@article{SandevDomazetoskiKocarevetal.2022, author = {Sandev, Trifce and Domazetoski, Viktor and Kocarev, Ljupco and Metzler, Ralf and Chechkin, Aleksei}, title = {Heterogeneous diffusion with stochastic resetting}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {55}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {7}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ac491c}, pages = {26}, year = {2022}, abstract = {We study a heterogeneous diffusion process (HDP) with position-dependent diffusion coefficient and Poissonian stochastic resetting. We find exact results for the mean squared displacement and the probability density function. The nonequilibrium steady state reached in the long time limit is studied. We also analyse the transition to the non-equilibrium steady state by finding the large deviation function. We found that similarly to the case of the normal diffusion process where the diffusion length grows like t (1/2) while the length scale xi(t) of the inner core region of the nonequilibrium steady state grows linearly with time t, in the HDP with diffusion length increasing like t ( p/2) the length scale xi(t) grows like t ( p ). The obtained results are verified by numerical solutions of the corresponding Langevin equation.}, language = {en} } @article{PetreskaPejovSandevetal.2022, author = {Petreska, Irina and Pejov, Ljupco and Sandev, Trifce and Kocarev, Ljupčo and Metzler, Ralf}, title = {Tuning of the dielectric relaxation and complex susceptibility in a system of polar molecules: a generalised model based on rotational diffusion with resetting}, series = {Fractal and fractional}, volume = {6}, journal = {Fractal and fractional}, number = {2}, publisher = {MDPI AG, Fractal Fract Editorial Office}, address = {Basel}, issn = {2504-3110}, doi = {10.3390/fractalfract6020088}, pages = {23}, year = {2022}, abstract = {The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately. The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation. In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process. The autocorrelation function and complex susceptibility are analysed in detail. We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate. The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.}, language = {en} } @article{BasnarkovTomovskiSandevetal.2022, author = {Basnarkov, Lasko and Tomovski, Igor and Sandev, Trifce and Kocarev, Ljupčo}, title = {Non-Markovian SIR epidemic spreading model of COVID-19}, series = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, volume = {160}, journal = {Chaos, solitons \& fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science}, publisher = {Elsevier}, address = {Oxford [u.a.]}, issn = {0960-0779}, doi = {10.1016/j.chaos.2022.112286}, pages = {8}, year = {2022}, abstract = {We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete-and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.}, language = {en} } @article{SandevIominKocarev2020, author = {Sandev, Trifce and Iomin, Alexander and Kocarev, Ljupco}, title = {Hitting times in turbulent diffusion due to multiplicative noise}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {102}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Institute of Physics}, address = {Woodbury, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.102.042109}, pages = {10}, year = {2020}, abstract = {We study a distribution of times of the first arrivals to absorbing targets in turbulent diffusion, which is due to a multiplicative noise. Two examples of dynamical systems with a multiplicative noise are studied. The first one is a random process according to inhomogeneous diffusion, which is also known as a geometric Brownian motion in the Black-Scholes model. The second model is due to a random processes on a two-dimensional comb, where inhomogeneous advection is possible only along the backbone, while Brownian diffusion takes place inside the branches. It is shown that in both cases turbulent diffusion takes place as the one-dimensional random process with the log-normal distribution in the presence of absorbing targets, which are characterized by the Levy-Smirnov distribution for the first hitting times.}, language = {en} } @article{PetreskadeCastroSandevetal.2020, author = {Petreska, Irina and de Castro, Antonio S. M. and Sandev, Trifce and Lenzi, Ervin K.}, title = {The time-dependent Schr{\"o}dinger equation in non-integer dimensions for constrained quantum motion}, series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, volume = {384}, journal = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, number = {34}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2020.126866}, pages = {9}, year = {2020}, abstract = {We propose a theoretical model, based on a generalized Schroedinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X x Y, comprising x-coordinate with a Hausdorff measure of dimension alpha(1) = D -1 (1 < D < 2) and y-coordinate with the Lebesgue measure of dimension of length (alpha(2) = 1). Geometric constraints are set at y = 0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D = 2, the solution for two-dimensional quantum motion on a comb is recovered. (C) 2020 Elsevier B.V. All rights reserved.}, language = {en} } @article{XuDengSandev2020, author = {Xu, Pengbo and Deng, Weihua and Sandev, Trifce}, title = {Levy walk with parameter dependent velocity}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {53}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {11}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/ab7420}, pages = {26}, year = {2020}, abstract = {To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Levy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Levy walk. Two approaches are compared for some already known results. We also consider the generalized Levy walk with parameter dependent velocity. Namely, we consider the Levy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Levy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement.}, language = {en} } @article{StojkoskiSandevBasnarkovetal.2020, author = {Stojkoski, Viktor and Sandev, Trifce and Basnarkov, Lasko and Kocarev, Ljupco and Metzler, Ralf}, title = {Generalised geometric Brownian motion}, series = {Entropy}, volume = {22}, journal = {Entropy}, number = {12}, publisher = {MDPI}, address = {Basel}, issn = {1099-4300}, doi = {10.3390/e22121432}, pages = {34}, year = {2020}, abstract = {Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.}, language = {en} } @article{SinghMetzlerSandev2020, author = {Singh, Rishu Kumar and Metzler, Ralf and Sandev, Trifce}, title = {Resetting dynamics in a confining potential}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {53}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {50}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/abc83a}, pages = {28}, year = {2020}, abstract = {We study Brownian motion in a confining potential under a constant-rate resetting to a reset position x(0). The relaxation of this system to the steady-state exhibits a dynamic phase transition, and is achieved in a light cone region which grows linearly with time. When an absorbing boundary is introduced, effecting a symmetry breaking of the system, we find that resetting aids the barrier escape only when the particle starts on the same side as the barrier with respect to the origin. We find that the optimal resetting rate exhibits a continuous phase transition with critical exponent of unity. Exact expressions are derived for the mean escape time, the second moment, and the coefficient of variation (CV).}, language = {en} } @article{SandevMetzlerChechkin2018, author = {Sandev, Trifce and Metzler, Ralf and Chechkin, Aleksei V.}, title = {From continuous time random walks to the generalized diffusion equation}, series = {Fractional calculus and applied analysis : an international journal for theory and applications}, volume = {21}, journal = {Fractional calculus and applied analysis : an international journal for theory and applications}, number = {1}, publisher = {De Gruyter}, address = {Berlin}, issn = {1311-0454}, doi = {10.1515/fca-2018-0002}, pages = {10 -- 28}, year = {2018}, abstract = {We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.}, language = {en} } @article{SandevTomovskiDubbeldametal.2018, author = {Sandev, Trifce and Tomovski, Zivorad and Dubbeldam, Johan L. A. and Chechkin, Aleksei V.}, title = {Generalized diffusion-wave equation with memory kernel}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {52}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {1}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aaefa3}, pages = {22}, year = {2018}, abstract = {We study generalized diffusion-wave equation in which the second order time derivative is replaced by an integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate the mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with a regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling the broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.}, language = {en} }