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Let $X$ be a one-dimensional diffusion and let $g\colon[0,T]\times\mathbb{R}\to\mathbb{R}$ be a payoff function depending on time and the value of $X$. The paper analyzes the inverse optimal stopping problem of finding a time-dependent…
We show that the tail distribution $U$ of the explosion time for a multidimensional diffusion (and more generally, a suitable function $\mathscr{U}$ of the Feynman-Kac type involving the explosion time) is a viscosity solution of an…
The author studies the diffusion problem $u_t=u_{xx},\ 0<x<1,\ t>0; \ u(x,0)=0,$ and $-u_x(0,t)=u_x(1,t)=\phi(t),$ where $\phi(t)$ is a control function that ensures that the total mass $\int_0^1 u(x,t_k)dx$ stays between two predetermined…
Generating accurate and stable long rollouts is a notorious challenge for time-dependent PDEs (Partial Differential Equations). Recently, motivated by the importance of high-frequency accuracy, a refiner model called PDERefiner utilizes…
Unruh-DeWitt particle detector models are studied in a variety of time-dependent and time-independent settings. We work within the framework of first-order perturbation theory and couple the detector to a massless scalar field. The…
This article focuses on a large family of cross-diffusion systems of the form $\partial$ t U-$\Delta$A(U) = 0, in dimension d $\in$ N * , and where U $\in$ R 2. We show that under natural conditions on the nonlinearity A, those systems have…
We present local existence and uniqueness results for the following $2+1$ dispersive diffusive equation due to P. Hall arising in modeling of river braiding: $$u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left (u^2…
We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive…
We consider diffusion processes in Hilbert spaces with constant non-degenerate diffusion operators and show that, under broad assumptions on the drift, the transition probabilities of the process are positive on ellipsoids associated with…
We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using cross-fluctuations, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic…
In [1], we showed contractivity of reaction-diffusion PDE: \frac{\partial u}{\partial t}({\omega},t) = F(u({\omega},t)) + D\Delta u({\omega},t) with Neumann boundary condition, provided \mu_{p,Q}(J_F (u)) < 0 (uniformly on u), for some 1…
We consider a non-Gaussian stochastic process where a particle diffuses in the $y$-direction, $dy/dt=\eta(t)$, subject to a transverse shear flow in the $x$-direction, $dx/dt=f(y)$. Absorption with probability $p$ occurs at each crossing of…
We study nonlinear pantograph-type reaction-diffusion PDEs, which, in addition to the unknown $u=u(x,t)$, also contain the same functions with dilated or contracted arguments of the form $w=u(px,t)$, $w=u(x,qt)$, and $w=u(px,qt)$, where $p$…
We first study crossing statistics in random connection models (RCM) built on marked Poisson point processes on $\mathbb R^d$. Under general assumptions, we show exponential tail bounds for the number of crossings of a box contained in the…
In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{N}, whose distribution is related to space-time fractional n-dimensional…
Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…
In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a…
This paper presents a minimum flow approach applicable to a wide range of doubly nonlinear diffusion problems. We introduce a minimum flow steepest descent algorithm that seeks an optimal traffic flow by minimizing an internal energy…
We extend Regularised Diffusion-Shock (RDS) filtering from Euclidean space $\mathbb{R}_2$ [1] to position-orientation space $\mathbb{M}_2 \cong \mathbb{R}^2 \times S^1$. This has numerous advantages, e.g. making it possible to enhance and…
We study the liberation process for projections: $(p,q)\mapsto (p_t,q)= (u_tpu_t^\ast,q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\{p,q\}$. Its action on the operator-valued angle $qp_tq$ between the…