Multifractal properties of sample paths of ground state-transformed jump processes

Abstract

We consider a class of Lévy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schrödinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a càdlàg version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local Hölder exponents of sample paths of ground state-transformed processes.

23 pages

References in corpus (5)

• Levy processes: Capacity and Hausdorff dimension
• Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes
• Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials
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Cited by in corpus (3)

• Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks
• Levy flights in steep potential wells: Langevin modeling versus direct response to energy landscapes
• Typical long time behaviour of ground state-transformed jump processes