of second order ordinary differential equations. Buslov, Makarov and Kolokoltsov [34, 10, 11, 31, 30] established a connection between the top spectrum of the diffusion’s generator and the spectrum of the matrix of the inverse mean life times of in the potential wells, found very exact approximations for the corresponding eigenfunctions and proved the existence of an -independent spectral gap. Bovier, Eckhoff, Gayrard and Klein in [20, 6, 7, 8] developed a potential theoretic approach to metastability of Markov chains and gradient diffusions and obtained very exact asymptotics of the top eigenvalues. Metastable behaviour of diffusions in a double-well potential was studied by probabilistic methods in [29, 23, 39]. Applications of the spectral theory to simulated annealing can be found in [16, 25, 26], to stochastic resonance in , and to analysis of molecular dynamics [44, 37]. Berglund and Gentz  studied the behaviour of small noise diffusions in potentials with non-quadratic extrema. We refer the reader to a recent review  by Berglund for further references on the subject.
In recent time, equations driven by non-Gaussian noise, especially -stable Lévy processes (Lévy flights), are being adopted for description and modelling of various real world phenomena (see e.g. ). Let be a symmetric -stable Lévy process with the characteristic function
where . Such a process has heavy tails and for convenience the constant is chosen to guarantee the following tail asymptotics:
For , the SDE
In the limit , the process enjoys the metastable behaviour in the following sense.
Theorem 1.1 (Theorem 1.1, ).
Let and . Then the process , , converges, as , in the sense of finite dimensional distributions to a Markov chain , , on the state space with the stable conservative generator given by
In other words, there is a time scale on which the process reminds of a finite state Markov chain . It was also shown in , that on slower time scales , , the process does not leave the potential well where it has started. On faster time scales , , one cannot obtain a meaningful limit of since its life times in the potential wells converge to zero and the process persistently jumps between different wells.
The generator of is the integro-differential operator
and the metastability result of Theorem 1.1 suggests that the top spectrum of should remind of the spectrum of the Markov chain . One can expect that the top eigenvalues of should be closely connected with the eigenvalues of the matrix and well separated from the rest of the spectrum by a spectral gap. Since the time scale in the metastability result is unique, the eigenvalues should be all of the same order .
Unfortunately not much can be said about the spectrum of . No weighted space is known where is self-adjoint and no general spectral theory can be applied in this case. Although it is known that the invariant distribution exists and is unique [41, 33], explicit formulae for its density can be obtained only in a very few particular cases (mainly for the Cauchy process and polynomial potentials, see [15, 18, 19])
In this paper we make the first step towards a better understanding of the spectral properties of by reduction of a jump-diffusion to a finite state discrete time Markov chain and analysing its spectral properties in the limit .
2 A discrete Lévy driven system and the main result
We construct a discrete-time Markov chain on a finite state space that mimic the metastable behaviour of the solution of (1.9) in the limit of small . The construction is based on the standard Euler scheme for SDEs.
For any there are a range parameter , a time step as well as the spacial mesh parameter such that and such that the following holds true.
All the local minima of belong to , . With fixed, let us redefine the points and for convenience.
There exists a finite set of points and a partition consisting of the intervals such that , each interval contains only one point from and , , , and for any
The set and the intervals can be constructed as follows. Let . Let .
First, we demand that all and set . Choose such that for all , all
For definiteness, let us construct the partition of the interval . Let be the solution of the equation , . Decompose the interval
into a finite disjoint union of the intervals , , denote by the middle point of each such an interval, and add to . Then due to the fact that on , we get . Denote the solution of the equation , and again decompose the interval a finite disjoint union of the intervals of the maximal length . Continue this procedure and assign the last interval to be .
For any let be the unique element such that
Let us denote by the mapping corresponding to the operation “ * ”, i.e. . The condition (2.1) implies that for whereas for .
On the set define a discrete time deterministic motion such that
For any the sequence is monotone.
The deterministic motion mimics the behaviour of the solutions of the deterministic ordinary differential equation . It can also be described as a discrete time Markov chain on with the matrix of one step transition probabilities given by
Eventually let us construct a discrete-time Markov chain on the state space which mimics the behaviour of the jump-diffusion and can be considered as a random perturbation of .
Denote and and define the matrix of one-step transition probabilities of as
as well as
By construction, the Markov chain gets reflected at the barriers and which mimics the very fast return from infinity of the trajectory of due to the fast increase of the potential at infinity.
First we have to study the metastable behaviour of as . Since is constructed in such a way that it resembles the Lévy driven jump diffusion one could expect that the demonstrates the same metastable behaviour. Indeed, the following analogue of the Theorem 1.1 holds true.
Let and . Then the process , , converges, as , in the sense of finite dimensional distributions to a Markov chain , , on the state space with the generator defined in (1.10).
The proof of this result essentially follows the arguments presented in  for the case of a one dimensional jump diffusion . The arguments are even easier since by construction, does not contain the local maxima of Furthermore thanks to the reflection condition at and the finiteness of , we do not have to consider returns of from infinity. Note that this convergence result does not depend on the size or the particular choice of the state space . For a detailed proof we refer the reader to Chapter 3 in . ∎
The Theorem 2.2 suggests that top spectrum of the generator of the Markov chain should be closely related to the spectrum of the matrix . Let denote the cardinality of the set . To compare the generators of and we note that
so that the matrix can be viewed as the discrete analogue of an infinitesimal generator. Taking into account the time scaling which appears in the metastability result (Theorem 2.2), we introduce the matrix
and denote by and the spectra of and , respectively. Then the following main theorem holds.
The spectrum can be divided into two disjoint parts and for which the following assertions hold:
contains precisely eigenvalues such that and
For any sequence such that , there is such that for all and all
The proof the Theorem 2.3 does not take unto account any a priori information about the eigenvalues of other than the fact that one of them is . However, we strongly believe that in the generic case, e.g. when the potential does not have any intrinsic symmetries, all eigenvalues of are real and simple. This conjecture is supported by numerous computer simulations of spectra of a matrix , when the entries of the form (1.10) are randomly generated.
Denote by , the indicator function (vector) of the set , i.e. for and elsewhere.
Assume that the eigenvalues of are real and simple. Let and let be the right eigenvector of associated with and normalized such that . Let be the right eigenvector of associated with and normalized such that . Then the following limit holds:
In particular, for , and .
Consider a three-well potential such that, for simplicity of simulations, outside of small neighbourhoods of the extrema, see Fig. 1, and fix . The matrix calculated according to (1.10) has the eigenvalues , , , and the corresponding eigenvectors , , . We construct the state space consisting of points from the interval and a Markov chain with the matrix of transition probabilities defined according to (2.8), (2.9), (2.10) with and . The spectrum of the matrix consists of 203 eigenvalues, see Fig. 2, and the top eigenvalues are , and , coincide with the spectrum of with a four decimal places precision. The eigenvector is constant on whereas the eigenvectors and converge to the values , , , on the parts of corresponding to the domains of attraction of , see Fig. 3.
3 Eigenvalues. Proof of the Theorem 2.3
In particular, for any fixed and there is such that in the limit
Let and the set and the partition be fixed. Then there is small enough and a constant such that for all the following estimates hold:
for any and any such that we have
for and we have
3.1 Proof of Theorem 2.3
In the first step we will analyse the structure of the matrix
which obviously plays the essential role in the investigation of the spectrum of .
First we observe that the matrices and possess a block structure. Indeed, recalling the decomposition of the state space , , , as a disjoint union of states belonging to different potential wells, we may write
The blocks of determine the transitions of the Markov chain between the wells and . Only the blocks , , that include the main diagonal depend on .
For the further analysis of the matrix it is helpful to study the limit behaviour of its entries as .
The following expansions follow immediately from Lemma 3.1.
Let , then
for and we have ;
for and we have , more precisely,
for we have ), more precisely,
for we have , more precisely,
As in Lemma 3.1, the bounds hidden in the Landau symbols are uniform over .
To get a better understanding of the structure of , let us for example take a detailed look at the block . If we number the states of in the increasing order by , then this block has the form
Note that the numbers that appear in the previous lemma are all positive.
Let be the generator of the Markov chain on the state space defined in (1.10). Then there exist constants and such that for every , every , ,