Related papers: Spectral analysis of the multi-dimensional diffusi…
Using an operator-theoretic framework in a Hilbert-space setting, we perform a detailed spectral analysis of the one-dimensional Laplacian in a bounded interval, subject to specific non-self-adjoint connected boundary conditions modelling a…
The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion…
We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T.…
This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a…
This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered…
We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the…
In this paper, half inverse spectral problem for diffusion operator with jump conditions dependent on the spectral parameter and discontinuoty coeffcient is considered. The half inverse problems is studied of determining the coeffcient and…
Let $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] %\[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second…
We develop an encounter-based approach for describing restricted diffusion with a gradient drift towards a partially reactive boundary. For this purpose, we introduce an extension of the Dirichlet-to-Neumann operator and use its eigenbasis…
In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case…
We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing/contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers…
A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert…
The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit process of…
We consider a class of diffusions controlled through the drift and jump size, and driven by a jump L\'evy process and a nondegenerate Wiener process, and we study infinite horizon (ergodic) risk-sensitive control problem for this model. We…
We present a model of anomalous diffusion consisting of an ensemble of particles undergoing homogeneous Brownian motion except for confinement by randomly placed reflecting boundaries. For power-law distributed compartment sizes, we…
Operator-valued concentration inequalities are foundational to the analysis of modern high-dimensional statistics and randomized algorithms. However, standard oracle bounds are frequently limited in practice: they require explicit a priori…
This paper presents a novel formula for the transition density of the Brownian motion on a sphere of any dimension and discusses an algorithm for the simulation of the increments of the spherical Brownian motion based on this formula. The…
We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class…
Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let…
We study boundary traces of shift-invariant diffusions: two-dimensional diffusions in the upper half-plane $\mathbb{R} \times [0, \infty)$ (or in $\mathbb{R} \times [0, R)$) invariant under horizontal translations. We prove that the…