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We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain,…

Consider a Markov chain on the space of rooted real binary trees that randomly removes leaves and reinserts them on a random edge and suitably rescales the lengths of edges. This chain was introduced by David Aldous who conjectured a…

This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily…

We consider large random graphs with prescribed degrees, such as those generated by the configuration model. In the regime where the empirical degree distribution approaches a limit $\mu$ with finite mean, we establish the systematic…

In this paper we investigate deterministic diffusion in systems which are spatially extended in certain directions but are restricted in size and open in other directions, consequently particles can escape. We introduce besides the…

For real world systems, nonuniform medium is ubiquitous. Therefore, we investigate the diffusion-limited-aggregation process on a two dimensional directed small-world network instead of regular lattice. The network structure is established…

Motivated by a down-up Markov chain on cladograms, David Aldous conjectured in 1999 that there exists a "diffusion on continuum trees" whose mass partitions at any finite number of branch points evolve as Wright-Fisher diffusions with some…

We analyse diffusion dynamics on weakly-coupled networks (interconnected networks) by means of separation of time scales. Using an adiabatic approximation we reduced the system dynamics to a Markov chain with aggregated variables and…

We present discrete-time approximation of optimal control policies for infinite horizon discounted/ergodic control problems for controlled diffusions in $\Rd$\,. In particular, our objective is to show near optimality of optimal policies…

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when…

Most large language models are autoregressive: they generate tokens one at a time. Discrete diffusion language models can generate multiple tokens in parallel, but sampling from them requires a denoising order: a strategy for deciding which…

Part I of this work [2] developed the exact diffusion algorithm to remove the bias that is characteristic of distributed solutions for deterministic optimization problems. The algorithm was shown to be applicable to a larger set of…

Diffusion-based planning, learning, and control methods present a promising branch of powerful and expressive decision-making solutions. Given the growing interest, such methods have undergone numerous refinements over the past years.…

Diffusion models have become the go-to method for many generative tasks, particularly for image-to-image generation tasks such as super-resolution and inpainting. Current diffusion-based methods do not provide statistical guarantees…

We extend a Discrete Time Random Walk (DTRW) numerical scheme to simulate the anomalous diffusion of financial market orders in a simulated order book. Here using random walks with Sibuya waiting times to include a time-dependent stochastic…

We propose a kind of reaction-diffusion equations for cell differentiation, which exhibits the Turing instability. If the diffusivity of some variables is set to be infinity, we get coupled competitive reaction-diffusion equations with a…

Reduced order models are computationally inexpensive approximations that capture the important dynamical characteristics of large, high-fidelity computer models of physical systems. This paper applies machine learning techniques to improve…

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a…

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter.…

Generative artificial intelligence offers a new paradigm to design matter in high-dimensional spaces. However, its underlying mechanisms remain difficult to interpret and limit adoption in computational mechanics. This gap is striking…