the amount of wireless ip data traffic is projected to grow by well over times within a decade ( from under exabytes in 2010 to more than exabytes by 2020 ) . to address such wireless data traffic demand ,there has been increasing effort to define the 5 g network in recent years .it is widely recognized that the 5 g network will be required to deliver data rate about to times the current 4 g technology , utilizing radical increase in wireless bandwidths at very high frequencies , extreme network densitification , and massive number of antennas .  distributed base station architecture and cloud radio access network ( c - ran ) will continue to be an important network architecture well into the future .therefore , there is a pressing need to drastically enhance the data rate of the common public radio interface ( cpri ) ( see fig .[ fig : cpri ] ) , which is the industry standard for interface between the baseband units ( bbu ) and the remote radio units ( rru ) .  one way to address the significant increase in the cpri data rate is to deploy more links ( typically fibers ) connecting the bbus and the rrus , but such deployment would incur extraordinary high cost .an alternative method , which can be much more cost effective , is to employ data compression over cpri links .it is impossible to utilize only cpri link compression to meet the cpri link data rate requirement .nevertheless , cpri link compression can greatly reduce the required cost when employed in conjunction with new links deployment .rate reduction between the bbu and the rru can also be achieved by moving some of the functions traditionally performed at the bbu to the rru , but this requires a significant change to the existing distributed base station or c - ran architecture .this paper focuses on cpri link data compression which can be employed with minimal change to the current distributed base station or c - ran architecture .cpri link compression techniques commonly found in the literature are based on scalar quantization , often with block scaling before quantization to adjust for the dynamic range of samples to be quantized . for lte networks , time - domain lte ofdm i/ q samples are carried over cpri links .cyclic prefix removal for downlink and decimation to remove the inherent redundancy in oversampled lte signals have been proposed to achieve additional compression . proposed a non - linear scalar quantization , whereby the quantizer is trained off - line using an iterative gradient algorithm .compression gain of times was reported with approximately evm for mhz downlink lte data . reported a compression gain of times with approximately evm distortion using decimation , an enhanced block scaling and a uniform quantizer .lloyd - max scalar quantization with noise shaping was considered in , and distributed compression was investigated in .our approach differs from previous approaches in that we consider a vector quantization ( vq ) based compression , rather than a scalar one .we exploit the fact that due to the ifft ( fft ) operation for downlink ( uplink ) , the i / q samples of an ofdm symbol are correlated over time .scalar quantizer is not capable of exploiting such time correlations .on the other hand , vector quantization , by mapping grouped samples into codewords , can explore such correlations and achieves better compression gain .from complex i / q samples , vectors need to be formed before quantization .there are several vectorization methods depending on how i / q samples are placed within the vectors that are formed .we investigate the performance of different vectorization methods for constructing vectors from i / q samples . for vector quantization codebook training ,lloyd algorithm is introduced . to further enhance performance , we propose a modified algorithm with different initialization step , where multiple trials work in serial to generate better codebook .low - complexity vector quantization algorithms in the form of multi - stage vector quantization ( msvq ) and unequally protected multi - group quantization ( upmgq ) are also considered in the paper .analysis and simulation result show that the proposed compression scheme can achieve times compression for uplink and times compression for downlink within evm distortion .the rest of this paper is organized as follows .section [ sec : algorithm ] describes the cpri compression algorithm in details , including the report on the serial initialization pattern to improve the performance of trained codebook from regular vector quantization as well as the discussion on universal compression .section [ sec : advanced ] discusses advanced quantization methods , in order to reduce the searching and storing complexities of regular vector quantization .after that , section [ sec : simulation ] contains all simulation results for both downlink and uplink , comparing different quantization methods mentioned in previous sections .finally , section [ sec : conclusion ] concludes the paper .a system framework for our vector quantization based cpri compression and decompression for both downlink and uplink is illustrated in fig .[ fig : framework ] . for downlink ,the input to the cpri compression module located at the bbu site is a stream of digital i / q samples from the bbu .the cpri compression module further contains modules of _ cyclic prefix removal _ , _ decimation _ , _ block scaling _ , _ vector quantizer _ and _ entropy encoding_. at the rru site , the cpri decompression module performs the reverse operations . for uplink ,the analog - to - digital converter ( adc ) output is the input to the cpri compression module located at the rru site and the cpri decompression module at the bbu site performs the reverse operations .    within these function blocks , _ cyclic prefix removal _ , _ decimation _ , and _block scaling _ are standard signal processing .a sketch of their roles and compression gains are summarized as follows ( please refer to appendix [ app : details ] for details of theses blocks ) :  * _ cp removal _ block , applicable for downlink only , aims to eliminate the time domain redundancy from cyclic prefix . the compression gain from this block ( i.e. , )can be expressed as where and denote ifft output symbol length and cyclic prefix length , respectively . *_ decimation _ block aims to reduce the redundancy in frequency domain because lte signal is oversampled .the compression gain from this block ( i.e. , ) can be expressed as where and denote downsampling and upsampling factors , respectively . *_ block scaling _ block aims to lower the resolution of the signal and maintain the dynamic range to be consistent with the downstream quantization codebook .there is no direct compression gain from this block .in contrast , extra signaling overhead of bits for every samples is required , where is the target resolution and is the number of samples forming a block ._ vector quantization _ and _ entropy coding _blocks are key techniques in our cpri compression algorithm . in section [ sec : algorithm : vq ] and section [ sec : algorithm : ec ] , we present their mechanisms and performances in detail .   as shown in fig .[ fig : quantization ] , vector quantization / dequantization is performed based on a vector quantizer codebook . the inputsto vector quantization module are the vectorized samples ( , where is the number of i / q samples and is the vector length ) , and the vector quantizer codebook which is a set of vector codewords ( , where is the codebook size ) .the codebook is trained off - line using training samples such that a specified distortion metric ( such as the euclidean distance ) is minimized .the vector quantizer maps a vector sample to one of the vector codewords which would minimize the specified distortion metric .each quantized sample vector is represented by bits .then , the compression gain from vector quantization is given by where is the uncompressed bitwidth of i or q component of each sample , and is the effective bitwidth of quantized samples . is typically , specified in .next , we discuss in detail the vectorization step and the codebook training .     before vector quantization , the vectorization of i / q samples is needed .the purpose of vectorization is to construct vectors from i / q samples that can capture correlation or dependency across i / q samples .a question of interest is : given a vector length , what is the best way to perform vectorization of i / q samples such that the compression gain of vector quantization can be maximized ? to this end , we consider the following three vectorization methods ( see fig . [ fig : vectorization ] for an illustration ) :  * method 1 : consecutive i components in time are grouped as vectors .similarly , consecutive q components are grouped as vectors .* method 2 : i and q components of the same time index are grouped as vectors .* method 3 : i and q components of all samples are randomly grouped as vectors .the three vectorization methods are compared using the entropy of the distribution of the constructed vectors of chosen length in euclidean orthants as metric . since lower entropy value indicates higher correlation between the components of the constructed vectors , the vectorization method with smaller entropy implies higher quantization compression gain .[ fig : vecmethod ] shows plots of entropy versus for the three vectorization methods .it is evident from the plots that for lte signals , method 1 has the smallest entropy among the three methods , which shall be assumed in the rest of this paper .our vector quantizer codebook is trained using lloyd algorithm ( a special case of lbg algorithm ) .lloyd algorithm is an iterative algorithm that can be utilized to construct codebook for vector quantization .it aims to find evenly - spaced sets of points ( as codewords ) in subsets of euclidean spaces , and to partition input samples into well - shaped and uniformly sized convex cells . in general , lloyd algorithm may start by randomly picking a number of input samples as initial codewords , and it then repeatedly executes samples partitioning and codebook updating in every iteration to reduce target distortion .a commonly used distortion metric is the euclidean distance .each time , the codeword points are left in a slightly more even distribution : closely spaced points move farther apart , and widely spaced points move closer together .finally , lloyd algorithm terminates at certain local optimal , after a proper stopping criterion is satisfied .classical lloyd algorithm is known to be quite sensitive to the initial choice of codewords , especially when input sample space does not have a smooth structure .in other words , if the input samples are concentrated in a particular space , lloyd algorithm easily converges to a local optimum which can be away from the global optimum . to this end, classical lloyd algorithm performs multiple independent trials and chooses the codebook that produces the lowest distortion metric from all trials to evade the initialization problem . a process chain for classical lloyd algorithm with multiple trialsis illustrated in fig .[ fig : classical_lloyd ] .    in the case of ltei / q samples , the sample values are observed to be highly concentrated in a narrow range .however , we also observe that independent trials is not effective to overcome the initialization problem described earlier , specifically we observe persistent convergence to similar local optimum despite multiple lloyd trials . to resolve this issue, we introduce a modified lloyd algorithm , whereby the output ( codebook ) from previous trial is utilized as the input ( the initial codebook ) to the next trial , after applying proper rescaling to the initial codebook magnitude .the procedure of rescaling is essential , because the output from previous trial is already a local optimum .rescaling helps to evade this local optimum and restart the search for a better codebook .the rescaling factor can be the square root of the average power of the i / q samples .the diagram illustrating the modified lloyd algorithm is given in fig .[ fig : modified_lloyd ] .the effectiveness of the modified lloyd algorithm is illustrated by comparing the codebooks trained using the classical lloyd algorithm and the modified lloyd algorithm as shown in fig .[ fig : codebook_compare_classical ] and fig . [ fig : codebook_compare_modified ] respectively . in this simulation ,lte uplink signals with snr , modulation , and awgn channel are considered as training sequences using vq with and .two axes of fig .[ fig : codebook_compare ] represent the two elements of the grouped vectors . each point onthe plot represents a codeword and the color corresponds to frequencies ( warmer color for higher frequency ) .it is observed that the codebook from modified algorithm better reflects the distribution of training sample vectors .this is also confirmed from the evm improvement of from the classical algorithm to from the modified lloyd algorithm for this particular case ( improvement ) .entropy coding is a lossless data compression scheme that utilizes more bits to represent sources with lower frequency ( higher entropy ) and fewer bits to represent sources with higher frequency ( lower entropy ) .the most common entropy coding technique is huffman coding . in huffman coding , the dictionary construction procedurecan be completed in linear time , and the final codeword for each symbol can be simply constructed from a splitting tree . according to shannon s lossless source coding theorem , the optimal average coded length for a source is its entropy , and huffman codes are proven to be optimal with linear time complexity for symbol - by - symbol coding .  without entropy coding, each codeword of vector quantization codebook requires exactly bits for representation .however , it is observed that the probability mass function ( pmf ) of codewords is not uniform ( see fig .[ fig : codebook_compare ] , where warmer color represents higher probability ) , implying potential entropy coding gain . to this end , huffman coding is applied based on the pmf of codewords . denoting the average length of huffman codes as , the compression gain from entropy coding can be expressed as    in this subsection, the aforementioned blocks in system are integrated together and performance of the whole cpri compression framework is evaluated .the compression gain and the evm distortion are investigated .we summarize the main results of the proposed cpri compression algorithm in a theorem as follows .[ thm : cr ] for a given vectorization method and a given vector length , the proposed vector quantization based lossy compression algorithm for cpri link can achieve the rate distortion trade - off given by _ _ and _ _ for large value of , where the compression gains in the denominator of are given by , , , and , respectively ; is the uncompressed bitwidth of i or q component of each sample ; and are parameters for block scaling ; is the input sequence to compression algorithm ; is the output from decompression algorithm ; is the number of input or output complex samples .  from the description of proposed algorithm , if complex samples are considered as input to the module , where each complex component is represented by bits , after compression , number of binary strings are transmitted on cpri link , where the average length for each string is . for _ block scaling _ block , number of extra bits are needed for every complex samples .hence , the final compression gain by cpri compression can be expressed as which further gives the expression in . note that if any of the blocks is not enabled / present , the corresponding compression gain is set to ( especially , if the block scaling is not enabled , is set to ) .equation is the standard form of error vector magnitude ( evm ) , which is a measure of deviation of constellation points from their ideal locations . in this study , evm is used to quantify the distortions introduced by compression .time - domain evm calculates the distortion over the whole bandwidth .however , for lte signals , only part of the bandwidth carries useful information .this motivates the use of frequency - domain evm instead of the time - domain evm as the distortion measure , i.e. , where and are transformed signals by fft for input and output respectively , and is the collection of indices corresponding to the utilized bandwidth .   in principle ,vector quantization codebook constructed from a particular set of training samples can produce the intended performance if the real samples do not deviate significantly from the training samples in statistical sense .however , in practice , system parameters or properties , such as channel types , snrs , and modulation schemes , may not be known perfectly , or may not always match those assumed for the training samples .this motivates the need for a universal compression technique , i.e. , a robust codebook that can provide reasonable or acceptable performances in all or a large range of system parameters , although the codebook is not necessarily optimal for a specific system setting .to understand if such a universal codebook exists , we investigate the performance of codebooks produced using mismatched training samples , in order to find the sensitivity factors impacting the performance .a key step is to discover the most robust parameter contributing to the universal codebook .if this procedure is infeasible , i.e. , such a parameter does not exist , a larger training sample pool mixing with different parameters should be considered , such that the trained codebook could reflect the distributions of all parameters .for example , for uplink lte samples with times target compression ratio , if vector length with decimation value is performed , constructed codebook is rather insensitive to modulation methods or channel types ( see table [ tab : level2_mismatch_modulation ] and table [ tab : level2_mismatch_channel ] ) , but is relatively more sensitive to snrs ( see table [ tab : level2_mismatch_snr ] ) .nevertheless , the codebook can be adopted as the universal codebook , since its evm performances are acceptable for all snr training samples .based on these observations , we conclude that the codebook obtained from snr , modulation , and awgn channel can be a suitable universal codebook , which we shall assume for further performance evaluation in section [ sec : simulation : ul ] .signal on cpri link can have a large dynamic range .for instance , for uplink cpri signals , different propagation path loss , shadowing , fading channel and mobility for different users may result in significant variance in general ._ block scaling _ , also known as automatic gain control ( agc ) , is employed to lower the resolution of signal and to maintain the dynamic range simultaneously .  in agc, a sequence of signal samples are grouped into blocks of consecutive samples , where the samples in each block are scaled by a scaling factor which can vary from block to block .more precisely , assume number of samples form a block , and the largest absolute value of i / q components for a particular block is determined by where is the output from the _ decimation _ block , and and represent the real and imaginary parts of a complex - valued sample , respectively .next , the corresponding scaling factor for block is determined by by this definition , is an integer that can be represented with bits .samples in block are then scaled to produce an output as follows . where denotes the number of bits per complex component for vector quantization .in essence , block scaling normalizes the input signals for vector quantization .there is a slight increase in signal processing latency associated with block scaling .for instance , in lte mhz system , by choosing , the latency due to block scaling is  with decimation value chosen as .d. samardzija , j. pastalan , m. macdonald , s. walker , and r. valenzuela , `` compressed transport of baseband signals in radio access networks , '' _ ieee transactions on wireless communications _11 , no . 9 , pp .32163225 , sep . 2012 .y. ren , y. wang , g. xu , and q. huang , `` a compression method for lte - a signals transported in radio access networks , '' in _ 2014 international conference on telecommunications ( ict 2014 ) _ , may 2014 , pp .293297 .k. f. nieman and b. l. evans , `` time - domain compression of complex - baseband lte signals for cloud radio access networks , '' in _ 2013 ieee global conference on signal and information processing ( globalsip 2013 ) _ , dec .2013 , pp . 11981201 .s. h. park , o. simeone , o. sahin , and s. shamai , `` robust and efficient distributed compression for cloud radio access networks , '' _ ieee transactions on vehicular technology _ , vol .62 , no . 2 ,692703 , feb .2013 .b. h. juang and a. gray jr , `` multiple stage vector quantization for speech coding , '' in _ 1982 ieee international conference on acoustics , speech , and signal processing ( icassp82 ) _ , vol . 7 , may 1982 , pp . 597600 .h. si , o. o. koyluoglu , and s. vishwanath , `` lossy compression of exponential and laplacian sources using expansion coding , '' in _ 2014 ieee international symposium on information theory ( isit 2014 ) _ , jun .2014 , pp . 30523056 .

Abstract: the future wireless network , such as centralized radio access network ( c - ran ) , will need to deliver data rate about to times the current 4 g technology . for c - ran based network architecture , there is a pressing need for tremendous enhancement of the effective data rate of the common public radio interface ( cpri ) . compression of cpri data is one of the potential enhancements . in this paper , we introduce a vector quantization based compression algorithm for cpri links , utilizing lloyd algorithm . methods to vectorize the i / q samples and enhanced initialization of lloyd algorithm for codebook training are investigated for improved performance . multi - stage vector quantization and unequally protected multi - group quantization are considered to reduce codebook search complexity and codebook size . simulation results show that our solution can achieve compression of times for uplink and times for downlink , within error vector magnitude ( evm ) distortion . remarkably , vector quantization codebook proves to be quite robust against data modulation mismatch , fading , signal - to - noise ratio ( snr ) and doppler spread .


despite many attempts to pin down the essential features of dynamics and relaxation in random systems , the problem is still far from its complete solution .it turns out that lvy flight models are adequate for the description of different random systems ranging from the motion of defects in disordered solids to the dynamics of assets in stock markets , see , e.g. .they are especially useful to model the random systems on the semi - phenomenological , mesoscopic level , when the ( often unknown ) details of their microscopic random behavior are substituted by a properly tailored ( e.g. based on experimental data ) ( gaussian or lvy ) noise .paradoxically , in disordered solids , the noise can promote order and organization , switching them between different equilibrium states .the latter situation emerges in disordered ferroelectrics , where the fluctuations of order parameter ( spontaneous polarization ) give rise to self - localization of charge carrier , generating a fluctuon , an analog of well - known polaron in disordered substance .these fluctuons make a substantial contribution into conductivity and optical properties of disordered dielectrics .many random processes admit a description based on stochastic differential equations . in such casethere is a routine passage procedure from microscopic random variables to macroscopic ( statistical ensemble ) data .the latter are encoded in the time evolution of an associated probability density function ( pdf ) which is a solution of a deterministic transport equation .a paradigmatic example is the so - called langevin modeling of diffusion - type and jump - type processes .the presumed microscopic model of the dynamics in external force fields is provided by the langevin ( stochastic ) equation whose direct consequence is the fokker - planck equation , and .we note that in case of jump - type processes the familiar laplacian ( wiener noise generator ) needs to be replaced by a suitable pseudo - differential operator ( fractional laplacian , in case of a symmetric lvy - stable noise ) .we pay a particular attention to jump - type processes which are omnipresent in nature ( see and references therein ) .their characterization is primarily provided by jump transition rates between different states of the system under consideration . in the present paper ,our major focus is on a specific class of random systems which are plainly incompatible with a straightforward langevin modeling of jump - type processes and , as such , are seldom addressed in the literature .  to this endwe depart from the concept , coined in an isolated publication , of lvy flights - driven models of disorder that , while at equilibrium , do obey detailed balance .the corresponding research line has been effectively initiated in refs .it has next been expanded in various directions , with a special emphasis on so - called lvy - schrdinger semigroup reformulation of the original probability density function ( pdf ) dynamics , and - , c.f .also .we note in passing that the familiar fokker - planck equation can be equally well reformulated in terms of the schrdinger semigroup and this property is universally valid in the standard theory of brownian motion , .its generalization to lvy flights is neither immediate nor obvious .it is often considered in the prohibitive vein following .in fact , in relation to lvy flights , a novel fractional generalization of the fokker - planck equation has been introduced in refs .- to handle systems that are randomized by symmetric lvy - stable drivers . in this case , contrary to the popular lore about properties of ( langevin - based ) lvy processes ( c.f . refs .- and ) , the pertinent random systems are allowed to relax to ( thermal ) equilibrium states of a standard boltzmann - gibbs form .the underlying jump - type processes , in the stationary ( equilibrium ) regime , respect the principle of detailed balance by construction .their distinctive feature , if compared with the standard langevin modeling of lvy flights , is that they have a built - in response _ not _ to external forces _ but rather to _ external force potentials .these potentials are interpreted to form confining `` potential landscapes '' that are specific to the environment .lvy jump - type processes appear to be particularly sensitive to environmental inhomogeneities , .we note in passing that confining lvy flights resolve themselves to the problem of truncated distributions , which can be addressed in many ways . beside a simple cut - off , by considering fast falling tails .the random walk theory predominantly employs to the master equation as the major tool to quantify random motion .inhomogeneous transition rates have been used in the literature before and the emergence of non - trivial target distributions has been demonstrated , c.f . .lvy flights are pure jump ( jump - type ) processes .therefore , it seems useful to indicate that various model realizations of standard jump processes ( jump size is bounded from below and above ) can be thermalized by means of a specific scenario of an energy exchange with the thermostat .it is based on the _ principle of detailed balance_. we have discussed this issue in some detail before along with an extension of this conceptual framework to lvy - stable processes .not to reproduce easily available arguments of past publications , we shall be very rudimentary in our motivations , see however .we quantify a probability density evolution , compatible with a jump - type process on ( this limitation may in principle be lifted in favor of ) , in terms of the master equation : ,\label{l1}\ ] ] where and are , respectively , the lower and upper bounds of jump size and }{|x - y|^{1+\mu}},\nonumber \\ \nonumber \\c_\mu&=&\frac{\gamma(1+\mu)\sin(\pi\mu/2)}{\pi } \label{l2}\end{aligned}\ ] ] is the jump transition rate from to .we stress that is a non - symmetric function of and .an implicit boltzmann - type weighting involves a square root of a target pdf  ] from a uniform distribution .5 .  using above and identities  corresponding to the `` transition channel '' . 6 .draw a new number from a uniform distribution .reset time label where  to a new value . 9 .return to the step 2 and repeat the procedure anew .in this section , we select the jump - type processes , which best demonstrate the capabilities of our algorithm of random paths generation . once suitable path ensemble data are collected , we ca n reconstruct the pdf dynamics and next verify whether statistical ( ensemble ) features of generated random trajectories are compatible with the asymptotic a priori imposed upon solutions of the master equation ( [ l1 ] ) . that refers to a control of an asymptotic behavior when .let us first consider an asymptotic invariant ( target ) pdf in the gaussian form : the corresponding -family of transition rates reads the c - codes for trajectory generating algorithm , , were employed to get the trajectory statistics data for lvy drivers with .second moment for 75 000 trajectories .inset visualize the oscillations smoothing in the asymptotic regime for ; figures near curves correspond to values . ]the results of our numerical simulations are reported in figs . 1 and 2. we note , that in fig .1 the second moment oscillates near its equilibrium value .a numerical convergence to is consistent with an analytic equilibrium value of the second moment of the chosen [ l6 ] .the rate of this convergence is higher for larger .clearly , for small the big jumps are frequent which enlarges the inferred time intervals in the gillespie s algorithm .thus , the relaxation to equilibrium is slow .it gets faster for larger , when big jumps are rare and time intervals are generically very small .  fig .2 displays a probability density evolution , inferred from the ensemble statistics of trajectories .all of them have started form the same point .although the data fidelity grows with the number of contributing paths , we have not found significant qualitative differences to justify a presentation of data for 100 000 , 200 000 , 250 000 and more trajectories . the relaxation time rate dependence on is clearly visible as well .it suffices to analyze differences between three curves for and/or .we observe a conspicuous lowering of their maxima with the growth of ( take care of different scales on the vertical axes on fig.2 panels ) .the simulated pdfs at are practically indistinguishable from an exact analytical asymptotic pdf [ l6 ] .the convergence of towards appears to be relatively fast irrespective of the chosen -driver .although our reasoning is definitely path - wise and all data have been extracted from simulated trajectory ensembles , our focus was on the inferred dynamics of .we do not reproduce figures visualizing generic sample paths .however , we indicate their ( paths ) distinctive qualitative typicalities , if one would compare e.g. motion scenarios corresponding to stability indices , and , the structural impact ( probability of occurrence ) of larger against smaller jumps , even on the visual level , conforms with standard simulations of lvy stable sample paths ( with no forces or potentials involved ) , c.f . .let us consider a long - tailed asymptotic pdf which is a special case of the one - parameter -family of equilibrium ( boltzmann - type ) states , associated with a logarithmic potential , , see : the transition rate [ l2 ] for any takes the form   second moment for 200 000 trajectories . ]simulation results for this case are displayed in figs .if we compare fig .3 with fig .1 we see the existence of small oscillations in the asymptotic regime about the value . those from fig.1 were relatively small , while those on fig . 3are more noticeable .this is related to much slower decay of transition rates ( [ l9 ] ) ( determined by slow - decaying asymptotic pdf ( [ l8 ] ) ) , as compared to those for gaussian case ( [ l7 ] ) .the second moment of the present , ( [ l8 ] ) , equals 1 and the convergence towards this value is clearly seen in fig .this convergence is much slower than in the gaussian ( harmonic confinement ) case which is not a surprise : ( [ l7 ] ) and ( [ l9 ] ) indicate that the present rate of convergence should be logarithmically slower . fig . 5 , quite alike fig .2 , convincingly demonstrates a convergence of to the asymptotic .for definitely large times around , and become practically indistinguishable . similarly to the gaussian case , the rate of convergence becomes larger with the growth of .for and trajectories , starting from .]   now we consider an asymptotic pdf of the form : in this case , the transition rate from to reads we consider cauchy driver corresponding to . in fig . 5, we report the time evolution of the statistically inferred for different time instants .an approach to the asymptotic pdf ( [ l20 ] ) is clearly seen , together with a convergence of a half - width to its asymptotic value 1 .it can be shown that the same convergence pattern is valid for cumulative probability distribution which approaches the asymptotic function .   to set firm grounds for future researchit is instructive to study our model for more complicated forms of confining potentials . in view of their physical relevance ,it is appealing to address an issue of confining ( trapping ) environments with a periodic spatial structure .this problem may be relevant to the random motion of defects in doped semiconductors and to phenomena like hopping conductivity .here , we encounter a major difficulty with integrability of the boltzmann - type weighting function as for periodic the corresponding integral is divergent .periodicity and integrability can here be reconciled by means of locally periodic potentials that take a definite confining form ( harmonic or polynomial ) for larger values of .let us consider the following asymptotic pdf where is a normalization constant .the transition rate from to reads },\ ] ] where the potential has the following locally confining form   inferred from trajectories at .the data for and trajectories ( not displayed ) do not show qualitative differences . ]time evolution of the inferred pdf is reported in fig . 6 for .all sample trajectories were started form which corresponds to the -type initial distribution .the probability density spreads out with time in conformity with the trapping ( confining ) properties of the locally periodic enclosure ( environment or `` potential landscape '' ) . for large running timest=400 the trajectory statistics produces data that are indistinguishable from those for the asymptotic pdf .we have checked that beginning from about 100 000 trajectories , further accumulation of the trajectories number like e.g. 200 000 ( displayed ) and 300 000 ( not displayed ) for the data statistics is inessential . in such casesthe curves are almost the same , we merely improve a fidelity of the statistics .if a random process does not admit the description in terms of a stochastic differential equation ( e.g. langevin modeling ) , its direct numerical simulation becomes impossible by means of existing popular algorithms . in the present paper , for the first time in the literature, we propose a working method to generate stochastic trajectories ( sample paths ) of a random jump - type process without resorting to any explicit ( or numerical ) solution of a stochastic differential equation .  to this endwe have modified the gillespie algorithm , normally devised for sample paths generation , if transition rates occur between a finite number of states of a system .the essence of our modification is that we take into account the continuum of possible transition rates , thereby changing the finite sums in the original gillespie algorithm into integrals .the corresponding procedure for stochastic trajectories generation has been changed accordingly .  in other words, here we `` extract '' the background sample paths of a jump process , whose pdf is to obey the generalized fokker - planck dynamics [ l1 ] , [ l2 ] , .we emphasize once more here , that we have focused our attention on those background jump - type processes that can not be modeled by any stochastic differential equation of the langevin type .however , our ultimate result was a reconstruction of the pdf dynamics from the path - wise data .thus , our modfication of the gillespie s algorithm serves two purposes : ( i ) getting access to path - wise data and ( ii ) reconstructing the pdf dynamics , compatible with the master equation in question , without resorting to other methods of its solution  although heavy - tailed lvy stable drivers were involved in the present considerations , we have clearly confirmed that an enormous variety of stationary target distributions is dynamically accessible in each particular case . that comprises not only a standard gaussian pdf , discussed usually in relation to the brownian motion like wiener process .we can reach cauchy pdf in terms of any driver as well , provided a steering environment is properly devised . in turn, the cauchy driver in a proper environment may lead to an asymptotic pdf with a finite number of moments , the gaussian case included ( ) .an example of the locally periodic environment has been considered as a toy model for more realistic physical systems .we mention possible generalizations of our method to the brownian motor concept ( see , e.g. , ref . for recent review ) to include a non - gaussian jumping component . in those systemsit is the properly tailored periodic `` potential landscape '' which enforces a conversion of a homogenous stochastic process ( brownian motion for reference ) into the directed motion of particles at nanometer scales .it is closely related to the problem of so - called sorting in periodic potentials .other problems to be addresses concern ultracold atoms in optical lattices subject to random potentials , which are promising not only from purely scientific point of view , but also for many technological applications .we note that the theoretical description of the above mentioned topics relies essentially on the langevin - like equation input .our approach offers prospects for generalizations , where systems with non - langevin response to external potentials may come into consideration , along with more traditional ones .what we actually need to implement our version of gillespie s algorithm is the knowledge of jump transition rates of those random systems only .our preliminary work ( in progress ) shows that an extension of our algorithm to higher dimensions is operational .in particular , the planar case is worth further analysis , possibly with more realistic `` potential landscapes '' .the direct numerical modeling in three - dimensional case can also be important , for instance , in the investigations of the dynamics of amorphous materials .theoretical studies of their physical properties have a longer history , starting in the eighties and even earlier .these materials may comprise conventional glasses ( see , e.g. , ) , so - called spin and orientational ( for example , dipole ) glasses as well as virtually any disordered solid like doped semiconductors . in these substances , the random motion of defects can be regarded as real jump - type process to be modeled numerically . moreover ,if defects have only rotational degrees of freedom ( like spins or electric / elastic dipoles ) , their dynamics can be well represented by some effective jump - type process .such representation can be complementary to existing monte - carlo modeling techniques which for disordered systems are very computer intensive .for instance , we hope to tackle the issue of interplay between exponential and long - time ( logarithmic or stretched - exponential ) relaxation in these substances .also , the 3d modification of our algorithm can be used in the simulation of switching processes in disordered ferroelectrics .similar fictitious jump - type process can be assigned to the dynamics of inhomogeneous broadening of resonant lines .this broadening occurs in condensed matter and/or biological species , in a number of spectroscopic manifestations like the electron paramagnetic resonance , nuclear magnetic resonance , optical and neutron scattering methods .it arises due to random electric and magnetic fields , strains and other perturbations from defects in a substance containing the centers whose resonant transitions between energy levels are studied . here also , the standard approaches can not describe all the details of experiment and we hope that our algorithm can improve the situation .let us finally note that in the 3d problems admitting full spherical symmetry ( like spin glasses with random exchange interactions ) , the problem becomes reducible to 1d case so that our algorithm might work in its present form .* acknowledgement * : p. g. would like to thank professor i. m. sokolov for focusing our attention on the gillespie s algorithm .all numerical simulations were completed by means of the facilities of the platon - science services platform of the polish pionier network .99 h. s. wio , r.r .deza , j.m .lopez , _ an introduction to stochastic processes and nonequilibrium statistical physics _ , ( world scientific publishing , singapore , 2012 ) .v.a.stephanovich , jetp letters * 65 * , 446 ( 1997 ) .h. risken , _ the fokker - plack equation _ , ( springer - verlag , berlin , 1989 ) s. jespersen , r. metzler and h. c. fogedby , phys .e 59 * , 2736 , ( 1999 ) r. metzler , j. klafter , j. phys . a : math* 37 * , r161 , ( 2004 ) l. chen and m. w. deem , phys* e 65 * , 011109 , ( 2001 ) d. brockmann and i. m. sokolov , chemical physics , * 284 * , 409 , ( 2002 ) d. brockmann and t. geisel , phys .lett . * 90 * , 170601 , ( 2003 ) v. v. belik and d. brockmann , new . j. phys .* 9 * , 54 , ( 2007 ) p. garbaczewski and r. olkiewicz , j. math .* 40 * , 1057 , ( 1999 ) p. garbaczewski and r. olkiewicz , j. math. phys . * 41 * , 6843 , ( 2000 ) p. garbaczewski and v. a. stephanovich , phys . rev . *e 80 * , 031113 , ( 2009 ) p. garbaczewski and v. stephanovich , open systems & information dynamics , * 17 * , 287 , ( 2010 ) p. garbaczewski and v. a. stephanovich , physica * a 389 * , 4410 , ( 2010 ) p. garbaczewski and v. stephanovich , phys . rev . *e 84 * , 011142 ( 2011 ) p. garbaczewski and v. a. stephanovich , acta phys .b * 43 * , 977 , ( 2012 ) j. lrinczi , f. hiroshima and v. betz , _ feyman - kac - type theorems and gibbs measures on path space _ , ( studies in mathematics * 34 * , walter de gruyter , berlin , 2011 ) r. vilela mendes , j. math. phys . * 27*,178 , ( 1986 ) s. eleutrio and r. vilela mendes , phys . rev . *b 50 * , 5035 , ( 1993 ) .i. eliazar and j. klafter , j. stat . phys . * 111 * , 739 , ( 2003 ) v. v. janovsky et al ., physica * a 282 * , 13 , ( 2000 ) a. dubkov and b. spagnolo , fluct .noise lett .* 5 * , l267 , ( 2005 ) h. touchette and e. g. d. cohen , phys .* e 76 * , 020101 , ( 2007 ) t. srokowski , physica * a 388 * , 1057 , ( 2009 ) t. d. gillespie , j. phys .chemistry , * 81 * ( 25 ) : 2340 , ( 1977 ) t. d. gillespie , j. comput .physics , * 22 * ( 4 ) : 403 , ( 1976 ) m. matsumoto and t. nishimura , acm trans . model .* 8 * , 3 ( 1998 ) r. n. mantegna and h. e. stanley , phys .lett . * 73 * , 2946 , ( 1994 ) i. m. sokolov , private communication c - code is available upon request a. janicki and a. weron , _ simulation and chaotic behavior of -stable stochastic processes _ , ( m. dekker , new york , 1994 ) p. hnggi , f. marchesoni , rev .phys . * 81 * , 387 , ( 2009 ) s. i. denisov , t. v. lyutyy _et al _ , phys . rev .e * 79 * , 051102 , ( 2009 ) h. gimperlein , s. wessel , j. schmiedmayer , and l. santos , phys .lett . * 95 * , 170401 , ( 2005 ) k. binder , a.p .young , rev .phys . * 58 * , 801 ( 1986 ) .hchli , k. knorr , a. loidl , adv .* 39 * , 405 ( 1990 ) .g. parisi , _ field theory , disorder and simulations _ , ( world scientific publishing , singapore , 1992 ) .d. kedzierski , e.v .kirichenko , v.a .stephanovich , phys .a * 375 * , 685 ( 2011 ) .a.m. stoneham , rev .phys . * 41 * , 1 ( 1969 ) .

Abstract: we analyze a specific class of random systems that are driven by a symmetric lvy stable noise , where langevin representation is absent . in view of the lvy noise sensitivity to environmental inhomogeneities , the pertinent random motion asymptotically sets down at the boltzmann - type equilibrium , represented by a probability density function ( pdf ) $ ] . here , we infer pdf based on numerical path - wise simulation of the underlying jump - type process . a priori given data are jump transition rates entering the master equation for and its target pdf . to simulate the above processes , we construct a suitable modification of the gillespie algorithm , originally invented in the chemical kinetics context . we exemplified our algorithm simulating different jump - type processes and discuss the dynamics of real physical systems where it can be useful .