document: the term ` stochastic resonance ' was introduced in the early  80s ( see @xcite and @xcite ) in the study of periodic advance of glaciers on earth . 
   the stochastic resonance is the effect of nonmonotone dependence of the response of a system on the noise when this noise ( for instance the temperature ) is added to a periodic input signal ( see e.g. @xcite , in which the author explains also differences and similarities with the notion of stochastic filtering ) . 
   an extensive review on stochastic resonance and its presence in different fields of applications can be found in @xcite . 
   following @xcite , as stochastic resonance we intend the phenomenon in which the transmission of a signal can be improved ( in terms of statistical quantities ) by the addition of noise . from the statistical point of view 
   the problem is to estimate a signal @xmath0 transmitted through a channel . 
   this signal has to be detected by a receiver that can reveal signals louder than a threshold @xmath1 . 
   if @xmath2 is bounded from above by @xmath1 , the signal is not observable and the problem has not a solution . 
   but , if some noise @xmath3 is added to the signal , the perturbed signal @xmath4 may be observable and inference can be done on @xmath2 . 
   too few noise is not sufficient to give good estimates and too much noise deteriorates excessively the signal . 
   the optimal  in some sense  level of the noise will be called stochastic resonance in this framework . 
   usually ( see @xcite ) the criterion applied to measure optimality of estimators are the shannon mutual information or the kullback divergence . 
   more recently the fisher information quantity have been also proposed ( see @xcite and @xcite ) . 
   here we are concerned with the fisher information quantity . 
   it happens that this quantity , as a function of the noise , can be maximized for certain noise structures . if there is only one global maximum , the corresponding noise level is the value for which we have stochastic resonance , if several local maxima are present , the phenomenon is called stochastic multi - resonance .    in this paper 
   we study the problem of estimation and hypotheses testing for the following model : we suppose to have a threshold @xmath5 and a subthreshold constant and non negative signal @xmath6 , @xmath7 . 
   we add , in continuous time , a noise that is a trajectory of a diffusion process @xmath8 and we observe the perturbed signal @xmath9 where @xmath10 is the level of the noise . 
   we propose two schemes of observations : _ 
   i ) _ we observe only the proportion of time spent by the perturbed signal over the threshold @xmath1 and _ ii ) _ we measure the energy of the perturbed signal when it is above the threshold . 
   the asymptotic is considered as time goes to infinity . 
   this approach differs from the ones in the current statistical literature basically for two reasons : the noise structure is an ergodic diffusion process and not a sequence of independent and identically distributed random variables and data are collected in continuous time . 
   this second aspect is a substantial difference but it is not a problem from the point of view of applications for the two schemes of observations proposed if one thinks at analogical devices . 
   we propose two different estimators for the schemes and we study their asymptotic properties . 
   we present an example where , in both cases , it emerges the phenomenon of stochastic resonance . for the same model we also solve the problem of testing the simple hypothesis @xmath11 against the simple alternative @xmath12 by applying the bayesian maximum a posterior probability criterion . 
   it emerges that the overall probability of error is nonmonotonically dependent on @xmath13 . 
   we show again that there exists a non trivial local minimum of this probability that is again the effect of stochastic resonance . 
   the presence of stochastic resonance in this context is noted for the first time here .    the paper is organized as follows . in section 
   [ sec : model ] we set up the regularity assumptions of the model . in sections 
   [ sec : time ] and [ sec : energy ] we prove some asymptotic properties estimators for the two schemes and we calculate numerically the points where the fisher information quantity attains its maximum for both models . 
   it turns out that the estimators proposed are asymptotically equivalent to the maximum likelihood estimators . 
   section [ sec : test ] is devoted to the problem of hypotheses testing . 
   all the figures are collected at the end of the paper . 
   let @xmath1 be the threshold and @xmath6 a constant signal . taking @xmath14 will not influence the calculations that follows but 
   may improve the exposition , so we use this assumption . 
   let @xmath15 be a given diffusion process solution to the following stochastic differential equation @xmath16 with non random initial value @xmath17 . 
   the process @xmath15 is supposed to have the ergodic property with invariant measure @xmath18 and invariant distribution function @xmath19)$ ] as @xmath20 . 
   the functions @xmath21 and @xmath22 satisfy the global lipschitz condition @xmath23 where @xmath24 is the lipschitz constant . under condition [ cond : c1 ] 
   , equation has a unique strong solution ( see e.g. @xcite ) but any equivalent condition to [ cond : c1 ] can be assumed because we do not use explicitly it in the sequel . 
   the following conditions are needed to ensure the ergodicity of the process @xmath15 . 
   if @xmath25 and @xmath26 then there exists the stationary distribution function @xmath27 and it takes the following form @xmath28 again , any other couple of conditions that imply the existence of @xmath27 can be used instead of [ cond : c3 ] and [ cond : c3 ] . 
   we perturb the signal @xmath6 by adding , proportionally to some level @xmath10 , the trajectory diffusion process @xmath15 into the channel 
   . the result will be the perturbed signal @xmath29 . 
   this new signal will be detectable only when it is above the threshold @xmath1 . 
   moreover , @xmath30 is still ergodic with trend and diffusion coefficients respectively @xmath31 and @xmath32 and initial value @xmath33 , but we will not use directly this process . we denote by @xmath34 the observable part of the trajectory of @xmath30 , being @xmath35 the indicator function of the set @xmath36 . 
   we consider two possibile schemes of observation :    * we observe only the proportion of time spent by @xmath30 over the threshold @xmath1 @xmath37 * we measure the energy of the signal @xmath38 
   @xmath39    in the next sections , for the two models we establish asymptotic properties of estimators given by the generalized method of moments . in @xcite different properties of the generalized method of moments for ergodic diffusion processes are studied . in this note 
   we follows the lines given in the paper of@xcite for the i.i.d . setting . 
   these results are interesting in themselves independently from the problem of stochastic resonance . 
   we give an example of stochastic resonance based on the process where the phenomenon of stochastic resonance appears pronounced and in which results in a closed form can be written down . 
   the random variable @xmath40 can be rewritten in terms of the process @xmath15 as @xmath41 by the ergodic property of @xmath15 we have that @xmath42 where @xmath43 has @xmath27 as distribution function . from it 
   derives that @xmath44 so that @xmath6 is a one - to - one continuous function of @xmath45 . from the glivenko - cantelli theorem ( see e.g. @xcite ) for the empirical distribution function ( edf ) defined by @xmath46 follows directly that @xmath40 is a @xmath47-consistent estimator of @xmath45 thus also @xmath48 is a @xmath49-consistent estimator for @xmath6 . 
   we can calculate the asymptotic variance of this estimator . 
   it is known that ( see @xcite and @xcite ) the edf is asymptotically gaussian and in particular @xmath50 where @xmath51 is the inverse of the analogue of the fisher information quantity in the problem of distribution function estimation : @xmath52 where @xmath53 and @xmath54 . 
   the quantity @xmath55 is also the minimax asymptotic lower bound for the quadratic risk associated to the estimation of @xmath56 , so that @xmath57 is asymptotically efficient in this sense . 
   the asymptotic variance @xmath58 of @xmath59 can be derived by means of the so - called @xmath60-method ( see e.g. @xcite ) : @xmath61 thus @xmath62 where @xmath2 is the density of @xmath27 . 
   the quantity @xmath58 can also be derived from the asymptotic minimal variance @xmath63 of the edf estimator . 
   in fact , with little abuse of notation , by putting @xmath64 and @xmath65 we have that @xmath66      we now show that @xmath40 also maximizes the approximate likelihood of the model . 
   in fact , for the central limit theorem for the edf we have @xmath67 where @xmath68 is a standard gaussian random variable . 
   thus @xmath69 where @xmath70 is the distribution function of @xmath68 . 
   we approximate the likelihood function of @xmath40 by @xmath71 that is maximal when @xmath72 thus , the maximum likelihood estimator of @xmath6 ( constructed on the approximated likelihood ) reads @xmath73 so if the approximation above is acceptable , one can infer the optimality property of @xmath40 of having minimum variance from being also the maximum likelihood estimator .      to view the effect of stochastic resonance on the fisher information we consider a particular example . by setting @xmath74 and 
   @xmath75 the noise become a standard process solution to the stochastic differential equation @xmath76 in such a case , the ergodic distribution function @xmath27 is the gaussian law with zero mean and variance 1/2 . 
   the asymptotic variance @xmath58 assumes the following form @xmath77 where @xmath78 is the classical error function . 
   in figure [ fig1 ] it is shown that for this model there exists the phenomenon of stochastic resonance . for a fixed level of noise @xmath13 the fisher information increases as the signal @xmath6 is closer to the threshold @xmath1 . 
   for a fixed value of the signal @xmath6 , the fisher information , as a function of @xmath13 , has a single maximum , that is the optimal level of noise . 
   for example , if @xmath79 then the optimal level is @xmath80 and for @xmath81 it is @xmath82 . 
   suppose that it is possibile to observe not only the time when the perturbed process is over the threshold but also its trajectory above @xmath1 , say @xmath83 , @xmath84 . 
   we now show how it is possibile to estimate the unknown signal @xmath6 from the equivalent of the energy of the signal for @xmath85 : literally from the quantity @xmath86 we use the following general result from @xcite on the estimation of functionals of the invariant distribution functions for ergodic diffusion processes . _ 
   let @xmath87 and @xmath88 be such that @xmath89 . 
   then @xmath90 is a @xmath49-consistent estimator for @xmath91 where @xmath43 is distributed according to @xmath27_. in our case @xmath92 and @xmath93 . 
   the estimator @xmath94 can be rewritten as @xmath95 and it converges to the quantity @xmath96 that is a continuous and increasing function of @xmath6 , @xmath97 . 
   its inverse @xmath98 allows us to have again @xmath99 . by applying the @xmath60-method once again 
   , we can obtain the asymptotic variance of @xmath100 from the asymptotic variance of @xmath94 . 
   @xmath101 where @xmath102 the asymptotic variance of @xmath94 is given by ( see @xcite ) @xmath103 where @xmath104 and its inverse is also the minimal asymptotic variance in the problem of estimation of functionals for ergodic diffusion . 
   thus , the asymptotic variance of @xmath100 is given by @xmath105    * remark : * _ by the asymptotic normality of @xmath94 follows that @xmath100 is also the value that maximizes the approximate likelihood function . 
   in fact , as in the previous example , if we approximate the density function of @xmath94 with @xmath106 it is clear that @xmath100 is its maximum . _      as before , we put in evidence the phenomenon of stochastic resonance by using the process as noise . 
   the quantities involved ( @xmath107 and @xmath108 ) transform into the following @xmath109 ( from which it appears that @xmath107 is an increasing function of @xmath6 ) and @xmath110 in figure [ fig1 ] it is plotted the fisher information of the model as a function of @xmath6 and @xmath13 . 
   also in this case there is evidence of stochastic resonance . for a fixed value of @xmath6 
   is then possibile to find the optimal noise level @xmath13 . 
   for example , taking @xmath111 then we have stochastic resonance at @xmath112 and for @xmath113 , @xmath114 . 
   we now study a problem of testing two simple hypotheses for the model discussed in the previous section . as in @xcite , we apply the maximum _ a posteriori _ probability ( map ) criterion . 
   we will see that the decision rules for our model are similar to the one proposed by chapeau - blondeau in the i.i.d . setting . 
   given the observation @xmath40 we want to verify the null hypothesis that the unknown constant signal is @xmath115 against the simple alternative @xmath116 , with @xmath117 : @xmath118 suppose that , before observing @xmath40 , we have a _ 
   prior _ information on the parameter , that is @xmath119 and @xmath120 . 
   the map criterion uses the following likelihood ratio @xmath121 and the decision rule is to accept @xmath122 whenever @xmath123 ( decision @xmath124 ) or refuse it otherwise ( decision @xmath125 ) . 
   the overall probability of error is @xmath126 let now be @xmath127 then , the likelihood @xmath128 appears as @xmath129 to write explicitly the decision rule and then study the effect of stochastic resonance we have to distinguish three cases : @xmath130 , @xmath131 and @xmath132 .    1 . 
   let it be @xmath130 , then put @xmath133 then , if @xmath134 accept @xmath122 and @xmath135 . 
   if @xmath136 , then if @xmath137 reject @xmath122 otherwise accept it . in both cases @xmath138 2 . 
   let it be @xmath131 , then put @xmath139 then , if @xmath134 reject @xmath122 and @xmath140 . 
   if @xmath136 , then if @xmath137 accept @xmath122 otherwise reject it . in both cases 
   @xmath141 3 . 
   let it be @xmath132 , then put @xmath142 then , if @xmath143 reject @xmath122 otherwise accept it . in both cases 
   @xmath144      as before , we apply this method to the model . in this case 
   the variance @xmath145 , for a fixed threshold @xmath1 and noise level @xmath13 , is a non decreasing function of @xmath6 being @xmath146 only a scale factor ( figure [ fig2 ] gives a numerical representation of this statement ) . 
   thus , the @xmath147 is , in general , given by formula . 
   what is amazing is the behavior of @xmath147 . in figure [ fig3 ] 
   it is reported the graph of @xmath147 as a function of @xmath13 and @xmath116 given @xmath148 and @xmath149 . for @xmath116 around 1/2 
   the @xmath147 shows the effect of stochastic resonance . 
   so it appears that in some cases the noise level @xmath13 can reduce sensibly the overall probability of making the wrong decision . 
   this kind of behavior is non outlined in the work of @xcite . 
   * remark : * _ following the same scheme , similar results can be obtained for the model _ 
   ii ) _ when we observe the energy @xmath94 . in this case 
   it is sufficient to replace in the values of @xmath150 and @xmath151 with the quantities @xmath152 and @xmath40 with @xmath94 in the decision rule . _ 
   the use of ergodic diffusions as noise in the problem of stochastic resonance seems quite powerful . 
   characterizations of classes of ergodic process that enhance the stochastic resonance can be done ( see e.g. @xcite ) but not in a simple way as in the i.i.d . 
   case as calculations are always cumbersome . 
   the problem of a parametric non constant signal can also be treated while the full nonparametric non constant signal requires more attention and will be object for further investigations . for i.i.d . 
   observations , @xcite and @xcite considered the problem of non parametric estimation for regression models of the form @xmath153 , @xmath154 . their approach can be applied in this context 
   .    other criterion of optimality than the fisher information quantity can be used as it is usually done in information theory ( e.g. shannon mutual information or kullback divergence ) . 
   the analysis of the overall probability of error seems to put in evidence something new with respect to the current literature ( see e.g. @xcite ) . 
   it is worth noting that in a recent paper @xcite models driven by ergodic diffusions have also been used but the effect of stochastic resonance is not used to estimate parameters . 
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   [ cols="^ " , ]      as a function of @xmath13 for different values of @xmath116 given @xmath148 and @xmath149 . from left - top 
   to right - bottom : @xmath116 = 0.1 , 0.25 , 0.5 , 0.75 , 0.9 , 0.95 . 
   the bottom plot is to show the effect of stochastic resonance for @xmath116 = 0.5 . ]
summary: a subthreshold signal is transmitted through a channel and may be detected when some noise  with known structure and proportional to some level  is added to the data . 
   there is an optimal noise level , called stochastic resonance , that corresponds to the highest fisher information in the problem of estimation of the signal . as noise we consider an ergodic diffusion process and the asymptotic 
   is considered as time goes to infinity . 
   we propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing . 
   we also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples . 
   * keywords : * stochastic resonance , diffusion processes , unobservable signal detection , maximum a posterior probability . 
   * msc : * 93e10 , 62m99 .