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 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.

Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

“…For the operators ( 5)-( 6) and more general non-local Schrödinger operators, ground state transformed processes and related stochastic differential equations with jumps have been analysed in [28]. It is, however, in general not trivial to obtain the necessary regularity of φp and the existence of a solution of (8) needs a proper study.…”

Typical long-time behavior of ground state-transformed jump processes

“…(1) As shown in [27,Th. 3.1], under the condition that x → ∇ log ϕ 0 (x) is locally bounded, a GST process ( X t ) t≥0 satisfies the SDE given in (1.4).…”

“…A first variant for jump processes has been obtained in [16,Th. 5.1] for GST processes derived from isotropic stable processes, but the argument is generic and it applies directly to the present settings, see for further details [27,Th. 2.1].…”

“…where ν is the Lévy intensity and A = σσ T is the diffusion matrix of (X t ) t≥0 , and where we use the notation σ∇ · σ∇f (x) = d i,j=1 (σσ T ) ij ∂ x i ∂ x j f (x). Under suitable conditions (see a discussion in [27]), the GST process satisfies a stochastic differential equation with jumps of the form where (B t ) t≥0 is standard Brownian motion, N is a Poisson random measure on [0, ∞) × R d × [0, ∞) with intensity dtν(z)dzdv, and N is the related compensated Poisson measure. From the above two observations it is seen that the potential V perturbing the Lévy process enters the GST process via the ground state ϕ 0 of the operator H, and in general gives rise to a position-dependent drift and a position-dependent bias in the jump kernel, i.e., a Lévy-type process.…”

Characterization of Generalizability of Spike Timing Dependent Plasticity Trained Spiking Neural Networks

“…Simsekli et al (2020a) identified the complexity of the fractals generated by a Feller process that approximates SGD. The intrinsic complexity of a fractal is typically characterized by a notion called the Hausdorff dimension (Le Guével, 2019;Lőrinczi and Yang, 2019), which extends the usual notion of dimension to fractional orders. Due to their recursive nature, Markov processes often generate random fractals (Xiao, 2003).…”