In this paper, we investigate pure jump symmetric processes in R whose jumping kernel is comparable to the one for truncated rotationally symmetric α-stable process where jumps of size larger than a fixed number κ > 0 is removed. We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for such jump-type processes. Moreover, we show that bounded functions that ar...

    Given the need to increase the efficiency of heat transfer in thermal systems, especially systems using nanofluids in microscale and nanoscale heat transfer equipment ideas to improve their performance is very good.In present study, the flow and heat transfer of Water-Cu nanofluid in micro-tube with slip regime with constant wall heat flux numerically simulated with low Reynolds numbers. Sl...

Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This report provides some tools for working with such processes. We develop a sample path formula for the cumulative jump height over a given time interval. From this formula an expression for the expected value of the cumulative jump random variable is d...

Let (Nt : t > 0) be the simple counting process associated with the number of jumps in (0, t]. Then, Sn and Xn are the respectively the arrival instant and the size of the nth jump, and we can write Zt = ∑t i=1 Xi. Let Fs = σ(Zu : u ∈ (0,s]), and F• = (Fs : s> 0) be the natural filtration associated with the process Z. Clearly, jump times (Sn : n∈N) are stopping times with respect to filtration...