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We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be…
We consider a population of particles with unit life length. Dying each particle produces offspring whose size depends on the random environment specifying the reproduction law of all particles of the given generation and on the number of…
This paper introduces the formalism required to analyze a certain class of stochastic control problems that involve a super diffusion as the underlying controlled system. To establish the existence of these processes, we show that they are…
Complex systems are often characterized by the interplay of multiple interconnected dynamical processes operating across a range of temporal scales. This phenomenon is widespread in both biological and artificial scenarios, making it…
We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous…
Stochastic Spatio-Temporal processes are prevalent across domains ranging from modeling of plasma to the turbulence in fluids to the wave function of quantum systems. This letter studies a measure-theoretic description of such systems by…
The paper is devoted to a systematic study of the duality of processes in the sense that $E f(X_t^x,y)=E f (x, Y_t^y)$ for a certain $f$. This classical topic has well known applications in interacting particles, intertwining,…
In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems…
The density hypothesis on random times becomes now a standard in modeling of risks. One of the basic reasons to introduce the density hypothesis is the desire to have a computable credit risk model. However, recent work shows that merely an…
We present new links between some remarkable martingales found in the study of the Binary Search Tree, or of the Bisection Problem, looking at them on the probability space of a continuous time binary branching process.
We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle…
We consider a Markov jump process on a general state space to which we apply a time-dependent weak perturbation over a finite time interval. By martingale-based stochastic calculus, under a suitable exponential moment bound for the…
We investigate the long-time evolution of branching diffusion processes (starting with a finite number of particles) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. In the…
The paper studies processes defined on time domains structured as oriented spatial graphs (or metric graphs, or oriented branched 1-manifolds). This setting can be used, for example, for forecasting models involving branching scenarios. For…
We are concerned with a stochastic mean curvature flow of graphs with extra force over a periodic domain of any dimension. Based on compact embedding method of variational SPDE, we prove the existence of martingale solution. Moreover, we…
In this paper, we investigate the well-posedness of the martingale problem associated to non-linear stochastic differential equations (SDEs) in the sense of McKean-Vlasov under mild assumptions on the coefficients as well as classical…
We study the spatio-temporal prediction problem and introduce a novel point-process-based prediction algorithm. Spatio-temporal prediction is extensively studied in Machine Learning literature due to its critical real-life applications such…
We characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and quadratic variation. The notion of stationarily local integrability plays a key role.
The analysis of spatial extremes requires the joint modeling of a spatial process at a large number of stations and max-stable processes have been developed as a class of stochastic processes suitable for studying spatial extremes. Spatial…
Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of…