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the astrophysically ubiquitous keplerian accretion disks should be unstable and turbulent in order to explain observed data , but are remarkably rayleigh stable . they are found in active galactic nuclei ( agns ) , around a compact object in binary systems , around newly formed stars etc . ( see , e.g. , * ? ? ? the main puzzle of accreting material in disks is its inadequacy of molecular viscosity to transport them towards the central object . thus the idea of turbulence and , hence , turbulent viscosity has been proposed . similar issue is there in certain shear flows , e.g. plane couette flow , which are shown to be linearly stable for any reynolds number ( @xmath0 ) but in laboratory could be turbulent for @xmath0 as low as @xmath1 . therefore , linear perturbation can not induce the turbulent viscosity to transport matter inwards and angular momentum outwards , in the keplerian disks . note that the issue of linear instability of the couette - taylor flow ( when accretion disks are the subset of it ) is a century old problem . although in the presence of vertical shear and/or stratification , keplerian flow may reveal rayleigh - taylor type instability ( e.g. @xcite ) , convective overstability ( @xcite ) and the zombie vortex instability ( @xcite ) , we intend here to solve the classic century old problem of the origin of linear instability with the exponential growth of perturbation in purely hydrodynamical rayleigh - stable flows with only radial shear . the convective overstability does not correspond to an indefinitely growing mode and it has some saturation ( @xcite ) . in addition , the zombie vortex instability is not sufficient to transport angular momentum significantly in a small domain of study . in fact , all of them could exhibit only smaller shakura - sunyaev viscosity parameter ( @xcite ) @xmath2 than that generally required to explain observation . the robustness of our work is that , it can explain the turbulent behavior of any kind of rayleigh - stable shear flows , starting from laboratory to astrophysical flows . while many realistic non - magnetized and keplerian flows could be stratified in both the vertical and radial directions of the disks , it is perhaps impossible to prove that all the non - magnetized accretion disks have significant amount of vertical shear and/or stratification to sustain the above mentioned instabilities . note that indeed many accretion disks are geometrically thin . moreover , the laboratory taylor - couette flows have no vertical shear and/or stratification .    in 1991 , with the application of magnetorotational instability ( mri ; * ? ? ? * ; * ? ? ? * ) to keplerian disks , @xcite showed that initial weak magnetic field can lead to the perturbations growing exponentially . within a few rotation times , such exponential growth could reveal the onset of turbulence . however , for charge neutral flows mri should not work . note also that for flows having strong magnetic fields , where the magnetic field is tightly coupled with the flow , mri is not expected to work ( e.g. @xcite ) . it is a long standing controversy ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ) , whether the matter in rayleigh stable astrophysical disks is stable or unstable . the answer has profound significance for our understanding of how stars and planets form . it is argued , however , that some types of rayleigh stable flows certainly can be destabilized @xcite . based on ` shearing sheet ' approximation , without @xcite and with @xcite explicit viscosity , some authors attempted to tackle the issue of turbulence in hot accretion disks . however , other authors argued for limitations in this work @xcite . based on the simulations including explicit viscosity , the authors could achieve @xmath3 and concluded that keplerian like flows could exhibit very weak turbulence in the absence of magnetic field . nevertheless , the recent experimental results by @xcite clearly argued for the significant level of transport from hydrodynamics alone . moreover , the results from direct numerical simulations @xcite and exploration of transient amplification , in otherwise linearly stable flows , with and without noise ( e.g. * ? ? ? * ; * ? ? ? * ) also argued for ( plausible ) hydrodynamic instability and turbulence at low @xmath0 . interestingly , accretion disks have huge @xmath0 ( @xmath4 ) @xcite , prompting to the belief that they are hydrodynamically unstable . we show here that linearly perturbed apparently rayleigh stable flows driven stochastically can be made unstable even in the absence of any magnetic field . we also argue , why stochastic noise is inevitable in such flows . they exist in the flows under consideration inherently . we develop our theory following the seminal concept based on fluctuating hydrodynamics of randomly stirred fluid , pioneered by @xcite and @xcite , which , however , was never applied in the context of accretion flows or other shear flows . this work provides a new path of linear hydrodynamic instability of shear flows , which will have vast applications from accretion disks to laboratory flows , for the first time . the plan of the paper is the following . in the next section , we introduce equations describing the system under consideration . then 3 describes the evolution of various perturbations in stochastically driven hydrodynamic flows . subsequently , we discuss the relevance of white noise in the context of shear flows in  4 . finally we summarize with conclusions in  5 . in appendix , we demonstrate in detail the generation of white noise from random walk , particularly in the present context . the linearized navier - stokes equation in the presence of background plane shear @xmath5 and angular velocity @xmath6 , when @xmath7 being the distance from the center of the system , in a small section approximated as incompressible flow with @xmath8 , has already been established @xcite . here , any length is expressed in units of the size @xmath9 of the system in the @xmath10direction , the time in units of @xmath11 , the velocity in @xmath12 ( @xmath13 ) , and other variables are expressed accordingly ( see , e.g. , @xcite , for detailed description of the choice of coordinate in a small section ) . hence , in dimensionless units , the linearized navier - stokes equation and continuity equation ( for an incompressible flow ) can be recasted into the well - known orr - sommerfeld and squire equations , but in the presence of stochastic noise and coriolis force @xcite , given by    @xmath14    @xmath15    where @xmath16 are the components of noise arising in the linearized system due to stochastic forcing such that @xmath17 @xcite , where @xmath18 is a constant for white noise and @xmath19 ; @xmath20 is the @xmath21-component of velocity perturbation vector and @xmath22 the @xmath21-component of vorticity perturbation vector . now , we can resort to a fourier series expansion of @xmath20 , @xmath22 and @xmath23 as @xmath24 where @xmath25 can be any one of @xmath20 , @xmath22 and @xmath23 ; @xmath26 and @xmath27 are the wavevector and frequency respectively in the fourier space such that @xmath28 and @xmath29=@xmath30 . writing down equations ( [ hydroorrv ] ) and ( [ hydroorrzeta ] ) in fourier space by using equation ( [ fourier ] ) , and taking ensemble average , we obtain the equations involving the evolution of mean values of perturbations in the presence of noise as @xmath31    @xmath32    where fourier transformations of @xmath16 are basically @xmath33 multiplied with a random number and on ensemble average it appears to be a constant @xmath34 which is the mean value of the white noise ( we get @xmath35 and @xmath36 when the drift coefficient of the brownian motion or wiener process corresponding to the white noise is zero and nonzero respectively , see appendix for details ) , and @xmath37 , @xmath38 are the fourier transforms of @xmath39 and @xmath40 which are the mean or the ensemble averaged values of @xmath20 and @xmath22 respectively . now let us take the trial solutions , @xmath41 , where @xmath42 are the constant , in general complex , amplitudes of perturbation and @xmath43 , is a vertical wavevector ( one should not confuse this @xmath44 with the shakura - sunyaev viscosity parameter ) . vertical wave vector is chosen since it will be unaffected by shear @xcite . this gives @xmath45 ( using equation ( [ fourier ] ) ) . substituting these trial solutions in equations ( [ hydrofourierumean ] ) and ( [ hydrofourierzetamean ] ) , integrating with respect to @xmath26 and @xmath27 we obtain    @xmath46    @xmath47      now eliminating @xmath34 and assuming @xmath48 we obtain the dispersion relation @xmath49 if we find any pair of @xmath44 and @xmath50 satisfying equation ( [ vertmeli ] ) for which the imaginary part of @xmath50 positive , then we can say that the mean value of perturbation is unstable . equation ( [ vertmeli ] ) is the hydrodynamic counter part of the dispersion relation obtained due to mri @xcite , leading to the avenue of pure hydrodynamic instability . for @xmath35 , from equations ( [ hydroudispersion ] ) and ( [ hydrozetadispersion ] ) , either @xmath51 and @xmath52 both turn out to be zero or there is no instability for non - trivial @xmath51 and @xmath52 . overall , @xmath35 gives rise to stable solutions like the zero magnetic field for mri . figure [ vertm ] shows the ranges of @xmath44 giving rise to linear instability . it is easy to understand that similar results could be obtained with the choice of unequal ensemble averages of white noise in equations ( [ hydroudispersion ] ) and ( [ hydrozetadispersion ] ) and a more general phase difference between @xmath52 and @xmath51 .     and the imaginary part of @xmath50 , for vertical perturbation in case i with @xmath53 . we consider @xmath54 , however we obtain almost the same results for other admissible values of @xmath55 . ]    1.5 cm      now for a given @xmath51 and @xmath34 , after eliminating @xmath52 from equations ( [ hydroudispersion ] ) and ( [ hydrozetadispersion ] ) , we obtain a dispersion relation between @xmath44 and @xmath50 as @xmath56 which is second order in @xmath50 and hence has two roots @xmath57 and @xmath58 . if we find any pair of @xmath44 and @xmath50 for which the imaginary part of @xmath50 positive , then we can say that the mean value of perturbation is unstable . for @xmath35 in equation ( [ dispersion ] ) , there is no instability , like the zero magnetic field for mri . @cc@   and the imaginary part of one of the solutions of @xmath50 , for vertical perturbation in case ii . we consider @xmath59 , however we obtain almost the same results for other admissible values of @xmath55 . other solution of @xmath50 is stable.,title="fig : " ]   and the imaginary part of one of the solutions of @xmath50 , for vertical perturbation in case ii . we consider @xmath59 , however we obtain almost the same results for other admissible values of @xmath55 . other solution of @xmath50 is stable.,title="fig : " ] +   +   +   +   and the imaginary part of one of the solutions of @xmath50 , for vertical perturbation in case ii . we consider @xmath59 , however we obtain almost the same results for other admissible values of @xmath55 . other solution of @xmath50 is stable.,title="fig : " ]   and the imaginary part of one of the solutions of @xmath50 , for vertical perturbation in case ii . we consider @xmath59 , however we obtain almost the same results for other admissible values of @xmath55 . other solution of @xmath50 is stable.,title="fig : " ]    1.5 cm in fig . [ unstable ] , for @xmath60 and different values of @xmath0 above a certain value , we show that for keplerian flows , there are modes for which the mean values of perturbation are unstable . if the amplitude of perturbations decreases , the value of @xmath61 increases for any fixed nonzero @xmath34 , leading to a larger range of @xmath44 for instability . however for @xmath35 , i.e. for the white noise with zero mean ( which also corresponds to the hydrodynamic accretion flows without any noise ) , we obtain no such unstable modes . while modes are stable for smaller @xmath0 , with the increase of @xmath0 they become unstable and range of @xmath44 giving rise to instability increases with increasing @xmath0 and for @xmath62 unstable modes arise all the way upto @xmath63 .    [ cols="^,^ " , ]      in fig . [ unstablemsp ] , we show how spherical perturbation modes in keplerian flows vary with @xmath64 . this is very similar to as shown in fig . [ unstablem ] for vertical perturbations , except that the modes are stable for a very small but non - zero @xmath64 , while for vertical perturbation the modes remain unstable for @xmath65 .      for plane couette flows , making @xmath66 in equation ( [ dispersionsp ] ) , we obtain the corresponding dispersion relation as @xmath67 while the first root always corresponds to the stable mode for a real @xmath44 , the second one will lead to the unstable solution for a negative @xmath34 satisfying @xmath68 .     and imaginary part of one of the solutions of @xmath50 , for spherical perturbation in case ii , where @xmath59 . it shows the unstable modes for negative @xmath34 ( drift velocity).,width=321 ]    figure [ negmsp ] shows that for spherical perturbations , the keplerian flows remain unstable upto @xmath69 , as shown in fig . [ negm ] for vertical perturbation cases . now we shall discuss that how relevant and how likely the white noise is to be present in shear flows . the rayleigh stable flows under consideration have a background shear profile , with some molecular viscosity however small that may be , and hence some drag ( e.g. , in protoplanetary disks , it could be due to the drag between gas and solid particles ) . for plane couette flow , such shear is driven in the fluids by moving the boundary walls by externally applied force . if the external force ( cause ) is switched off , the shearing motion ( effect ) dies out . similarly , in accretion disks , the central gravitational force plays the role of driving force ( cause ) producing differential velocity ( shear ) in the flow . hence , by fluctuation - dissipation theorem of statistical mechanics ( see , e.g. , * ? ? ? * ; * ? ? ? * ) , there must be some thermal fluctuations in such flows , with some temperature however low be , and that cause the fluid particles to have brownian motion . therefore the time variation ( derivative ) of this brownian motion , which is defined as white noise , plays the role of extra stochastic forcing term in the orr - sommerfeld equations ( equations ( [ hydroorrv ] ) , ( [ hydroorrzeta ] ) ) which are present generically , in particular when perturbation is considered . now , due to the presence of background shear in some preferential direction , it is very likely for the fluid particles to have brownian motion with nonzero drift , however small it may be . the detailed technical description of generation of white noise ( with zero and nonzero mean ) from brownian motion has been included in appendix . therefore , if @xmath70 is the random displacement variable of a brownian motion with drift coefficient @xmath34 , its probability density function @xmath71 can be written as @xmath72 ,   \label{brownian0}\ ] ] where @xmath73 is the standard deviation of the distribution and @xmath74 the time . taking the stochastic time derivative of @xmath70 , we obtain the white noise process which we denote by @xmath75  ( @xmath76 ) . since the stochastic variable @xmath70 is not differentiable in the usual sense , we consider a finite difference approximation of @xmath75 using a time interval of width @xmath77 as @xmath78 therefore , the presence of infinitesimal molecular viscosity ( and shear ) , which is there always , would be enough just to give rise to a nonzero ( infinitesimal ) temperature , leading to thermal noise which can do the rest of the job of governing instability . note that a very tiny mean noise strength , due to tiny asymmetry in the system , is enough to lead to linear instability , as demonstrated in previous sections . here , the externally applied force ( for plane couette ) or the force arising due to the presence of strongly gravitating object ( accretion disk ) introduces the asymmetry in the system , just like , e.g. , the brownian ratchets which has several applications in soft condensed matter and biology ( see , e.g. , @xcite ) . the measure of asymmetry and drag determines the value of @xmath34 , which furthermore controls the growth rate of perturbation . the corresponding power spectrum appears to be almost flat / constant ( for ideal white noise it is purely flat ) . although in our chosen shearing box , the azimuthal direction is assumed to be periodic , every such small box always encounters drag and hence thermal fluctuation , which assures the presence of nonzero mean noise . as a result , every such sharing box reveals exponential growth of perturbation . we have shown that linearly perturbed hydrodynamic apparently rayleigh stable rotating shear flows , including accretion disks , and plane couette flow , driven stochastically , can indeed be unstable , since the averaged values of the perturbations grow exponentially . due to background shear and hence drag , thermal fluctuations arise in these flows which induce brownian motion of the fluid particles and hence stochastic forcing by white noise . therefore the accretion flows , in particular due to perturbation , are inevitably driven by white noise which can not be neglected . it is indeed shown in experiments that the stochastic details decide whether turbulence will spread or eventually decay @xcite , which furthermore argues for the determining factor played by stochastic forcing , which we demonstrate here for the first time . since the forcing term in this system is a random variable , the solutions of the perturbations @xmath79 are also random variables and hence have some distributions whose averaged values are investigated . hence , we have shown that even in the absence of magnetic field , accretion disks can be made unstable and plausibly turbulent if they are driven by stochastic noise which is very likely to be present in the disks due to thermal fluctuations . in fact , we argue that neglecting the stochastic noise in accretion flows and any other shear flows is vastly an inappropriate assumption . this is because , some shear is always there ( because those are always driven externally by definition ) , which leads to some temperature ( however be the magnitude ) and a small temperature is enough to reveal stochastic noise , which is the basic building block of our work . hence , the presence of ( asymmetric ) drag and stochastic noise in shearing flows is inherent . hence , this work inevitably presents the origin of pure hydrodynamic instability of rotating shear flows and plane couette flow . therefore , this sheds enormous light on to the understanding of formation of planets and stars . evidently this mechanism works for magnetized shear flows as well , because thermal fluctuations are available there also . for example , a background field of the order of unity with @xmath80 can easily lead to unstable modes of perturbation for @xmath81 in the limit of very large @xmath0 and @xmath82 which is the case in accretion disks . in future , we will report this result in detail . indeed , earlier we studied stochastically driven magnetized flows and showed them to be plausibly unstable and turbulent by calculating the correlation functions of perturbations ( @xcite ) . hence the pure hydrodynamic instability explored here is generic . this is , to the best of our knowledge , the first solution to the century old problem of hydrodynamic instability of apparently rayleigh stable flows . in due courses , one has to investigate how exactly the required value of stochastic forcing strength could be arised in real systems and if the growth rates of unstable modes could adequately explain data . in certain cases , only high @xmath0 reveals instability which might be difficult to achieve in laboratory experiments and numerical simulations as of now . we have assumed here that the white noise has a nonzero mean value . the term _ white noise _ is ambiguous . to shed light on this matter , here we point out the two definitions of white noise . @xcite defines it as +   + . . . a stationary random process having a constant spectral density function .. +   + @xcite defines it as +   + we shall say that a process @xmath83 is white noise if its values @xmath84 and @xmath85 are uncorrelated for every @xmath86 and @xmath87 : @xmath88 .. +   + the following subsections explore the implications of each definition with respect to the mean of the resulting process . let @xmath70 is an _ ergodic stochastic process _ with the property that it has a constant power spectral density , i.e. @xmath89 where @xmath90 is the power spectral density of the random variable @xmath70 and @xmath44 is a constant . then the corresponding autocorrelation function for the process is @xmath91=\phi_{xx}(\tau)=\alpha \delta(\tau ) , % ~{\rm by~taking~inverse~fourier } \\   % \nonumber \label{autocorr}\end{aligned}\ ] ] by taking inverse fourier transform of @xmath90 , where @xmath92 $ ] denotes the expectation value . now let us assume that @xmath70 is a zero mean white noise process and @xmath93 is a nonzero mean process . then @xmath94=e[(x(t)+m)(x(t+\tau)+m)]=\alpha\delta(\tau)+m^2 . \\   % \label{brownian}\end{aligned}\ ] ] therefore @xmath95 which is not constant , thus @xmath96 violates the requirement of the white noise process by this definition .      representing the papoulis definition of white noise in our notations , we can write , a stochastic process @xmath70 is called a white noise process if any two distinct random variables of this stochastic process are independent and uncorrelated , i.e. , the autocovariance function @xmath97 when @xmath98 . in mathematical notation , @xmath99 \\ \nonumber = e[x(t)]e[x(t+\tau)]-m_tm_{t+\tau } \\   = m_tm_{t+\tau}-m_tm_{t+\tau}=0 , \label{cov}\end{aligned}\ ] ] where @xmath100 and @xmath101 are the corresponding mean values of the random variables @xmath70 and @xmath102 respectively . we can write the second equality in equation ( [ cov ] ) since @xmath70 , for different values of @xmath74 , are independent random variables by definition . _ thus it is not necessary that a white noise process always has to have a zero mean , from papoulis definition_. that is , a stochastic process having nonzero mean can be a white noise process according to this definition . _ in the present work , we have used papoulis definition of white noise which can indeed have a non - zero mean_. now let us explain why we have chosen papoulis definition over brown s definition . let us consider a signal @xmath103 with constant power spectral density @xmath104 . that is , @xmath105 , \label{corrf}\end{aligned}\ ] ] and the fourier transform of @xmath106 is @xmath107 a constant . now the _ parseval s theorem _ tells us that @xmath108 where @xmath109 is the fourier transform of @xmath103 . since @xmath104 and consequently @xmath110 has a constant positive value according to brown s definition of white noise , the equation ( [ energyf ] ) tells us that the total power of the signal is infinity ( see , e.g. , @xcite ) . in mathematical terminology , the energy norm of the signal @xmath103 is infinity and hence the function @xmath103 is not @xmath111-integrable . therefore , driving a system by a stochastic noise with constant power spectral density is same as injecting infinite amount of energy into the system , which is unphysical @xcite .      in this section , we outline the derivation of white noise starting from the random walk , via brownian motion . 1.5 cm figure [ rwdiagram ] shows an array of positions @xmath112 where @xmath113 etc . and @xmath114 is the spacing between points . at each interval of time , @xmath115 , a hop is made with probability @xmath116 to the right and @xmath117 to the left . the distribution of @xmath7 , of hops to the right , in @xmath118 steps is given by the bernoulli distribution @xmath119 the first moment ( mean ) and the second moment ( variance ) of the bernoulli distribution in equation ( [ bar ] ) is given by @xmath120 a particle that started at @xmath121 and took @xmath7 steps to the right and @xmath122 steps to the left arrives at the position @xmath123 with mean value @xmath124 notice that , if @xmath125 , or equal probability to jump to the right and the left , the average position after @xmath118 steps will remain @xmath121 . the second moment about the mean is given by @xmath126 therefore , from the central limit theorem , the limiting distribution after many steps is gaussian , with the first and second moments just obtained ( in equations ( [ meanposition ] ) and ( [ varposition ] ) ) , given by @xmath127 ^ 2}{8npq}\right\rbrace . \label{distposition}\end{aligned}\ ] ] if we introduce the position and time variables by the relations @xmath128 the moments of @xmath21 are given by @xmath129 the factor @xmath130 in the definition of diffusion coefficient @xmath131 is appropriate for one dimension , and would be replaced by @xmath132 if we consider the random walk in a space of dimension @xmath133 . thus the distribution moves with a driftvelocity @xmath134 and spreads with a diffusion coefficient defined by @xmath135 thus the probability distribution of the displacement @xmath21 of a particle under this random walk is @xmath136 , \label{distdisp}\end{aligned}\ ] ] where @xmath137 . a stochastic process in which the random variables @xmath70s are stationary and independent and have distribution as in equation ( [ distdisp ] ) , is called a brownian motion or wiener process . it is very clear from the equation ( [ driftvel ] ) that , when @xmath125 , then the drift velocity is @xmath121 , which means if some random walk is fully symmetric without any bias , then only we obtain the zero drift velocity of the corresponding brownian motion ( which is known as standard brownian motion in literature ) . however , if some process has any asymmetry ( for example hydrodynamic flows with shear in a particular direction , bulk hydrodynamic flows , flows under gravity etc . ) , the random walk of particles in that process will have some bias ( i.e. @xmath138 ) , which eventually introduces a brownian motion with _ nonzero drift velocity_.      if we take stochastic time derivative of a brownian motion or wiener process , we obtain a _ white noise _ process . if @xmath70 is the random displacement variable of a brownian motion with drift velocity @xmath34 , its probability density function @xmath71 can be written as ( using equation ( [ distdisp ] ) ) @xmath139 , \label{browniansupp}\end{aligned}\ ] ] where @xmath73 is the standard deviation of the distribution and @xmath74 the time . taking the stochastic time derivative of @xmath70 , we obtain the white noise process which we denote by @xmath75  ( @xmath76 ) . since the stochastic variable @xmath70 is not differentiable in the usual sense , we consider a finite difference approximation of @xmath75 using a time interval of width @xmath77 as @xmath140 since the stochastic random variables @xmath70 corresponding to a brownian motion process are stationary and independent , from equations ( [ browniansupp ] ) and ( [ whitenoisesupp ] ) we obtain that the white noise process has mean / averaged value @xmath34 and variance @xmath141 . as @xmath142 , the variance @xmath143 , and this white noise tends to the ideal white noise having a constant power spectral density @xcite . however , since brownian motion is not differentiable anywhere , the ideal white noise does not exist , as also explained above from the energy norm point of view . now we will show that the white noise defined in equation ( [ whitenoisesupp ] ) satisfies the papoulis definition of white noise , i.e. , the process is an uncorrelated stochastic process . to establish this , let us first note that if @xmath70 and @xmath144 are two random variables of a brownian motion with @xmath145 , then @xmath146 \\ \nonumber = e[\lbrace(x(t)-mt)-(x(s)-ms)+(x(s)-ms)\rbrace ( x(s)-ms ) ] \\   \nonumber = e[\lbrace(x(t)-x(s))-(mt - ms)\rbrace(x(s)-ms)]+e[(x(s)-ms)^2 ] \\   = 0+\sigma^2 s=\sigma^2 { \rm min}\lbrace t , s\rbrace . \label{browncov}\end{aligned}\ ] ] the third equality is possible since @xmath147 and @xmath144 are independent random variables for a brownian motion . having the result of equation ( [ browncov ] ) in hand , we now calculate the autocovariance of white noise . it is very easy to verify that the autocovariance function @xmath148 of two random variables @xmath149 and @xmath150 is a linear function in both of its arguments . therefore , @xmath151 \\   \nonumber = \frac{1}{\delta t^2 } \left[c(x(t+\delta t),x(s+\delta t))-c(x(t+\delta t),x(s ) ) \right . \\   \left . -c(x(t),x(s+\delta t))+c(x(t),x(s))\right ] . \label{whitecov}\end{aligned}\ ] ] when @xmath152 , i.e. @xmath153 , then using equation ( [ browncov ] ) , from equation ( [ whitecov ] ) we obtain @xmath154 now let us consider the cases when @xmath155 , i.e. when @xmath156 or @xmath157 . for @xmath156 , equations ( [ browncov ] ) and ( [ whitecov ] ) imply @xmath158 and also for @xmath157 , @xmath159 therefore , @xmath160 i.e. @xmath161 and @xmath162 are uncorrelated . let us define @xmath163    @cc@   for two small values of @xmath77.,title="fig : " ] +   +   for two small values of @xmath77.,title="fig : " ] +   +    figure [ twodelta ] shows the variation of @xmath164 for two different small values of @xmath77 . the function @xmath164 defined in equation ( [ whitecovdelta ] ) is an approximation of the well known delta function @xmath165 , because @xmath166 as it is seen from fig . also the function @xmath164 satisfies the integral property of the delta function as shown below , @xmath167 hence , when @xmath142 , from equation ( [ whitecovfinal ] ) we obtain @xmath168 therefore , the noise with nonzero mean , obtained from the stochastic time derivative of brownian motion with nonzero drift , is a white noise process according to the papoulis definition and also has the correlators 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we provide the possible resolution for the century old problem of hydrodynamic shear flows , which are apparently stable in linear analysis but shown to be turbulent in astrophysically observed data and experiments . 
 this mismatch is noticed in a variety of systems , from laboratory to astrophysical flows . 
 there are so many uncountable attempts made so far to resolve this mismatch , beginning with the early work of kelvin , rayleigh , and reynolds towards the end of the nineteenth century . here 
 we show that the presence of stochastic noise , whose inevitable presence should not be neglected in the stability analysis of shear flows , leads to pure hydrodynamic linear instability therein . 
 this explains the origin of turbulence , which has been observed / interpreted in astrophysical accretion disks , laboratory experiments and direct numerical simulations . 
 this is , to the best of our knowledge , the first solution to the long standing problem of hydrodynamic instability of rayleigh stable flows .