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Davar Khoshnevisan is a mathematician whose work concerns probability theory, particularly random fields and stochastic partial differential equations. His institutional page lists research on long-time behavior and well-posedness of randomly forced equations, as well as geometric properties of their solutions.^[ref]

His Topics in Probability: Lévy Processes by Davar Khoshnevisan develops how processes with stationary independent increments are decomposed and described by their exponents. For instance, the notes distinguish increasing jump processes through the Laplace exponent of a subordinator of a subordinator, connecting an abstract stochastic process to an explicitly computable transform.^[ref]

The jump activity of a Lévy process describes whether it has finitely or infinitely many jumps in a bounded time interval. Its Lévy measure $\nu$ decides this: finite activity corresponds to $\nu ({\mathbb{R}}^d\setminus \{0\})<\infty$, in which case the jump component is a Compound Poisson process. Infinite mass gives infinitely many small jumps in every positive-length time interval, almost surely. For every $\epsilon >0$, however, jumps larger than $\epsilon$ still have a finite rate.

Activity and total variation are different. With zero Brownian component, sample paths have finite variation on compact time intervals exactly whenFor example, $\nu (dx)=c{x}^{-1-\alpha }dx$ on $(0,\infty )$, with $c>0$ and $0<\alpha <1$, has infinite total mass but a finite integral of $x$ near zero. It therefore permits infinite jump activity with finite variation. Conversely, Brownian motion has no jumps but infinite total variation on every positive-length time interval, almost surely.^[ref]

For a finite-valued subordinator $T$ starting at zero, the Laplace exponent of a subordinator is the function $\Phi$ determined byIts Lévy–Khintchine formula takes the formwhere $d$ is the deterministic drift coefficient and $\nu$ is the Lévy measure. This representation separates steady motion of the clock from its positive jumps.^[ref]

For a deterministic clock $T_t=dt$, $\Phi (\lambda )=d\lambda$. For a Poisson process of rate $a$ with unit jumps, $\Phi (\lambda )=a(1-e^{-\lambda })$. For a sum of independent subordinators, the Laplace exponent of a subordinator is the sum of the individual functions, because the Laplace transforms multiply. The transform at $\lambda =0$ gives $\Phi (0)=0$.