Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

Tempered fractional processes

Farzad Sabzikar

Tempered fractional Brownian motion (TFBM) is defined by exponentially
tempering the power law kernel in the moving average representation
of a fractional Brownian motion (FBM). TFBM is a Gaussian process with
stationary increments, and we call those increments tempered fractional
Gaussian noise (TFGN). TFGN exhibits semi-long range dependence. That
is, its autocovariance function closely resembles that of fractional
Gaussian noise on an intermediate scale, but then it eventually falls
off more rapidly. The spectral density of TFGN resembles a negative
power law for low frequencies, but eventually converges to zero at very
low frequencies. This behavior of the spectral density is consistent
with the Davenport spectrum that extends the $5/3$ Kolmogorov theory of
turbulence beyond the inertial range. TFBM is a linear combination of
tempered fractional integrals (or derivatives) of a white noise. Using
that fact, we developed the theory of stochastic integration for
TFBM. Replacing the Gaussian random measure in the moving average or
harmonizable representation of TFBM by a stable random measure, we
obtained a linear tempered fractional stable motion (LTFSM), or a real
harmonizable tempered fractional stable motion (HTFSM), respectively.