Relativistic Α-stable Processes Research Articles

Relativistic stable processes in quasiballistic heat conduction in semiconductors.

In this article, we show how relativistic α-stable processes can be used to explain quasiballistic heat conduction in semiconductors. This is a method that can fit experimental results of ultrafast laser heating in alloys. It also provides a connection to a rich literature on the Feynman-Kac formalism and random processes that transition from a stable Lévy process on short time and length scales to the Brownian motion at larger scales. This transition was captured by a heuristic truncated Lévy distribution in earlier papers. The rigorous Feynman-Kac approach is used to derive sharp bounds for the transition kernel. Future directions are briefly discussed.

Trace Estimates for Relativistic Stable Processes

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C 1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t 2/α t −d/α for C 1,1 open sets and of order t 1/α t −d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t −d/α and t (2−d)/α for C 1,1 open sets, and, when α ∈ (0, 1), between the orders t −d/α and t (1−d)/α for Lipschitz open sets.

Two-term trace estimates for relativistic stable processes

We prove trace estimates for the relativistic α-stable process extending the result of Bañuelos and Kulczycki (2008) in the stable case.

Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane

We study α-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for α ∈ (1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for α-harmonic functions. We also give explicit formulas for the Martin kernel of the complement of the sphere and for the harmonic measure, Green function and Martin kernel of the complement of the hyperplane for the symmetric α-stable Levy processes. Some extensions for the relativistic α-stable processes are discussed.

Global heat kernel estimate for relativistic stable processes in exterior open sets

In this paper, sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m∈(0,1] in C1,1 exterior open sets are established for all time t>0. These transition densities are also the Dirichlet heat kernels of m−(m2/α−Δ)α/2 with m∈(0,1] in C1,1 exterior open sets. The estimates are uniform in m in the sense that the constants are independent of m∈(0,1]. As a corollary of our main result, we establish sharp two-sided Green function estimates for relativistic α-stable processes with mass m∈(0,1] in C1,1 exterior open sets.

Global Heat Kernel Estimates for Relativistic Stable Processes in Half-space-like Open Sets

In this paper, by using probabilistic methods, we establish sharp two-sided large time estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] (i.e., for the Dirichlet heat kernels of m − (m 2/α − Δ) α/2 with m ∈ (0, 1]) in half-space-like C 1, 1 open sets. The estimates are uniform in m in the sense that the constants are independent of m ∈ (0, 1]. Combining with the sharp two-sided small time estimates, established in Chen et al. (Ann Probab, 2011), valid for all C 1, 1 open sets, we have now sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets for all times. Integrating the heat kernel estimates with respect to the time variable, one can recover the sharp two-sided Green function estimates for relativistic α-stable processes with mass m ∈ (0, 1] in half-space-like C 1, 1 open sets established recently in Chen et al. (Stoch Process their Appl, 2011).

ON A SUFFICIENT CONDITION FOR LARGE DEVIATIONS OF ADDITIVE FUNCTIONALS

We prove the large deviation principle for continuous additive functionals under certain assumptions. The underlying symmetric Markov processes include Brownian motion and symmetric and relativistic α-stable processes.

Two-Sided Optimal Bounds for Green Functions of Half-Spaces for Relativistic α-Stable Process

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α -stable process with parameter m on ℝ d space. This process has an infinitesimal generator of the form mI–(m 2/α I–Δ) α/2, where 0 0, and reduces to the isotropic α-stable process for m=0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in Byczkowski et al. (Bessel Potentials, Hitting Distributions and Green Functions (2006) (preprint). http://arxiv.org/abs/math/0612176 ), where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d≥3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough.

Differentiability of Spectral Functions for Relativistic α-Stable Processes with Application to Large Deviations

We consider the relativistic α-stable process, a pure jump Markov process generated by $\mathcal{H}^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha /2}-m$ . Let −C(λ) be the bottom of spectrum of Schrodinger type operator $\mathcal{H}^{\lambda \mu} = \mathcal{H}^{\alpha} - \lambda \mu$ , where μ is a signed Kato measure. We prove the differentiability of C(λ). As an application of it, we establish a large deviation principle for the additive functional $A_t^{\mu}$ corresponding to the measure μ.

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G ( x , y ) and G D ( x , y ) be the Green functions of rotationally invariant symmetric α-stable process in R d and in an open set D, respectively, where 0 < α < 2 . The inequality G D ( x , y ) G D ( y , z ) / G D ( x , z ) ⩽ c ( G ( x , y ) + G ( y , z ) ) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c = c ( D ) ∈ ( 0 , ∞ ) , then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form G D ( x , y ) G D ( z , w ) / G D ( x , w ) , where y may be different from z. When y = z , we recover the usual 3G. The 3G form G D ( x , y ) G D ( z , w ) / G D ( x , w ) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α = 1 ) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.

Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes

Let X t X_t be the relativistic α \alpha -stable process in R d \mathbf {R}^d , α ∈ ( 0 , 2 ) \alpha \in (0,2) , d &gt; α d &gt; \alpha , with infinitesimal generator H 0 ( α ) = − ( ( − Δ + m 2 / α ) α / 2 − m ) H_0^{(\alpha )}= - ((-\Delta +m^{2/\alpha })^{\alpha /2}-m) . We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup T t T_t for this process with generator H 0 ( α ) − V H_0^{(\alpha )} - V , V ≥ 0 V \ge 0 , V V locally bounded. We prove that if lim | x | → ∞ V ( x ) = ∞ \lim _{|x| \to \infty } V(x) = \infty , then for every t &gt; 0 t &gt;0 the operator T t T_t is compact. We consider the class V \mathcal {V} of potentials V V such that V ≥ 0 V \ge 0 , lim | x | → ∞ V ( x ) = ∞ \lim _{|x| \to \infty } V(x) = \infty and V V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V V in the class V \mathcal {V} we show that the semigroup T t T_t is IU if and only if lim | x | → ∞ V ( x ) / | x | = ∞ \lim _{|x| \to \infty } V(x)/|x| = \infty . If this condition is satisfied we also obtain sharp estimates of the first eigenfunction ϕ 1 \phi _1 for T t T_t . In particular, when V ( x ) = | x | β V(x) = |x|^{\beta } , β &gt; 0 \beta &gt; 0 , then the semigroup T t T_t is IU if and only if β &gt; 1 \beta &gt;1 . For β &gt; 1 \beta &gt;1 the first eigenfunction ϕ 1 ( x ) \phi _1(x) is comparable to \[ exp ⁡ ( − m 1 / α | x | ) ( | x | + 1 ) ( − d − α − 2 β − 1 ) / 2 . \exp (-m^{1/{\alpha }}|x|) \, (|x| + 1)^{(-d - \alpha - 2 \beta -1 )/2}. \]

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C 1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman–Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.

Estimates of Green Function for Relativistic α-Stable Process

We call an R d -valued stochastic process X t with characteristic function exp {−t{(m 2/α+ξ2)α/2−m}},ξ∈R d ,m>0, the relativistic α-stable process. In the paper we derive sharp estimates for the Green function of the relativistic α-stable process on C 1,1 domains. Using these estimates we provide lower and upper bounds for the Poisson kernel. As another application we derive 3G Theorem and Boundary Harnack Principle for C 1,1 domains.