A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Ito algebra. A representation theorem, giving the pseudo-Hilbert dilation for the germ-matrix of the differential, is proved. This suggests the general form of quantum stochastic evolution equations with respect to the Poisson (jumps), Wiener (diffusion) or general quantum noise.
A characterisation of quantum stochastic positive definite (PD) exponent is given in terms of the conditional positive definiteness (CPD) of their form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The structure of quasi-Poisson stochastic generators giving rise to a quantum stochastic birth processes is studied.
Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=Δu+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $α=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $α=4k,k \in \bN_{+}$.
We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics. Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the alpha-th positive semidefinite power of the process itself with 0.5<alpha<1 and obtain existence conditions for them. In the case of a…
We propose a definition of branching-type stationary stochastic processes on rooted trees and related definitions of hyper-positivity for functions on the unit circle and functions on the set of non-negative integers. We then obtain (1) a necessary and sufficient condition on a rooted tree for the existence of non-trivial branching-type stationary stochastic processes on it, (2) a complete criterion of the hyper-positive functions in the setting of rooted homogeneous trees in terms of a variant of the classical Herglotz-Bochner Theorem, (3) a prediction theory result for branching-type station…
This paper present a construction and the analysis of a class of non-Gaussian positive-definite matrix-valued homogeneous random fields with uncertain spectral measure for stochastic elliptic operators. Then the stochastic elliptic boundary value problem in a bounded domain of the 3D-space is introduced and analyzed for stochastic homogenization.
We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general given positive definite kernel, but are then subsequently specialized to the above mentioned applications. Given a positive definite kernel $K$ on $S\times S$ where $S$ is a fixed set, we first study families of factorizations of $K$. By a factorization (or representation) we mean a probability space $\left(B,μ\right)$ and an associated stoc…
This is a survey of results about permanental processes, real valued positive processes which are a generalization of squares of Gaussian processes. In a certain sense the symmetric positive definite function that determines a Gaussian process is replaced by a function that is not necessarily symmetric nor positive definite, but that nevertheless determines a stochastic process. This is a new avenue of research with very many open problems.
With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $K$ we analyze associated Gaussian processes $V$. Properties of the Gaussian processes will be derived from certain factorizations of $K$, arising as a covariance kernel of $V$. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $K$. Our results entail an …
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 variate generalization of the chi-squared distribution. Since Wishart processes are defined as a solution to a stochastic differential equation, the existence and uniqueness of strong solutions will be discussed comprehensively. It is also shown that some solutions of the stochastic differential equation can be expressed as squares of matrix variate Ornstein-…
We introduce and give a more or less complete study of a family of branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional analogues of such operators form a very natural family of structured sparse matrices called branching-Toeplitz matrices and will also be investigated. The branching-Toeplitz operators/matrices in this paper should be viewed as natural generalizations of the standard Toeplitz operators/matrices. We will apply our results to construct a family of determinantal point processes on hom…
Let $X_t$ be a reversible and positive recurrent diffusion in $R^d$ described by \begin{equation}\nonumber X_t=x+σb(t)+\int_0^tm(X_s)\dif s, \end{equation} where the diffusion coefficient $σ$ is a positive-definite matrix and the drift $m$ is a smooth function. Let $X_t(A)$ denote the image of a compact set $A\subset R^d$ under the stochastic flow generated by $X_t$. If the divergence of the drift is strictly negative, there exists a set of functions $u$ such that \[\lim_{t\to\infty} \int_{X_t(A)}u(x)\dif x=0\quad{a.s.} \] A characterization of the functions $u$ is provided, as well as lower a…
Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a $\si$-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(μ)$. By using an Itô type inequality for the $H$-norm and an integrability condition for the hyperbound of the semigroup $P_t:=\e^{-Lt}$, general extinction results are derived for a class of continuous adapted processes on $H$. Main applications include stochastic and deterministic fast diffusion equations with fractional Laplacians. Furthermore, we prove exponential integrability of the extinction time for all spa…
We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein-Uhlenbeck process with Levy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein-Uhlenbeck process, where in particular we must restrict to operator-valued Levy processes with "non-decreasing paths". It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Orns…
Given a fixed sigma-finite measure space $\left(X,\mathscr{B},ν\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $ν$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\left(\cdot,\cdot\right)$ with the two variables from the subset of the sigma-algebra $\mathscr{B}$, sets having finite $ν$ measure. Our setting and results are motivated by applications. The latter are covered …
In this paper, we study the stochastic partial differential equation with multiplicative noise $\frac{\partial u}{\partial t} =\mathcal L u+u\dot W$, where $\mathcal L$ is the generator of a symmetric Lévy process $X$ and $\dot W$ is a Gaussian noise. For the equation in the Stratonovich sense, we show that the solution given by a Feynman-Kac type of representation is a mild solution, and we establish its Hölder continuity and the Feynman-Kac formula for the moments of the solution. For the equation in the Skorohod sense, we obtain a sufficient condition for the existence and uniqueness of the…
We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valued fields on a lattice and exploit an approximate gauge-invariance that is restored when taking the continuum limit. This analysis is applied to show the equivalence between Wilson loop expectations computed using partial axial-gauge, complete axial-gauge, and the Migdal-Witten l…
