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We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Lévy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Lévy processes, of which Brownian Motion is a special case, have become increasingly popular. Lévy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Lévy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.

We formally introduce and study locally-balanced Markov jump processes (LBMJPs) defined on a general state space. These continuous-time stochastic processes with a user-specified limiting distribution are designed for sampling in settings involving discrete parameters and/or non-smooth distributions, addressing limitations of other processes such as the overdamped Langevin diffusion. The paper establishes the well-posedness, non-explosivity, and ergodicity of LBMJPs under mild conditions. We further explore regularity properties such as the Feller property and characterise the weak generator of the process. We then derive conditions for exponential ergodicity via spectral gaps and establish comparison theorems for different balancing functions. In particular we show an equivalence between the spectral gaps of Metropolis--Hastings algorithms and LBMJPs with bounded balancing function, but show that LBMJPs can exhibit uniform ergodicity on unbounded state spaces when the balancing function is unbounded, even when the limiting distribution is not sub-Gaussian. We also establish a diffusion limit for an LBMJP in the small jump limit, and discuss applications to Monte Carlo sampling and

We introduce a novel approach to studying properties of processes in the π-calculus based on a processes-as-formulas interpretation, by establishing a correspondence between specific sequent calculus derivations and computation trees in the reduction semantics of the recursion-free π-calculus. Our method provides a simple logical characterisation of deadlock-freedom for the recursion- and race-free fragment of the π-calculus, supporting key features such as cyclic dependencies and an independence of the name restriction and parallel operators. Based on this technique, we establish a strong completeness result for a nontrivial choreographic language: all deadlock-free and race-free finite π-calculus processes composed in parallel at the top level can be faithfully represented by a choreography. With these results, we show how the paradigm of computation-as-derivation extends the reach of logical methods for the study of concurrency, by bridging important gaps between logic, the expressiveness of the π-calculus, and the expressiveness of choreographic languages.

Surrogate modeling techniques have become indispensable in accelerating the discovery and optimization of high-entropy alloys(HEAs), especially when integrating computational predictions with sparse experimental observations. This study systematically evaluates the fitting performance of four prominent surrogate models conventional Gaussian Processes(cGP), Deep Gaussian Processes(DGP), encoder-decoder neural networks for multi-output regression and XGBoost applied to a hybrid dataset of experimental and computational properties in the AlCoCrCuFeMnNiV HEA system. We specifically assess their capabilities in predicting correlated material properties, including yield strength, hardness, modulus, ultimate tensile strength, elongation, and average hardness under dynamic and quasi-static conditions, alongside auxiliary computational properties. The comparison highlights the strengths of hierarchical and deep modeling approaches in handling heteroscedastic, heterotopic, and incomplete data commonly encountered in materials informatics. Our findings illustrate that DGP infused with machine learning-based prior outperform other surrogates by effectively capturing inter-property correlations

Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between zeros at fixed pre-specified times. Their additive randomisation allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomised arcade processes (RAPs) can thus be interpreted as a generalisation of anticipative stochastic bridges. The filtrations generated by these processes are utilised to construct a class of martingales that interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally Markov randomised arcade processes, the dynamics of FAMs are informed by Bayesian updating. The same ideas are applied to filtered arcade reverse-martingales, which are constructed in a similar fashion, using reverse-filtrations of RAPs, instead. Several explicit examples for RAPs and FAMs are provided and simulated. This paper concludes with an outlook on potential connections betwee

Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. While explicit solutions for these equations are often elusive, valuable insights can be gained through their asymptotic analysis with respect to relevant parameters. This paper is a brief survey of some recent progress in the study of such equations related to processes with fractional covariance structure.

This paper establishes the precise small-time asymptotic behavior of the spectral heat content for isotropic Lévy processes on bounded $C^{1,1}$ open sets of $\mathbb{R}^{d}$ with $d\ge 2$, where the underlying characteristic exponents are regularly varying at infinity with index $α\in (1,2]$, including the case $α=2$. Moreover, this asymptotic behavior is shown to be stable under an integrable perturbation of its Lévy measure. These results cover a wide class of isotropic Lévy processes, including Brownian motions, stable processes, and jump diffusions, and the proofs provide a unified approach to the asymptotic behavior of the spectral heat content for all of these processes.

We introduce a class of hybrid marked point processes, which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process interacting with its state process. We show the existence and uniqueness of hybrid marked point processes under general assumptions, extending the results of Massoulié (1998) on interacting point processes.

We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form $$ (\mathcal Lf)(x) = \int_{\mathbb R^d \setminus \{0\}} \left(f(x+h)-f(x) - ( abla f(x) \cdot h)1_{\|h\| < 1}\right)K(x,h) dh, $$ where $K(x,\cdot)$ is a jump kernel of the form $K(x,h) \sim \frac{l(\|h\|)}{\|h\|^d}$ for each $x \in \mathbb R^d,\|h\|<1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $α$-geometric stable processes, $α\in (0,2]$.

We study the optimal control of path-dependent piecewise deterministic processes. An appropriate dynamic programming principle is established. We prove that the associated value function is the unique minimax solution of the corresponding non-local path-dependent Hamilton-Jacobi-Bellman equation. This is the first well-posedness result for nonsmooth solutions of fully nonlinear non-local path-dependent partial differential equations.

We obtain local weak limits in probability for Collapsed Branching Processes (CBP), which are directed random networks obtained by collapsing random-sized families of individuals in a general continuous-time branching process. The local weak limit of a given CBP, as the network grows, is shown to be a related continuous-time branching process stopped at an independent exponential time. The proof involves the construction of an explicit coupling of the in-components of vertices with the limiting object. We also show that the in-components of a finite collection of uniformly chosen vertices locally weakly converge (in probability) to i.i.d. copies of the above limit, reminiscent of propagation of chaos in interacting particle systems. We obtain as special cases novel descriptions of the local weak limits of directed preferential and uniform attachment models. We also outline some applications of our results for analyzing the limiting in-degree and PageRank distributions. In particular, upper and lower bounds on the tail of the in-degree distribution are obtained and a phase transition is detected in terms of the growth rate of the attachment function governing reproduction rates in t

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (2002) studied the polynomial transformations of Gaussian FARIMA(0,d,0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (1996, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p,d,q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails.

We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot-Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.

Superpositions of Ornstein-Uhlenbeck type (supOU) processes provide a rich class of stationary stochastic processes for which the marginal distribution and the dependence structure may be modeled independently. We show that they can also display intermittency, a phenomenon affecting the rate of growth of moments. To do so, we investigate the limiting behavior of integrated supOU processes with finite variance. After suitable normalization four different limiting processes may arise depending on the decay of the correlation function and on the characteristic triplet of the marginal distribution. To show that supOU processes may exhibit intermittency, we establish the rate of growth of moments for each of the four limiting scenarios. The rate change indicates that there is intermittency, which is expressed here as a change-point in the asymptotic behavior of the absolute moments.

We propose a novel fused Gromov-Wasserstein alignment method to jointly learn the Hawkes processes in different event spaces, and align their event types. Given two Hawkes processes, we use fused Gromov-Wasserstein discrepancy to measure their dissimilarity, which considers both the Wasserstein discrepancy based on their base intensities and the Gromov-Wasserstein discrepancy based on their infectivity matrices. Accordingly, the learned optimal transport reflects the correspondence between the event types of these two Hawkes processes. The Hawkes processes and their optimal transport are learned jointly via maximum likelihood estimation, with a fused Gromov-Wasserstein regularizer. Experimental results show that the proposed method works well on synthetic and real-world data.

Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0; 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.

We consider random walks associated with conductances on Delaunay triangulations, Gabriel graphs and skeletons of Voronoi tilings which are generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on point processes and conductances, we show that, for almost any realization of the point process, these random walks are recurrent if $d=2$ and transient if $d\geq 3$. These results hold for a large variety of point processes including Poisson point processes, Mat{é}rn cluster and Mat{é}rn hardcore processes which have clustering or repulsive properties. In order to prove them, we state general criteria for recurrence or almost sure transience which apply to random graphs embedded in $\mathbb{R}^d$.

We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using algebraic techniques. Markov cubature rules aid the tractability of path-dependent tasks such as American option pricing in models where the underlying factors are polynomial processes.