Related papers: Strong solutions to a beta-Wishart particle system

We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov-R\"ockner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion.…

Existence and uniqueness of weak solutions to the collision-induced breakage and coag-ulation equation are shown when coagulation is the dominant mechanism for small volumes. The collision kernel may feature a stronger singularity for small…

We consider the problem of existence and uniqueness of strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \]…

This paper is concerned with the existence and uniqueness of random periodic solutions for stochastic differential equations (SDEs), where the drift terms involved need not to be uniformly dissipative. On the one hand, via the reflection…

The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete,…

We give a proof of the strong existence and the regularity of stochastic differential equations driven by a Brownian motion and a measurable, Markovian drift without no regularity hypothesis except that the Girsanov exponential associated…

We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and…

The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations…

We derive the numerical schemes for the strong order integration of the set of the stochastic differential equations (SDEs) corresponding to the non-stationary Parker transport equation (PTE). PTE is 5-dimensional (3 spatial coordinates,…

We construct a series of stochastic differential equations of the form $dX_t = b(t, X_t) dt + dB_t$ which exhibit nonuniqueness in the path-by-path sense while having a unique adapted solution in the sense of stochastic processes, i.e.…

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et…

In this paper, we study the following supercritical McKean-Vlasov SDE, driven by a symmetric non-degenerate cylindrical $\alpha$-stable process in $\mathbb{R}^d$ with $\alpha \in (0,1)$: $$ \mathord{{\rm d}} X_t = (K *…

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-time solutions is proved by an extension of the Ovsyannikov method. This…

Based on a student research project this article gives a short review on Wishart processes. A Wishart procces is a matrix valued continuous time stochastic process with a marginal Wishart distribution. The Wishart distribution is a matrix…

In this article, the existence of global classical solutions to the discrete coagulation equations with collisional breakage is established for collisional kernel having linear growth whereas the uniqueness is shown under additional…

This paper investigates the solvability and optimal control of a class of impulsive stochastic differential equations (SDEs) within a Hilbert space setting. First, we establish the existence and uniqueness of mild solutions for the proposed…

Motivated by a recent publication by Ishiwata and Nakata (2022), we prove that sufficiently regular stochastic delay differential equations (SDDEs) with a single discrete delay have blow up solutions if and only if their undelayed…

This paper considers a general one-dimensional stochastic differential equation (SDE). A particular attention is given to the SDEs that may be transformed (via Ito's formula) into:$$d X\_t = ( \bar{B} (X\_t) - b X\_t) d t + \sqrt{X\_t} d…

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary…

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs.…