TY  - JOUR
A1  - Grebenkov, Denis S.
A1  - Sposini, Vittoria
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
A1  - Seno, Flavio
T1  - Exact distributions of the maximum and range of random diffusivity processes
JF  - New Journal of Physics
N2  - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.
KW  - random diffusivity
KW  - extremal values
KW  - maximum and range
KW  - diffusion
KW  - Brownian motion
Y1  - 2021
U6  - https://doi.org/10.1088/1367-2630/abd313
SN  - 1367-2630
VL  - 23
PB  - Dt. Physikalische Ges.
CY  - Bad Honnef
ER  - 
TY  - GEN
A1  - Grebenkov, Denis S.
A1  - Sposini, Vittoria
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
A1  - Seno, Flavio
T1  - Exact distributions of the maximum and range of random diffusivity processes
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - We study the extremal properties of a stochastic process xt defined by the Langevin equation ẋₜ =√2Dₜ ξₜ, in which ξt is a Gaussian white noise with zero mean and Dₜ is a stochastic‘diffusivity’, defined as a functional of independent Brownian motion Bₜ.We focus on threechoices for the random diffusivity Dₜ: cut-off Brownian motion, Dₜt ∼ Θ(Bₜ), where Θ(x) is the Heaviside step function; geometric Brownian motion, Dₜ ∼ exp(−Bₜ); and a superdiffusive process based on squared Brownian motion, Dₜ ∼ B²ₜ. For these cases we derive exact expressions for the probability density functions of the maximal positive displacement and of the range of the process xₜ on the time interval ₜ ∈ (0, T).We discuss the asymptotic behaviours of the associated probability density functions, compare these against the behaviour of the corresponding properties of standard Brownian motion with constant diffusivity (Dₜ = D0) and also analyse the typical behaviour of the probability density functions which is observed for a majority of realisations of the stochastic diffusivity process.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1142 
KW  - random diffusivity
KW  - extremal values
KW  - maximum and range
KW  - diffusion
KW  - Brownian motion
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-503976
SN  - 1866-8372
IS  - 1142
ER  - 
TY  - JOUR
A1  - Sposini, Vittoria
A1  - Grebenkov, Denis S.
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
A1  - Seno, Flavio
T1  - Universal spectral features of different classes of random-diffusivity processes
JF  - New Journal of Physics
N2  - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.
KW  - diffusion
KW  - power spectrum
KW  - random diffusivity
KW  - single trajectories
Y1  - 2020
U6  - https://doi.org/10.1088/1367-2630/ab9200
SN  - 1367-2630
VL  - 22
IS  - 6
PB  - Dt. Physikalische Ges.
CY  - Bad Honnef
ER  - 
TY  - GEN
A1  - Sposini, Vittoria
A1  - Grebenkov, Denis S.
A1  - Metzler, Ralf
A1  - Oshanin, Gleb
A1  - Seno, Flavio
T1  - Universal spectral features of different classes of random-diffusivity processes
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - Stochastic models based on random diffusivities, such as the diffusing-diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random-diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random-diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal, or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic 1/f²-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random-diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 999 
KW  - diffusion
KW  - power spectrum
KW  - random diffusivity
KW  - single trajectories
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-476960
SN  - 1866-8372
IS  - 999
ER  -