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Ivan Biočić

Publications and source records attributed to Ivan Biočić.

9 recordsLinked to original sources
arXiv2026

Monte Carlo Approximations of Time-Nonlocal Diffusions in Bounded Domains

We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.

math.PR↗
arXiv2026

Modelling Anomalous Diffusion: The Role of CTRWs and Non-Local Dynamics

These notes provide a self-consistent summary of the stochastic approach to anomalous diffusion based on Continuous Time Random Walks (CTRWs) and their scaling limits. An introduction to CTRWs and their relationship to the theory of semi-Markov processes is provided. A general technique to study scaling limits of CTRWs is then described, and the semi-Markov property of the limiting processes is discussed. With this at hand, the connection of limit processes with non-local (fractional-type) equations is introduced, and the most recent (and general) contributions, going far beyond fractional equations, are also described. Indeed, the theory presented here includes very general non-local evolution equations as abstract Cauchy problems as well as pointwise non-local fractional diffusion equations in bounded domains.

math.PR↗
arXiv2026

Nonhomogeneous boundary condition for spectral non-local operators

We study semilinear non-local elliptic problems driven by spectral-type operators of the form $ψ(-L_{|D})$ in a bounded $C^{1,1}$ domain $D\subset \mathbb{R}^d$ with a nonhomogeneous boundary condition. Here $ψ$ is a Bernstein function satisfying a weak scaling condition at infinity, and $L_{|D}$ is the generator of a killed Lévy process. This general framework covers and extends the theory of the interpolated fractional Laplacian. A key novelty in this setting is the analysis of the nonhomogeneous boundary condition formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. We establish sharp boundary estimates for Green and Poisson potentials, introduce a weak $L^1$ trace-like boundary operator, and provide existence results for solutions under quite general nonlinearities, including sign-changing and non-monotone cases. The methodology combines stochastic process techniques, potential theory, and spectral analysis, and expresses the boundary behavior of the solution in terms of the renewal function and the distance to the boundary, suggesting a possible unified treatment of semilinear boundary problems in non-local settings.

math.AP↗
arXiv2025

Continuous Branching Processes with Settlement in Cancer Metastasis: Stochastic Modelling and the Feller Property

Motivated by models of cancer metastasis, this paper introduces a type of (multi-type) branching process that records the positions of particles, representing tumor cells or clusters. Particles may be absorbed (removed from the state space), move, or settle. The process is rigorously constructed, and the Markov property is established via embedding into a multidimensional process that tracks the labels, positions, and phases (moving or resting) of living particles. The Feller property for the associated semigroup is investigated. It is proved for a simplified model that tracks the number of particles in each class, and an explicit generator is derived, enabling Feynman-Kac-type formulas in this framework.

math.PR↗
arXiv2025

Sampling inverse subordinators and subdiffusions

In this paper, a method to exactly sample the trajectories of inverse subordinators (in the sense of the finite-dimensional distributions), jointly with the undershooting or overshooting process, is provided. The method applies to general strictly increasing subordinators. The (random) running times of these algorithms have finite moments and explicit bounds for the expectations are provided. Additionally, the Monte Carlo approximation of functionals of subdiffusive processes (in the form of time-changed Feller processes) is considered where a central limit theorem and the Berry-Esseen bounds are proved. The approximation of time-changed Itô diffusions is also studied. The strong error, as a function of the time step, is explicitly evaluated demonstrating the strong convergence, and the algorithm's complexity is provided. The Monte Carlo approximation of functionals and its properties for the approximate method is studied as well. An application of our algorithms in the context of weak ergodicity breaking of subdiffusion is also discussed.

math.PR↗
arXiv2025

Large solutions for subordinate spectral Laplacian

We find a large solution to a semilinear Dirichlet problem in a bounded $C^{1,1}$ domain for a non-local operator $ϕ(-Δ\vert_{D})$, an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained, and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the nonlinearity. Additionally, we prove interior higher regularity results for this operator.

math.AP↗
arXiv2025

Harmonic problems arising from continuous time random walks limit processes

In this paper, we develop a universal method that identifies the (non-local) governing evolution equations for Continuous Time Random Walks' (CTRWs) limit processes. Given one of these processes, our method provides the form of a non-local operator, acting on space and time variables jointly, such that the (generalized) harmonic problem associated with it represents an evolution governing equation for this process. Then, the well-posedness of this problem must be established case by case. In this paper, we establish well-posedness when the process is a Feller process (on a general Polish space $E$) time-changed with the overshooting of a subordinator. Also, we will show how our method applies to several cases when the equation and its well-posedness are already known, hence unifying several different approaches in the literature.

math.PR↗