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V. Knopova

Publications and source records attributed to V. Knopova.

10 recordsLinked to original sources
arXiv2026

Invariant Measures of L\'evy-driven Stochastic Differential Equations

We study the structure and regularity of (infinitesimally) invariant measures of the solutions to stochastic differential equations $dX_t = b(X_t)\,dt + dZ_t$, where $(Z_t)_{t\geq 0}$ is a L\'evy process. We show, in particular, that the invariant measure has to satisfy a Volterra-type convolution equation; since we can obtain the kernels explicitly, we are able to apply regularity methods from harmonic analysis. As an application, we get a very short proof -- in any dimension -- of a classic result due to Sato and Yamazato on the form of the invariant measure of a generalized Ornstein--Uhlenbeck process.

math.PR↗
arXiv2020

Construction and heat kernel estimates of general stable-like Markov processes

A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in $\mathbb{R}^d$ and whose generator behaves, locally, like an $α$-stable Lévy process, but the index $α$ and all other characteristics may depend on the state space. More precisely, the jump measure need not to be symmetric and it strongly depends on the current state of the process; moreover, we do not require the gradient term to be dominated by the pure jump part. Our approach is to understand the above phenomena as suitable microstructural perturbations. We show that the corresponding martingale problem is well-posed, and its solution is a strong Feller process which admits a transition density. For the transition density we obtain a representation as a sum of an explicitly given principal term -- this is essentially the density of an $α$-stable random variable whose parameters depend on the current state $x$ -- and a residual term; the $L^\infty\otimes L^1$-norm of the residual term is negligible and so is, under an additional structural assumption, the $L^\infty\otimes L^\infty$-norm. Concrete examples illustrate the relation between the assumptions and possible transition density estimates.

math.PR↗
arXiv2015

Accuracy of discrete approximation for integral functionals of Markov processes

The article is devoted to the estimation of the rate of convergence of integral functionals of a Markov process. Under the assumption that the given Markov process admits a transition probability density which is differentiable in $t$ and the derivative has an integrable upper bound of a certain type, we derive the accuracy rates for strong and weak approximations of the functionals by Riemannian sums. Some examples are provided.

math.PR↗
arXiv2014

A note on the existence of transition probability densities for Lévy processes

We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for Lévy processes and isotropic Lévy processes. Under some mild conditions on the characteristic exponent we calculate the asymptotic behaviour of the transition density as $t\to 0$ and $t\to\infty$ and show a ratio-limit theorem.

math.PR↗
arXiv2013

On the small-time behaviour of Lévy-type processes

We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.

math.PR↗
arXiv2011

A geometric interpretation of the transition density of a symmetric Lévy Process

We study for a class of symmetric Lévy processes with state space $\rn$ the transition density $p_t(x)$ in terms of two one-parameter families of metrics, $(d_t)_{t>0}$ and $(δ_t)_{t>0}$. The first family of metrics describes the diagonal term $p_t(0)$; it is induced by the characteristic exponent $ψ$ of the Lévy process by $d_t(x,y)=\sqrt{tψ(x-y)}$. The second and new family of metrics $δ_t$ relates to $\sqrt{tψ}$ through the formula $$ \exp(-δ_t^2(x,y)) = \Ff[\frac{e^{-tψ}}{p_t(0)}](x-y) $$ where $\Ff$ denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: $p_t(x)=p_t(0) e^{-δ_t^2(x,0)}$ where $p_t(0)$ corresponds to a volume term related to $\sqrt{tψ}$ and where an "exponential" decay is governed by $δ_t^2$. This gives a complete and new geometric, intrinsic interpretation of $p_t(x)$.

math.PR↗
arXiv2010

Transition density estimates for a class of Lévy and Lévy-type processes

We show on- and off-diagonal upper estimates for the transition densities of symmetric Levy and Levy-type processes. To get the an-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal estimates we assume that the characteristic function of a Levy (type) process is analytic, which allows to apply the complex analysis technique.

math.PR↗