Related papers: Bismut-Elworthy-Li formulae for Bessel processes

This article describes a class of pseudo-differential operators \begin{equation*} (\mathcal{A}^{\alpha}\varphi)(x)=\mathcal{F}^{-1}_{\xi \rightarrow…

After proving the equivalence of the Bessel $K$-functional and the corresponding spherical modulus of smoothness we define fractional Bessel-Sobolev spaces. As an analog of the classical one the imbedding relation of fractional…

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a…

In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise,…

Let $\lambda>0$ and $\triangle_\lambda:=-\frac{d^2}{dx^2}-\frac{2\lambda}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We show that the oscillation operator ${\mathcal O(P^{[\lambda]}_\ast)}$ and variation operator…

We generalise the so-called Bismut-Elworthy-Li formula to a class of stochastic differential equations whose coefficients might depend on the law of the solution. We give some examples of where this formula can be applied to in the context…

It is demonstrated how to use certain family of commutative hypergroups to provide a universal construction of Biane's quantum Bessel processes of all dimensions not smaller than 1. The classical Bessel processes BES$(\delta)$ are…

For a (non-symmetric) strong Markov process $X$, consider the Feynman--Kac semigroup \[T_t^Af(x):=\mathbb {E}^x\bigl[e^{A_t}f(X_t)\bigr],\quad x\in {\mathbb {R}^n}, t>0,\] where $A$ is a continuous additive functional of $X$ associated with…

We study the distribution of the exponential functional $I(\xi,\eta)=\int_0^{\infty} \exp(\xi_{t-}) \d \eta_t$, where $\xi$ and $\eta$ are independent L\'evy processes. In the general setting using the theories of Markov processes and…

We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{M}, \tau)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we…

In this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the…

In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras $\b F = \b R, \b C$ or $\b H$ which interpolate the convolution algebras of radial bounded…

Consider the one-dimensional elliptic operator given by \begin{equation*} (L_\epsilon f)(x) \;=\; b (x) \, f'(x) \,+\, \epsilon\, a (x)\, f''(x) \;, \end{equation*} where the drift $b\colon R \to R$ and the diffusion coefficient $a\colon R…

In this survey we discuss derivatives of the Wright functions (of the first and the second kind) with respect to parameters. Differentiation of these functions leads to infinite power series with coefficient being quotients of the digamma…

Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It\^{o}'s formula for…

We consider the stochastic differential equations of the form \begin{equation*} \begin{cases} dX^ x(t) = \sigma(X(t-)) dL(t) \\ X^ x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where $\sigma:\mathbb{R}^ d\to \mathbb{R}^ d$ is…

Consider the Leibenson equation \begin{equation*} \partial_t u = \Delta_p u^q, \end{equation*} where $\Delta_p f = div(|\nabla f|^{p-2}\nabla f)$ for $p>1$ and $q>0$, which is a simultaneous generalization of the porous media and the…

We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^{\alpha}u=\phi(\Delta) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $\alpha\in (0,1)$, $\partial_{t}^{\alpha}$ is the Caputo fractional…

The derivatives with respect to order {\nu} for the Bessel functions of argument x (real or complex) are studied. Representations are derived in terms of integrals that involve the products pairs of Bessel functions, and in turn series…

The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered,…