Subordinate Brownian Motion Research Articles

Estimates concerning the heat content for the Schrödinger operator related to a subordinate Brownian motion

In this paper, we study the heat content for the Schrödinger operator related to a subordinate Brownian motion and we also establish its small time asymptotic behavior for suitable potentials V. The case V=c1Ω for c>0 and Ω a Borel set of finite measure is investigated in detail.

Harnack inequalities for McKean-Vlasov SDEs driven by subordinate Brownian motions

Existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.

Spectral upper bound for the torsion function of symmetric stable processes

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in R d \mathbb {R}^d . Our bound is explicit and captures the correct order of growth in d d , improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song’s [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion.

Representation of harmonic functions with respect to subordinate Brownian motion

In this article we prove a representation formula for non-negative generalized harmonic functions with respect to a subordinate Brownian motion in a general open set D⊂Rd. We also study oscillation properties of quotients of Poisson integrals and prove that oscillation can be uniformly tamed.

Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps

We study existence of densities for solutions to stochastic differential equations with H\"older continuous coefficients and driven by a $d$-dimensional L\'evy process $Z=(Z_{t})_{t\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(\alpha_{i})_{i=1,\dots,d}\subset(0,2)$ and $C>0$, \begin{align*} \limsup\limits _{t \to 0}t^{1/\alpha_{i}}\int\limits _{\mathbb{R}^{d}}|f_{t}(z+e_{i}h)-f_{t}(z)|dz\leq C|h|,\ \ h\in \mathbb{R},\ \ i=1,\dots,d. \end{align*} Here $e_{1},\dots,e_{d}$ denote the canonical basis vectors in $\mathbb{R}^{d}$. The latter condition covers anisotropic $(\alpha_{1},\dots,\alpha_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \citep{DF13}.

Harnack Inequalities for Functional SDEs Driven by Subordinate Brownian Motions

Using coupling by change of measure and an approximation technique, Wang’s Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.

On overdetermined problems for a general class of nonlocal operators

We study the overdetermined problem for a large family of non-local operators given by generators of subordinate Brownian motions. In particular, this family includes the fractional Laplacian, relativistic stable operators etc. We consider these problems in bounded domains, exterior domains, and in annular domains and we show that under suitable conditions, the domains and solutions are both radially symmetric. Our method uses both analytic and probabilistic tools.

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of L\xe9vy Processes

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

Spectral and analytic properties of some non-local Schrödinger operators and related jump processes

We discuss recent developments in the spectral theory of non-local Schrodinger operators via a Feynman-Kac-type approach. The processes we consider are subordinate Brownian motion and a class of jump Levy processes under a Kato-class potential. We discuss some explicitly soluble specific cases, and address the spatial decay properties of eigenfunctions and the number of negative eigenvalues in the general framework of the processes we introduce.

Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues

We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the It\^o equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy process with independent one-dimensional symmetric $\alpha$-stable components, or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $\alpha$-stable L\'evy process as a special case. We identify conditions on the parameters in the drift, the L\'evy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds.

Schauder Estimates for Equations Associated with Lévy Generators

We study the regularity of solutions to the integro-differential equation $Af-\lambda f=g$ associated with the infinitesimal generator $A$ of a L\'evy process. We show that gradient estimates for the transition density can be used to derive Schauder estimates for $f$. Our main result allows us to establish Schauder estimates for a wide class of L\'evy generators, including generators of stable L\'evy processes and subordinate Brownian motions. Moreover, we obtain new insights on the (domain of the) infinitesimal generator of a L\'evy process whose characteristic exponent $\psi$ satisfies $\text{Re} \, \psi(\xi) \asymp |\xi|^{\alpha}$ for large $|\xi|$. We discuss the optimality of our results by studying in detail the domain of the infinitesimal generator of the Cauchy process.

On the Boundary Theory of Subordinate Killed Lévy Processes

Let $Z$ be a subordinate Brownian motion in ${\mathbb R}^d$, $d\ge 2$, via a subordinator with Laplace exponent $\phi$. We kill the process $Z$ upon exiting a bounded open set $D\subset {\mathbb R}^d$ to obtain the killed process $Z^D$, and then we subordinate the process $Z^D$ by a subordinator with Laplace exponent $\psi$. The resulting process is denoted by $Y^D$. Both $\phi$ and $\psi$ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of $Y^D$, in particular the boundary theory. First, in case that $D$ is a $\kappa$-fat bounded open set, we show that the Harnack inequality holds. If, in addition, $D$ satisfies the local exterior volume condition, then we prove the Carleson estimate. In case $D$ is a smooth open set and the lower weak scaling index of $\psi$ is strictly larger than $1/2$, we establish the boundary Harnack principle with explicit decay rate near the boundary of $D$. On the other hand, when $\psi(\lambda)=\lambda^{\gamma}$ with $\gamma\in (0,1/2]$, we show that the boundary Harnack principle near the boundary of $D$ fails for any bounded $C^{1,1}$ open set $D$. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of $Y^D$. Under an additional condition on $\psi$, we establish sharp two-sided estimates of the jumping kernel of $Y^D$ which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set $E$ strictly contained in $D$, showing that the behavior of $Y^D$ in the interior of $D$ is determined by the composition $\psi\circ \phi$.

On Estimates of Transition Density for Subordinate Brownian Motions with Gaussian Components in C1,1-open Sets

We consider a subordinate Brownian motion $X$ with Gaussian components when the scaling order of purely discontinuous part is between $0$ and $2$ including $2$. In this paper we establish sharp two-sided bounds for transition density of $X$ in ${\mathbb R}^d$ and $C^{1,1}$-open sets. As a corollary, we obtain a sharp Green function estimates.

Stochastic flows for Lévy processes with Hölder drifts

In this paper, we study the following stochastic differential equation (SDE) in \mathbb R^d : \mathrm d X_t= \mathrm d Z_t + b(t, X_t)\,\mathrm d t, \quad X_0=x, where Z is a Lévy process. We show that for a large class of Lévy processes Z and Hölder continuous drifts b , the SDE above has a unique strong solution for every starting point x\in\mathbb R^d . Moreover, these strong solutions form a C^1 -stochastic flow. As a consequence, we show that, when Z is an \alpha -stable-type Lévy process with \alpha\in (0, 2) and b is a bounded \beta -Hölder continuous function with \beta\in (1- {\alpha}/{2},1) , the SDE above has a unique strong solution. When \alpha \in (0, 1) , this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for \nabla \mathbb E_x f(X_t) when Z is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Hölder continuous b and f : \partial_t u+\mathscr L u+b\cdot \nabla u+f=0,\quad u(1, \cdot )=0, where \mathscr L is the generator of the Lévy process Z .

Heat kernel asymptotics of the subordinator and subordinate Brownian motion

For a class of Laplace exponents, we consider the transition density of the subordinator and the heat kernel of the corresponding subordinate Brownian motion. We derive explicit approximate expressions for these objects in the form of asymptotic expansions: via the saddle point method for the subordinator’s transition density and via the Mellin transform for the subordinate heat kernel. The latter builds on ideas from index theory using zeta functions. In either case, we highlight the role played by the analyticity of the Laplace exponent for the qualitative properties of the asymptotics.

Lifschitz singularity for subordinate Brownian motions in presence of the Poissonian potential on the Sierpiński gasket

We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman–Kac functionals.

Estimates of Dirichlet heat kernels for subordinate Brownian motions

In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and the Laplace exponent of the corresponding subordinator only.

Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by α-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.

On purely discontinuous additive functionals of subordinate Brownian motions

Let At=∑s≤tF(Xs−,Xs) be a purely discontinuous additive functional of a subordinate Brownian motion X=(Xt,Px). We give a sufficient condition on the non-negative function F that guarantees that finiteness of A∞ implies finiteness of its expectation. This result is then applied to study the relative entropy of Px and the probability measure induced by a purely discontinuous Girsanov transform of the process X. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

We derive the off-diagonal short-time asymptotics of the heat kernels of functions of generalised Laplacians on a closed manifold. As an intermediate step we give an explicit asymptotic series for the kernels of the complex powers of generalised Laplacians. Each asymptotic series is formulated in terms of the geodesic distance. The key application concerns upper bounds for the transition density of subordinate Brownian motion. The approach is highly explicit and tractable.