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Geometric approximations to transition densities of Jump-type Markov processes

  • Yuanying Zhuang and Xiao Song EMAIL logo
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

Keywords: Levy-type processes; pseudo-differential operator; heat kernels
MSC 2010: 60J75

1 Introduction

Feller semigroups [1] and sub-Markovian semigroups [2] are basic objects to construct stochastic processes. In case that the transition functions of these semigroups have a density with respect to Lebesgue measure, estimates for these transition functions lead immediately to estimates for probabilities. However, in general not much is known about off-diagonal estimates for these transition functions when dealing with Levy-type processes, the generators of which are pseudo-differential operators with negative definite symbols. We refer to [3,4,5] and in particular to [6] and [7] in which basic existence results and some properties were discussed. The situation for diagonal estimates is better. Therfore, general results presented in [8] and comparison results, see for example [9], could be used.

Recently, in [10,11,12], a geometric interpretation for the transition function of a Levy process was suggested. The basic idea is to try to get bounds of Gaussian type but with different metrics. Metric measure spaces are since some time employed to study diffusions, see [13] or [14] and references therein. More recently also jump processes were considered, see [15] and references therein. However, in these cases the metric is given and the process already relates to that metric. A basic reference for the analysis of metric measure space is given in [16].

The idea in [10] is different. Using the square root of a (nice) negative definite function gives rise to a metric and that these are the only metrics which induce a metric space isometric to a Hilbert space, see [17,18] or [19]. In [10], it was suggested to express the diagonal estimate in terms of the metric induced by the symbol. Moreover, since Fourier transform of “Gaussians” should be “Gaussians” it was also suggested to express the off-diagonal term as a function decaying exponentially with respect to (the square of) another metric. All this was discussed for Levy processes. Only vague indications were made for generators with variable coefficients.

The original aim of our investigations was to settle the case with generators of variable coefficients as elliptic diffusions can be treated using the Riemannian metric associated with the principal symbol of the generator. But as we proceed, almost all basic concepts break down when trying to transfer from the local to non-local case. However, we succeeded to make several case studies in non-trivial situations which we think shed some light on the problems we encounter in the general situations.

Compared with the previous results, the contributions lie in the following three aspects.

  1. We summarize the classical comparison results of heat kernels and give a concrete diagonal estimates of transition function and Cauchy semigroup and subordinate Cauchy semigroup.

  2. We estimate the general probability moving from one metric ball to another and discuss the special situation in which metric balls can be decoupled into the combination of smaller metric balls. Furthermore, we list a few special examples to show the actual effects of our decoupling effects and their geometric interpretation.

  3. Graphical experiments have been given in this paper, which displays the effects of parametric variation of metric balls on the transition functions.

The rest of this paper is organized as follows. In Section 2, we introduce some classical diagonal estimates for the corresponding transition functions and give a concrete example of diagonal estimates of transition function of Cauchy semigroup and its subordination. In Section 3, we present our main results and their analytical proof. In Section 4, we make graphical experiments to show characteristics of our investigations. Finally, in Section 5, we summarize the paper with our key results and add some remarks.

2 Heat kernels and some of their estimates

Throughout the paper, we will use the following notations. R n denotes the n -dimensional Euclidean space, and C ∞ ( R n ) denotes the set of continuous functions that vanish at infinity. L p ( R n ) denotes the set of measurable functions ∣ f ∣ p , { 1 ≤ p < ∞ } are Lebesgue integrable in R n . B ( n ) denotes the Borel sets in R n . B b ( R n ) denotes the bounded Borel measurable functions in R n . ε x denotes the Dirac measure at x ∈ R n .

In this section, we will discuss estimates for heat kernels. We discuss mainly symmetric Feller semigroups since we can use Hilbert space methods to obtain regularity results.

First of all, let ( T t ) t ≥ 0 be a Feller semigroup or sub-Markovian semigroup on C ∞ ( R n ) ∩ L 2 ( R n ) . From the definition of Feller semigroup or sub-Markovian semigroup, for x ∈ R n and t ≥ 0 fixed, the mapping u ↦ T t u ( x ) is a linear continuous and positive functional on C ∞ ( R n ) ∩ L 2 ( R n ) . From Riesz’s representation theorem, it follows that there exists a Borel measure p t ( x , d y ) on ℬ ( n ) , which is uniquely determined and

(2.1) T t u ( x ) = ∫ R n u ( y ) p t ( x , d y )

holds for u ∈ C ∞ ( R n ) ∩ L 2 ( R n ) . And we can extend ( T t ) t ≥ 0 to all constant functions using the lemma below. The following result is taken from Lemma 4.6.24 in [3] without proof.

Lemma 2.1

Let ( T t ) t ≥ 0 be a Feller semigroup or sub-Markovian semigroup. Then we may extend T t , t > 0 , to all constant functions x ↦ a ∈ R , and for a ≥ 0 we have

(2.2) T t a ≤ a .

By monotone convergence we find for any measurable functions u ν → 1 in R n using (2.2)

1 ≥ ( T t 1 ) ( x ) = lim ν → ∞ ∫ R n u ν ( y ) p t ( x , d y ) = ∫ R n 1 p t ( x , d y ) .

Therefore, each of the measures p t ( x , d y ) , x ∈ R n , t ≥ 0 , is a sub-probability measure. Hence, we may extend T t to B b ( R n ) just by defining the operator T ˜ t by

(2.3) T ˜ t u ( x ) ≔ ∫ R n u ( y ) p t ( x , d y )

for all u ∈ B b ( R n ) . Moreover, we find that

∣ T ˜ t u ( x ) ∣ ≤ ∫ R n ∣ u ( y ) ∣ p t ( x , d y ) ≤ ‖ u ‖ ∞

or

‖ T ˜ t u ‖ ∞ ≤ ‖ u ‖ ∞ ,

i.e. T ˜ t : B b ( R n ) → { u : R n → R ∣ ‖ u ‖ ∞ < ∞ } is a positive preserving contraction. So we claim the following theorem.

Theorem 2.2

Let ( T t ) t ≥ 0 be a Feller semigroup or sub-Markovian semigroup on C ∞ ( R n ) ∩ L 2 ( R n ) and define p t ( x , d y ) by (2.1). Then p t ( x , d y ) is a sub-Markovian kernel for every t ≥ 0 .

Now we are in the position to prove also the semigroup property of the extended family ( T ˜ t ) t ≥ 0 of the Feller semigroup or sub-Markovian semigroup ( T t ) t ≥ 0 . For u ∈ C ∞ ( R n ) and t , s ≥ 0 we can use Fubini’s theorem

∫ R n p t + s ( x , d z ) u ( z ) = T t + s u ( x ) = T t ( T s u ) ( x ) = ∫ R n p t ( x , d y ) ∫ R n p s ( y , d z ) u ( z ) = ∫ R n ∫ R n p t ( x , d y ) p s ( y , d z ) u ( z )

and the uniqueness part of Riesz’s representation theorem gives

(2.4) p t + s ( x , A ) = ∫ A p t ( x , d y ) p s ( y , A )

for all t , s ≥ 0 , x ∈ R n and A ∈ ℬ ( n ) . The equations (2.4) are called the Chapman-Kolmogorov equations, and it follows that

(2.5) T ˜ t + s = T ˜ t ∘ T ˜ s = T ˜ s ∘ T ˜ t .

In addition, since T 0 = id , we find that

(2.6) u ( x ) = T 0 u ( x ) = ∫ R n u ( y ) p 0 ( x , d y ) ,

implying that

(2.7) p 0 ( x , d y ) = ε x ( d y )

for all x ∈ R n , i.e. T ˜ 0 = id . It proves the following theorem.

Theorem 2.3

Let ( T t ) t ≥ 0 be a Feller semigroup or sub-Markovian semigroup on C ∞ ( R n ) ∩ L 2 ( R n ) . Then there exists a family ( p t ( ⋅ , ⋅ ) ) t ≥ 0 of sub-Markovian kernels on R n × ℬ ( n ) satisfying the Chapman-Kolmogorov equations (2.4).

For L 2 -sub-Markovian semigroups, the situation is more complicated since x ↦ T t ( 2 ) u ( x ) in this case is only almost everywhere defined. In the situations we are interested in, however, we can assume the existence of sub-Markovian kernels with similar properties as discussed for Feller semigroups. For details we refer to Chapter 3 in [4] and Chapter 6 in [5].

Definition 2.4

An L 2 -sub-Markovian semigroup ( T t ) t ≥ 0 is conservative if T t 1 = 1 a.e. for all t > 1 .

Let ( T t ) t ≥ 0 be a symmetric Feller semigroup or an L 2 -sub Markovian semigroup with representing kernels p t ( x , d y ) , i.e.

T t u ( x ) = ∫ u ( y ) p t ( x , d y ) .

We want to find conditions for p t ( x , d y ) having a density with respect to Lebesgue measure, i.e.

p t ( x , d y ) = p t ( x , y ) λ ( n ) ( d y ) .

And then we long for estimates for the density p t ( x , y ) . The following result, due to Dunford and Pettis [20], is a tool for getting the existence of densities.

Theorem 2.5

Let K op : L p ( G ) → L ∞ ( G ) , 1 ≤ p < ∞ , be a bounded linear operator. Then there exists a kernel function K : G × G → C , K ∈ L ∞ ( G ) ⊗ L p ′ ( G ) , 1 p + 1 p ′ = 1 , such that

(2.8) K op u ( x ) = ∫ G K ( x , y ) u ( y ) d y .

Conversely, every operator defined by (2.8) with a kernel function K : G × G → C , K ∈ L ∞ ( G ) ⊗ L p ′ ( G ) , 1 p + 1 p ′ = 1 , defines a bounded linear operator from L p ( G ) into L ∞ ( G ) . Furthermore, the operator norm of K op is given by

‖ K op ‖ = ess sup x ∈ G ‖ K ( x , ⋅ ) ‖ L p ′ .

In case that the semigroup under consideration is also a conservative symmetric L 2 -Markovian semigroup the next result taken from Varopoulos et al. [8] gives estimates for the operator norms of T t , and these estimates are diagonal estimates for the corresponding densities.

Theorem 2.6

Let ( T t ) t ≥ 0 be a symmetric conservative sub-Markovian semigroup on L 2 ( R n ) with related regular Dirichlet form ( ℰ , D ( ℰ ) ) . Moreover, let p > 2 and N ≔ 2 p p − 2 > 2 . The following estimates are equivalent

(2.9) ‖ u ‖ L p 2 ≤ c ℰ ( u , u ) f o r a l l u ∈ D ( ℰ ) ;

(2.10) ‖ u ‖ L 2 2 + 4 / N ≤ c ℰ ( u , u ) ‖ u ‖ L 1 4 / N f o r a l l u ∈ D ( ℰ ) ∩ L 1 ( R n ) ;

(2.11) ‖ T t ‖ L 1 − L ∞ ≤ c ′ t − N / 2 f o r a l l t > 0 ,

where ‖ B ‖ X − Y denotes the operator norm of B : X → Y .

Note that (2.9) is a Sobolev-type inequality and (2.10) is a Nash-type inequality. For more information about Dirichlet form, readers can refer to [21] and [22].

An application of Theorem 2.5 to (2.11) yields that T t has a Kernel representation, i.e.

(2.12) T t u ( x ) = ∫ R n u ( y ) K t ( x , y ) d y , K t ( x , y ) ≥ 0

and

ess sup x , y ∈ R n K t ( x , y ) ≤ c 1 t − N / 2 .

And when we have instead of (2.9) a Gårding-type inequality

‖ u ‖ L p 2 ≤ c 2 ℰ λ ( u , u ) = c 2 ( ℰ ( u , u ) + λ ( u , u ) 0 ) , λ > 0 ,

we obtain instead of (2.11)

‖ T t ‖ L 1 − L ∞ ≤ c 3 ( λ ) e λ t t N / 2

and

ess sup x , y ∈ R n K t ( x , y ) ≤ c 4 ( λ ) e λ t t N / 2 .

Finally, we will give examples of diagonal estimates of transition functions of subordinate sub-Markovian or Feller semigroup. First of all, we need to state a theorem which is Theorem 1 of Schilling and Wang [23].

Theorem 2.7

Let ( T t ) t ≥ 0 be a strong continuous contraction semigroup of symmetric operators on L 2 ( X , m ) and assume that for each t ≥ 0 , T t ∣ L 2 ( X , m ) ∩ L 1 ( X , m ) has an extension which is a contraction on L 1 ( X , m ) , i.e. ‖ T t u ‖ 1 ≤ ‖ u ‖ 1 for all u ∈ L 1 ( X , m ) ∩ L 2 ( X , m ) . Suppose that the generator ( A , D ( A ) ) satisfies the following Nash-type inequality:

(2.13) ‖ u ‖ 2 2 B ( ‖ u ‖ 2 2 ) ≤ ⟨ A u , u ⟩ , u ∈ D ( A ) , ‖ u ‖ 1 = 1 ,

where B : ( 0 , ∞ ) → ( 0 , ∞ ) is an increasing function. Then, for any Bernstein function f , the generator f ( A ) of the subordinate semigroup satisfies

(2.14) ‖ u ‖ 2 2 2 f B ‖ u ‖ 2 2 2 ≤ ⟨ f ( A ) u , u ⟩ , u ∈ D ( f ( A ) ) , ‖ u ‖ 1 = 1 .

The following example is at first the result of diagonal estimate of transition function in the context of Cauchy semigroup. Then we deduce the diagonal estimates of transition function of subordinate Cauchy semigroup in the end.

Example 2.8

Let ( X t ) t ≥ 0 be the two-dimensional Cauchy process whose symbol is ψ ( ξ , η ) = ξ 2 + η 2 and the transition function is

(2.15) p t ψ ( x , y ) = 1 2 π t ( ( x 2 + y 2 ) + t 2 ) 3 / 2 .

Therefore, we have ‖ T t ‖ L 1 − L ∞ = p t ( 0 ) = 1 2 π t ( ( x 2 + y 2 ) + t 2 ) 3 / 2 x = 0 , y = 0 = 1 2 π t − 2 . Using Theorem 2.6, we know from (2.15) that N = 4 in the inequality (2.11) which implies

(2.16) ‖ u ‖ L 2 3 ≤ c ℰ ( u , u ) , u ∈ D ( ℰ ) ∩ L 1 ( R 2 ) , ‖ u ‖ L 1 = 1 .

In order to apply Theorem 2.7, we rewrite (2.16) as

‖ u ‖ L 2 2 B ( ‖ u ‖ L 2 2 ) ≤ ⟨ A u , u ⟩ where B ( x ) = 1 c x 1 / 2 ,

therefore, we can deduce from Theorem 2.7 that if we let f ( x ) = x α , ( 0 < α < 1 ) , we have

(2.17) ‖ u ‖ L 2 2 f B ‖ u ‖ 2 2 2 ≤ ⟨ A f u , u ⟩ , u ∈ D ( A f ) , ‖ u ‖ L 1 = 1 , ‖ u ‖ L 2 2 f 1 c ‖ u ‖ 2 2 2 1 2 ≤ ⟨ A f u , u ⟩ , ‖ u ‖ L 2 2 ( 1 / c α ) ‖ u ‖ 2 2 2 α 2 ≤ ⟨ A f u , u ⟩ , ‖ u ‖ L 2 2 + α ≤ c ′ ⟨ A f u , u ⟩ .

Using Theorem 2.6, this implies

‖ T t f ‖ L 1 → L ∞ ≤ c ″ t − N ′ 2 , t > 0 , N ′ = 4 / α ,

which is equivalent to p t f ( 0 ) ≤ c ″ t − N ′ 2 .

So as to introduce our next example, we also need to state three theorems. The first theorem is taken from [4]. We introduce two symbol classes Λ and S ρ m , ψ ( R n ) and Theorem 2.6.9 which we take from [4].

Definition 2.9

We say that a continuous negative definite function ψ : R n → R belongs to the class Λ if it satisfies

(2.18) ∣ ∂ ξ α ( 1 + ψ ( ξ ) ) ∣ ≤ c ∣ α ∣ ( 1 + ψ ( ξ ) ) 2 − ρ ( ∣ α ∣ ) 2 ,

where ρ : N 0 → N 0 , k ↦ ρ ( k ) ≔ k ∧ 2 , c ∣ α ∣ is a constant related to ∣ α ∣ , where α belongs to set of all multiindices.

Definition 2.10

Let m ∈ R and ψ ∈ Λ . We call a C ∞ -function q : R n × R n → C a symbol in the class S ρ m , ψ ( R n ) if for all α , β ∈ N 0 n there are constants c α , β ≥ 0 such that

(2.19) ∣ ∂ ξ α ∂ x β q ( x , ξ ) ∣ ≤ c α , β ( 1 + ψ ( ξ ) ) m − ρ ( ∣ α ∣ ) 2

holds for all x ∈ R n and ξ ∈ R n and ρ is as defined in Definition 2.9. We call m ∈ R the order of the symbol q ( x , ξ ) .

Theorem 2.11

Let ψ : R n → R be a continuous negative definitive function in the class Λ which we define above, and suppose in addition ψ satisfies ψ ( ξ ) ≥ c 0 ∣ ξ ∣ r 0 for some c 0 > 0 , r 0 > 0 and all ξ , ∣ ξ ∣ ≥ R . If q ( x , ξ ) is a negative definitive symbol belonging to S ρ 2 , ψ ( R n ) and satisfies

(2.20) q ( x , ξ ) ≥ δ ( 1 + ψ ( ξ ) )

for some δ > 0 and all ξ ∈ R n , ∣ ξ ∣ sufficiently large, then − q ( x , D ) defined on C 0 ∞ ( R n , R ) is closable in C ∞ ( R n ; R ) and its closure is a generator of a Feller semigroup.

The second theorem is taken from the paper [9]. For i = 1 , 2 , let ( ℰ ( i ) , D ( ℰ ( i ) ) ) be a Dirichlet form on L 2 and { T t ( i ) : t > 0 } the symmetric strongly continuous Markovian semigroup on L 2 associated with ℰ ( i ) .

Theorem 2.12

Assume that D ( ℰ ( 1 ) ) ⊃ D ( ℰ ( 2 ) ) and there is a positive number C such that

(2.21) ℰ ( 1 ) ( u , u ) ≤ C ℰ ( 2 ) ( u , u ) , u ∈ D ( ℰ ( 2 ) ) ,

and moreover suppose

(2.22) ‖ T t ( 1 ) ‖ 1 → ∞ ≤ g ( t ) , t > 0 ,

where g is a right continuous non-increasing function and satisfies the following condition: H ( ξ ) ≡ ∫ ξ ∞ { G ( η ) / g ( G ( η ) ) } d η < + ∞ , ξ > 0 , G being the left continuous inverse function of g . If T t ( 2 ) 1 = 1 m -a.e., t > 0 , then it holds that, for the inverse function h of H ,

(2.23) ‖ T t ( 2 ) ‖ 1 → ∞ ≤ 2 h ( t / 2 C ) , t > 0 .

In particular, if t g ( t ) is continuous and non-decreasing, then

(2.24) ‖ T t ( 2 ) ‖ 1 → ∞ ≤ 2 g ( t / 2 C ) , t > 0 .

Therefore, if t ‖ T t ( 1 ) ‖ 1 → ∞ is continuous and non-decreasing in t , then

‖ T t ( 2 ) ‖ 1 → ∞ ≤ 2 ‖ T t / 2 C ( 1 ) ‖ 1 → ∞ , t > 0 .

The third theorem is taken from [24], see [24] for Theorem 5.2.

Theorem 2.13

Let ( T t ) t ≥ 0 be a Feller semigroup with generator ( A , D ( A ) ) such that C c ∞ ( R n ) ⊂ D ( A ) and A ∣ C c ∞ ( R n ) = − q ( x , D ) with symbol q ( x , ξ ) satisfying sup x ∈ R n ∣ q ( x , ξ ) ∣ ≤ c ( 1 + ∣ ξ ∣ 2 ) . The semigroup { T t } t ≥ 0 is conservative, if q ( x , 0 ) ≡ 0 .

Conversely, if { T t } t ≥ 0 is conservative and q ( ⋅ , 0 ) continuous, then q ( x , 0 ) ≡ 0 .

In the above theorem, the symbol q ( x , ξ ) can be expressed as follows:

(2.25) q ( x , y , ξ , η ) = ψ ( ξ , η ) + a 2 ( y ) ψ 1 ( ξ ) + a 1 ( x ) ψ 2 ( η ) ,

where ψ : R n × R m → R , ψ 1 : R n → R and ψ 2 : R m → R are three continuous negative definite functions and a 1 : R n → R and a 2 : R m → R are two continuous functions such that 0 ≤ a 1 ( x ) ≤ ‖ a 1 ‖ L ∞ < ∞ and 0 ≤ a 2 ( y ) ≤ ‖ a 2 ‖ L ∞ < ∞ .

Example 2.14

(continues) Let ( T t ) t ≥ 0 be the stochastic process associated with a negative definitive symbol as (2.25) and we let the functions a 1 and a 2 be infinitely differentiable. We take ψ ( ξ , η ) = ( ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 / 2 , ψ 1 ( ξ ) = ∣ ξ ∣ 1 / 2 , ψ 2 ( η ) = ∣ η ∣ 1 / 2 , first of all, we need to check that ( T t ) t ≥ 0 is a Feller semigroup using Theorem 2.11. We let ψ ( ξ , η ) = ( ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 2 in Theorem 2.11. Since ( ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 2 is equivalent to ( 1 + ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 2 − 1 , we have q ( x , y , ξ , η ) = ( 1 + ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 2 − 1 + a 2 ( y ) ∣ ξ ∣ 1 2 + a 1 ( x ) ∣ η ∣ 1 2 . Moreover, we can choose ∣ ξ ∣ and ∣ η ∣ large enough so that we have a 2 ( y ) ∣ ξ ∣ 1 2 + a 1 ( x ) ∣ η ∣ 1 2 − 1 ≥ 0 . Furthermore, q ( x , y , ξ , η ) ≥ ( 1 + ( ∣ ξ ∣ 2 + ∣ η ∣ 2 ) 1 2 ) satisfies condition (2.20). Therefore, − q ( x , y , D ξ , D η ) is a generator of a Feller semigroup. Second, we can use Theorem 2.13 to verify that the Feller semigroup associated with our generator − q ( x , y , D ξ , D η ) is conservative. To this end, we need to verify that sup x , y ∈ R ∣ q ( x , y , ξ , η ) ∣ ≤ c ( 1 + ∣ ξ ∣ 2 + ∣ η ∣ 2 ) , which is obvious since 0 ≤ a 1 ( x ) ≤ ‖ a 1 ‖ L ∞ < ∞ and 0 ≤ a 2 ( y ) ≤ ‖ a 2 ‖ L ∞ < ∞ . Then we can use Theorem 2.12 to get an estimate of our symbol we stated beforehand. In Theorem 2.12, we take ℰ ( 1 ) as the Dirichlet form associated with ψ ( ξ , η ) and ℰ ( 2 ) as the Dirichlet form associated with q ( x , y , ξ , η ) . From Example 2.8, we can get that ‖ T t ‖ 1 → ∞ = 1 π t − 2 , which means we can take g ( t ) = 1 π t − 2 and H ( ξ ) = 1 2 ξ − 1 / 2 and h ( x ) = 1 4 x − 1 / 2 . Therefore ‖ T t ( 2 ) ‖ ≤ 2 C 2 t − 2 . Then we can use Theorems 2.6 and 2.7 to get the estimate of subordination of our symbol which is the same as Example 2.8 except the coefficients.

Unfortunately, so far this approach does not provide examples for off-diagonal estimates.

3 Main results

We study first transition functions of certain Levy processes with state space R n and then we move to a more concrete (class of) example(s) in order to get a better understanding of the non-isotropic behaviour of certain (classes of) Levy process(es). The next three definitions and theorem are taken from [10], which we will use after.

Definition 3.1

A metric measure space is a triple ( X , d , μ ) , where ( X , d ) is a metric space and μ is a measure on the Borel sets of the space X .

Definition 3.2

Let ψ : R n → R be a non-periodic, continuous negative definite function with ψ ( 0 ) = 0 such that for any ξ and η in R n , we can define d ψ ( ξ − η ) as ψ ( ξ − η ) , which generates a metric on R n . Then we call ψ to be the metric generating on R n and this class is denoted by ℳCN ( R n ) , i.e. ψ ∈ ℳCN ( R n ) .

Definition 3.3

Let ( X , d , μ ) be a metric measure space. We say that ( X , d , μ ) or μ has the volume doubling property if there exists a constant c 2 such that

(3.1) μ ( B d ( x , 2 r ) ) ≤ c 2 μ ( B d ( x , r ) )

holds for all metric balls B d ( x , r ) = { y ∈ X : d ( y , x ) < r } ⊂ X . If (3.1) holds only for all balls with radii r < ρ for some fixed ρ > 0 , we say that ( X , d , μ ) (or μ ) is locally volume doubling. If volume doubling holds, then for R ≥ 1 ,

μ ( B d ( x , R ) ) ≤ c 2 log 2 R μ ( B d ( x , 1 ) ) = R log 2 c 2 μ ( B d ( x , 1 ) ) ,

i.e. R ↦ μ ( B d ( x , R ) ) has at most power growth.

Theorem 3.4

Let ψ ∈ ℳCN ( R n ) and assume that e − t ψ ∈ L 1 ( R n , λ ) . Then

(3.2) p t ( 0 ) = ( 2 π ) − n ∫ 0 ∞ λ ( B d ˜ ψ ( 0 , r / t ) ) e − r d r , r > 0 .

If the metric measure space ( R n , d ψ , λ ) has the volume doubling property, then e − t ψ ∈ L 1 ( R n , λ ) and

(3.3) p t ( 0 ) ≍ λ ( B d ˜ ψ ( 0 , 1 / t ) ) f o r a l l t > 0 .

Let ( X t ψ ) t ≥ 0 be a symmetric Levy process with state space R n and transition function p t which we assume to exist as element in C ∞ ( R n ) given by

(3.4) p t ( x ) = ( 2 π ) − n ∫ R n e i x ξ e − t ψ ( ξ ) d ξ .

Furthermore, we assume that p t has the form

(3.5) p t ( x − y ) = p t ( 0 ) e − δ ψ , t 2 ( x , y )

with p t ( 0 ) as in (3.2) and δ ψ , t ( ⋅ , ⋅ ) : R n × R n → R is a (translation invariant) metric for every t ∈ ( 0 , ∞ ) , i.e.

(3.6) p t ( 0 ) = ( 2 π ) − n ∫ 0 ∞ λ ( n ) B d ψ 0 , r t e − r d r ,

where d ψ ( ξ , η ) = ψ 1 / 2 ( ξ − η ) is assumed to be a metric. In fact, we assume that the metric measure space ( R n , d ψ , λ ( n ) ) has the volume doubling property, and hence we have

(3.7) p t ( 0 ) ≍ λ ( n ) B d ψ 0 , 1 t t > 0 .

Moreover, we assume that δ ψ , t ( ⋅ , ⋅ ) is a metric on R n .

Let the process start at z ∈ R n and try to find estimate, i.e. the probability

(3.8) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 )

for Borel sets C 1 , C 2 ⊂ R n . Using the very definition and properties of (Levy) processes we find, see Schilling and Partzsch [25],

(3.9) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = T t 1 ( χ C 1 T t 2 − t 1 χ C 2 ) ( z )

or

(3.10) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = ∫ R n p t 1 ( z − x ) χ C 1 ( x ) ∫ R n p t 2 − t 1 ( x − y ) χ C 2 ( y ) d y d x .

Now we use the representation (3.5) for p t to find

(3.11) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = ∫ R n χ C 1 ( x ) p t 1 ( 0 ) e − δ ψ , t 1 2 ( z , x ) ∫ R n χ C 2 ( y ) p t 2 − t 1 ( 0 ) e − δ ψ , t 2 − t 1 2 ( x , y ) d y d x = p t 1 ( 0 ) p t 2 − t 1 ( 0 ) ∫ R n ∫ R n χ C 1 ( x ) χ C 2 ( y ) e − δ ψ , t 1 2 ( z , x ) e − δ ψ , t 2 − t 1 2 ( x , y ) d y d x = p t 1 ( 0 ) p t 2 − t 1 ( 0 ) ∫ C 1 ∫ C 2 e − δ ψ , t 1 2 ( z , x ) e − δ ψ , t 2 − t 1 2 ( x , y ) d y d x .

For bounded Borel sets C 1 and C 2 , the formula (3.11) allows us to obtain further estimates. For this observe that

∫ C 1 ∫ C 2 e − δ ψ , t 1 2 ( z , x ) e − δ ψ , t 2 − t 1 2 ( x , y ) d y d x ≥ e − sup x ∈ C 1 , y ∈ C 2 ( δ ψ , t 1 2 ( z , x ) + δ ψ , t 2 − t 1 2 ( x , y ) ) λ ( n ) ( C 1 ) λ ( n ) ( C 2 ) .

With (3.7) we obtain (with constants depending only on ψ and the volume growth function)

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≥ c 0 λ ( n ) B d ψ 0 , 1 t 1 λ ( n ) B d ψ 0 , 1 t 2 − t 1 × e − sup x ∈ C 1 , y ∈ C 2 ( δ ψ , t 1 2 ( z , x ) + δ ψ , t 2 − t 1 2 ( x , y ) ) λ ( n ) ( C 1 ) λ ( n ) ( C 2 ) .

Furthermore, we have

P z ( X t 1 ∈ C 1 , X t 1 ∈ C 2 ) ≤ c 1 λ ( n ) B d ψ 0 , 1 t 1 λ ( n ) B d ψ 0 , 1 t 2 − t 1 × e − inf x ∈ C 1 , y ∈ C 2 ( δ ψ , t 2 2 ( z , x ) + δ ψ , t 2 − t 1 2 ( x , y ) ) λ ( n ) ( C 1 ) λ ( n ) ( C 2 ) .

Of course, we now can specify the sets C 1 and C 2 . In particular, when prescribing both sets with the metrics δ ψ , t 2 and δ ψ , t 2 − t 1 , respectively, we obtain purely geometric bounds.

As C 1 , C 2 are considered as sets in the state space it would be natural to characterize both sets with the metric δ ψ . For example, we may take

(3.12) C 1 = B δ ψ , t 1 ( x 0 , r 1 ) ,

(3.13) C 2 = B δ ψ , t 2 − t 1 ( y 0 , r 2 ) ,

and we may assume C 1 ∩ C 2 = ∅ , z ∉ C 1 ∪ C 2 .

We want to use these considerations to get some more ideas on what happens in non-isotropic situations. For this we split R n = R n 1 × R n 2 and consider the symbol

(3.14) ψ : R n 1 × R n 2 → R ψ ( ξ , η ) = ψ 1 ( ξ ) + ψ 2 ( η ) ,

where ψ j corresponds to a Levy process ( X t ψ j ) t ≥ 0 with state space R n j , j = 1 , 2 , and for the corresponding transition functions we assume representations analogous to (3.5), i.e.

(3.15) P t ( j ) ( x j − y j ) = P t ( j ) ( 0 ) e − δ ψ j , t 2 ( x j , y j ) .

For this case, it follows now with C 1 , C 2 ⊂ R n 1 + n 2

(3.16) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = T t 1 ( χ C 1 T t 2 − t 1 χ C 2 ) ( z ) = ∫ R n p t 1 ( z , x ) χ C 1 ( x ) ∫ R n p t 2 − t 1 ( x , y ) χ C 2 ( y ) d y d x .

Using our previous estimates we obtain

(3.17) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≥ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) ∫ R n 1 + n 2 χ C 1 ( x 1 , x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) × ∫ R n 1 + n 2 e − sup x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) χ C 2 ( y 1 , y 2 ) d y 1 d y 2 d x 1 d x 2 ≥ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − sup x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × ∫ R n 1 + n 2 χ C 1 ( x 1 , x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) d x 1 d x 2 ≥ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − sup x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) × e − sup x ∈ C 1 ( δ t 2 − t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 2 − t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) λ 2 ( C 2 )

and

(3.18) P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≤ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) ∫ R n 1 + n 2 χ C 1 ( x 1 , x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) × ∫ R n 1 + n 2 e − inf x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) χ C 2 ( y 1 , y 2 ) d y 1 d y 2 d x 1 d x 2 ≤ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − inf x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × ∫ R n 1 + n 2 χ C 1 ( x 1 , x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) d x 1 d x 2 ≤ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − inf x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) × e − inf x ∈ C 1 ( δ t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) λ 2 ( C 2 ) .

We now want to specialize further. We consider the case where C j = C j , 1 × C j , 2 , j = 1 , 2 , and C j , i ⊂ R n i

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) ∫ R n 1 + n 2 χ C 1 ( x 1 , x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) × ∫ R n 1 + n 2 e − δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) e − δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) χ C 2 ( y 1 , y 2 ) d y 1 d y 2 d x 1 d x 2 = p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) ∫ R n 1 + n 2 χ C 11 ( x 1 ) χ C 12 ( x 2 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) × ∫ R n 1 + n 2 e − δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) e − δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) χ C 21 ( y 1 ) χ C 22 ( y 2 ) d y 1 d y 2 d x 1 d x 2 = p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) ∫ R n 1 χ C 11 ( x 1 ) e − δ t 1 ( 1 ) 2 ( z 1 , x 1 ) ∫ R n 2 χ C 12 ( x 2 ) e − δ t 1 ( 2 ) 2 ( z 2 , x 2 ) × ∫ R n 1 e − δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) χ C 21 ( y 1 ) ∫ R n 2 e − δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) χ C 22 ( y 2 ) d y 1 d y 2 d x 1 d x 2 = P z 1 ( X t 1 ψ 1 ∈ C 11 , X t 2 ψ 1 ∈ C 12 ) P z 2 ( X t 1 ψ 2 ∈ C 21 , X t 2 ψ 2 ∈ C 22 ) .

Now for P z 1 ( X t 1 ψ 1 ∈ C 11 , X t 2 ψ 1 ∈ C 12 ) and P z 2 ( X t 1 ψ 2 ∈ C 21 , X t 2 ψ 2 ∈ C 22 ) we can apply the previous results.

Remark 3.5

The process ( X t ≥ 0 ψ ) associated with the characteristic exponent, i.e. symbol, (3.14) splits into the processes ( X t ψ 1 ) t ≥ 0 and ( X t ψ 2 ) t ≥ 0 , i.e.

( X t ψ ) t = ( ( X t ψ 1 , X t ψ 2 ) ) t ≥ 0 .

In case we consider product sets C 1 × C 2 ⊂ R n 1 × R n 2 the process ( X t ψ ) t ≥ 0 is decoupled. We need to only consider the components to obtain results for ( X t ψ ) t ≥ 0 . This is of course the content of the last calculation. From the geometric point of view, matters are easy too since we are dealing with products only. However, in case that C ⊂ R n 1 × R n 2 is not of product structure, we start to observe a type of coupling of ( X t ψ 1 ) t ≥ 0 and ( X t ψ 2 ) t ≥ 0 of both processes. In particular, due to the non-isotropy we discover that while the products of balls in ( R n 1 × R n 2 , δ t ψ 1 ⊗ δ t ψ 2 ) and the balls in ( R n , δ t ψ ) give rise to the same topological space, their geometric impact (e.g. on estimates) is quite different.

If C j ≠ C j 1 × C j 2 , j = 1 , 2 , in general, it is complicated to estimate if we take arbitrary C j , but if we take C 1 = B r 1 δ t 1 ( 0 ) , C 2 = B r 2 δ t 1 − t 2 ( 0 ) , where δ t 1 2 = δ t 1 ( 1 ) 2 + δ t 1 ( 2 ) 2 and δ t 1 − t 2 2 = δ t 1 − t 2 ( 1 ) 2 + δ t 1 − t 2 ( 2 ) 2 , therefore, we have

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) = ∫ R n p t 1 ψ ( z , x ) χ C 1 ( x ) ∫ R n p t 2 − t 1 ψ χ C 2 ( y ) d y d x = p t 1 ( 0 ) p t 2 − t 1 ( 0 ) ∫ R n e − δ t 1 2 ( z , x ) χ C 1 ( x ) ∫ R n e − δ t 2 − t 1 2 ( x , y ) χ C 2 ( x ) d y d x .

So we have upper bound and lower bound of P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ,

(3.19) p t 1 ( 0 ) p t 2 − t 1 ( 0 ) λ n ( B r 1 δ t 1 ( 0 ) ) λ n ( B r 2 δ t 2 − t 1 ( 0 ) ) ≤ P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≤ p t 1 ( 0 ) p t 2 − t 1 ( 0 ) e − r 1 2 λ n ( B r 1 δ t 1 ( 0 ) ) e − r 2 2 λ n ( B r 2 δ t 2 − t 1 ( 0 ) ) .

Example 3.6

If we take C 1 = B 1 ψ 1 ( 0 ) × B 1 ψ 2 ( 0 ) , C 2 = B 1 ψ 1 ( a 1 ) × B 1 ψ 2 ( a 2 ) , where ψ 1 ( ξ ) = ∣ ξ ∣ 2 and ψ 2 ( η ) = ∣ η ∣ , therefore, from (3.17) and (3.18), we have

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≥ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − sup x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × e − sup x ∈ C 1 ( δ t 2 − t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 2 − t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) ≥ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 × e − 2 4 ( t 2 − t 1 ) − 2 ln ( ( t 2 − t 1 ) 2 ) + ln ( ( t 2 − t 1 ) 2 + max { ∣ a 1 − 1 ∣ , ∣ 1 + a 1 ∣ } 2 ) + ln ( ( t 2 − t 1 ) 2 + max { ∣ a 2 − 1 ∣ , ∣ a 2 + 1 ∣ } 2 ) × 4 × e − z 1 2 + z 2 2 4 ( t 2 − t 1 ) + 2 ln 1 + ( t 2 − t 1 ) 2 ( t 2 − t 1 ) 2 × 4

and

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≤ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − inf x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × e − inf x ∈ C 1 ( δ t 2 − t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 2 − t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) ≤ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 × e − 2 4 ( t 2 − t 1 ) − 2 ln ( ( t 2 − t 1 ) 2 ) + ln ( ( t 2 − t 1 ) 2 + max { sgn ( ( a 1 − 1 ) ( 1 + a 1 ) ) , 0 } min { ∣ a 1 − 1 ∣ , ∣ 1 + a 1 ∣ } 2 ) × e − ln ( ( t 2 − t 1 ) 2 + max { sgn ( ( a 2 − 1 ) ( a 2 + 1 ) ) , 0 } min { ∣ a 2 − 1 ∣ , ∣ a 2 + 1 ∣ } 2 ) × 4 × e − z 1 2 + z 2 2 4 ( t 2 − t 1 ) + 2 ln 1 + ( t 2 − t 1 ) 2 ( t 2 − t 1 ) 2 × 4 .

Example 3.7

If we take C 1 = B 1 ψ ( 0 , 0 ) , C 2 = B 1 ψ ( a 1 , a 2 ) , where ψ ( ξ , η ) = ∣ ξ ∣ 2 + ∣ η ∣ , therefore, from (3.17) and (3.18), we have

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≥ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − sup x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × e − sup x ∈ C 1 ( δ t 2 − t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 2 − t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) ≥ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 × e − 2 4 ( t 2 − t 1 ) − 2 ln ( ( t 2 − t 1 ) 2 ) + ln ( ( t 2 − t 1 ) 2 + max { ∣ a 1 − 1 ∣ , ∣ 1 + a 1 ∣ } 2 ) + ln ( ( t 2 − t 1 ) 2 + max { ∣ a 2 − 1 ∣ , ∣ a 2 + 1 ∣ } 2 ) × 8 2 3 × e − z 1 2 + z 2 2 4 ( t 2 − t 1 ) + 2 ln 1 + ( t 2 − t 1 ) 2 ( t 2 − t 1 ) 2 × 8 2 3

and

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≤ p t 1 ( 0 , 0 ) p t 2 − t 1 ( 0 , 0 ) e − inf x ∈ C 1 , y ∈ C 2 ( δ t 2 − t 1 ( 1 ) 2 ( x 1 , y 1 ) + δ t 2 − t 1 ( 2 ) 2 ( x 2 , y 2 ) ) λ 2 ( C 2 ) × e − inf x ∈ C 1 ( δ t 2 − t 1 ( 1 ) 2 ( z 1 , x 1 ) + δ t 2 − t 1 ( 2 ) 2 ( z 2 , x 2 ) ) λ 2 ( C 1 ) ≤ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 × e − 2 4 ( t 2 − t 1 ) − 2 ln ( ( t 2 − t 1 ) 2 ) + ln ( ( t 2 − t 1 ) 2 + max { sgn ( ( a 1 − 1 ) ( 1 + a 1 ) ) , 0 } min { ∣ a 1 − 1 ∣ , ∣ 1 + a 1 ∣ } 2 ) × e − ln ( ( t 2 − t 1 ) 2 + max { sgn ( ( a 2 − 1 ) ( a 2 + 1 ) ) , 0 } min { ∣ a 2 − 1 ∣ , ∣ a 2 + 1 ∣ } 2 ) × 8 2 3 × e − z 1 2 + z 2 2 4 ( t 2 − t 1 ) + 2 ln 1 + ( t 2 − t 1 ) 2 ( t 2 − t 1 ) 2 × 8 2 3 .

Example 3.8

If we take C 1 = B 1 δ t 1 ( 0 ) , C 2 = B 1 δ t 1 − t 2 ( 0 ) , where δ is related to ψ ( ξ , η ) = ∣ ξ ∣ 2 + ∣ η ∣ , which is,

δ ( x 1 , x 2 ) = x 1 2 4 t − ln ( t 2 ) + ln ( t 2 + x 2 2 ) ,

from (3.19), we have

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≥ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 λ 2 ( B 1 δ t 1 ( 0 ) ) λ 2 ( B 1 δ t 2 − t 1 ( 0 ) )

and

P z ( X t 1 ∈ C 1 , X t 2 ∈ C 2 ) ≤ 1 2 π t 1 × 1 π × 1 t 1 × 1 2 π ( t 2 − t 1 ) × 1 π × 1 t 2 − t 1 e − 1 λ 2 ( B 1 δ t 1 ( 0 ) ) e − 1 λ 2 ( B 1 δ t 2 − t 1 ( 0 ) ) .

Since the volume of the metric balls is hard to calculate based on the computing conditions, we cannot calculate the exact estimates of the above.

Remark 3.9

Examples 3.6, 3.7 and 3.8 show the greatest extent where our estimations of the transition probabilities can be calculated. The negative definite functions ψ 1 ( ξ ) = ∣ ξ ∣ 2 and ψ 2 ( η ) = ∣ η ∣ are related to Brownian motion and Cauchy process, respectively. Therefore, understanding them and their combinations has great significance in not only theories but also applications of Levy-type processes. Example 3.6 is the most special case where the general negative definite function ψ can be spilt into two special cases we are familiar with. Therefore, the transition probabilities can be calculated in details; Example 3.7 shows that the best estimates we can get if we cannot separate the general negative definite function into any further due to the inseparable situation of the domains; Example 3.8 is the special case of Example 3.7 where the domains are special metric balls. In this case, the estimates of transition functions can be calculated and represented by metric balls. Since the metric balls are generally hard to calculate analytically, in the following section, the graphical experiments are mainly simulations of variable coefficients version of Example 3.8 in two-dimensional case.

4 Graphical experiments

We have seen that in many cases of Levy processes there is a natural geometric interpretation of the transition functions involving two t -dependent families of metrics from Section 3. These metrics are in general non-isotropic.

In case we are dealing with Markov processes generated by non-translation invariant operators, i.e. by pseudo-differential operators having an x -dependent symbol, in general we do have neither explicit formula for the transition function nor so far geometric interpretations of these transition functions. However, using some symbolic calculus or other approximation procedures, compare [23] or [26], it is possible for small time and locally with respect to some point x 0 in the state space to compare such transition function with those of the corresponding Levy process obtained when freezing the coefficients of the generator, i.e. the x -dependence, at x 0 . Thus, one might have the idea to approximate a probability such as

P x 0 ( X t 1 ∈ G 1 , X t 2 ∈ G 2 )

by probabilities calculated with the help of transition functions of certain Levy processes. Thus, it is a natural question to study, i.e. to compare the transition functions (and the underlying metrics) of Levy processes obtained by freezing the coefficients of a given generator at different point, i.e. to compare transition functions

P t x j ( x − y ) = ( 2 π ) − n / 2 ∫ R n e i ( x − y ) ξ e − t q ( x j , ξ ) d ξ

for a sequence ( x j ) j ∈ N in state space.

We know that in general we have to handle non-isotropic operators (metrics, transition functions) and hence it is important to have some knowledge how the non-isotropy develops when x j runs through a set of points.

In order to have concrete functions and an affordable amount of calculations we restrict the case study to the symbol

(4.1) q ( x , y , ξ , η ) = a ( y ) ∣ ξ ∣ 2 + b ( x ) ∣ η ∣ ,

where x , y ∈ R as are ξ , η ∈ R . And from [3] we know that the symbol ∣ ξ ∣ 2 is related to Brownian motion and ∣ η ∣ is related to Cauchy process. Therefore, the symbol can be seen as the variable coefficient extension of Brownian motion and Cauchy process. As we mention in Remark 3.9, (4.1) is the variable coefficient version of Example 3.8. Freezing the coefficients leads to consider the characteristic functions

a ( y 0 ) ∣ ξ ∣ 2 + b ( x 0 ) ∣ η ∣

and the corresponding Levy processes. In this case from (3.15) and the calculation of Example 3.8 we find

p t ( x 0 , y 0 ) ( x 1 − x 2 , y 1 − y 2 ) = p t ( x 0 , y 0 ) ( 0 , 0 ) e − δ t , ( x 0 , y 0 ) 2 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) ,

where

p t ( x 0 , y 0 ) ( 0 , 0 ) = 1 2 π 3 / 2 t 3 / 2 a ( y 0 ) b ( x 0 )

with corresponding metric d t , ψ ( x 0 , y 0 ) ( ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) ) ≔ t ψ ( ξ − η ) and

(4.2) δ t , ( x 0 , y 0 ) 2 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = ( x 2 − x 1 ) 2 4 t a ( y 0 ) − ln ( t 2 b ( x 0 ) 2 ) + ln ( t 2 b ( x 0 ) 2 + ( y 2 − y 1 ) 2 ) .

The following plots show the balls with respect to the metric d t , ψ ( x j , y j ) , the balls with respect to the metric δ t , ψ ( x j , y j ) and the transition function p t ( x j , y j ) .

In order to have a good understanding of the Levy-type transition function p ( x 1 , x 2 ) = p ( 0 , 0 ) e − δ t 2 ( x 1 , x 2 ) , we need to understand the metric ball

(4.3) B t ψ ( 0 ) = { ( ξ 1 , ξ 2 ) ∈ R 2 ∣ a ( y 0 ) ∣ ξ 1 ∣ 2 + b ( x 0 ) ∣ ξ 2 ∣ ≤ t }

and the metric ball B δ t ( x 1 , x 2 ) ( 0 , 1 ) . First, we show the metric ball B t ψ ( 0 ) in the following graphics. We fix t = 0.3 and change the value b and a alternatively, which are shown in Figures 1 and 2. Second, from (4.2), we can calculate that

(4.4) δ t ( x 1 , x 2 ) = x 1 2 4 t a ( y ) − ln ( t 2 b ( x ) 2 ) + ln ( t 2 b ( x ) 2 + x 2 2 ) .

Now we deal this metric through three situations using graphs. First, we fix t = 0.3 , 0.5 , 0.75 and a = 1 and change the values of b , which are shown in Figures 3, 4 and 5.

Figure 1 The shape of metric ball B t ψ ( 0 ) {B}_{t}^{\psi }\left(0) when we fix t = 0.3 t=0.3 , a = 1 a=1 , and b b takes the values from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 t=0.3,a=1,b=1 ; (b) t = 0.3 , a = 1 , b = 1.5 t=0.3,a=1,b=1.5 ; (c) t = 0.3 , a = 1 , b = 2 t=0.3,a=1,b=2 ; (d) t = 0.3 , a = 1 , b = 2.5 t=0.3,a=1,b=2.5 ; (e) t = 0.3 , a = 1 , b = 3 t=0.3,a=1,b=3 ; (f) t = 0.3 , a = 1 , b = 3.5 t=0.3,a=1,b=3.5 .
Figure 1

The shape of metric ball B t ψ ( 0 ) when we fix t = 0.3 , a = 1 , and b takes the values from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 ; (b) t = 0.3 , a = 1 , b = 1.5 ; (c) t = 0.3 , a = 1 , b = 2 ; (d) t = 0.3 , a = 1 , b = 2.5 ; (e) t = 0.3 , a = 1 , b = 3 ; (f) t = 0.3 , a = 1 , b = 3.5 .

Figure 2 The shape of metric ball B t ψ ( 0 ) {B}_{t}^{\psi }\left(0) when we fix t = 0.3 t=0.3 , b = 1 b=1 and a a takes the values from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 t=0.3,a=1,b=1 ; (b) t = 0.3 , a = 1.5 , b = 1 t=0.3,a=1.5,b=1 ; (c) t = 0.3 , a = 2 , b = 1 t=0.3,a=2,b=1 ; (d) t = 0.3 , a = 2.5 , b = 1 t=0.3,a=2.5,b=1 ; (e) t = 0.3 , a = 3 , b = 1 t=0.3,a=3,b=1 ; (f) t = 0.3 , a = 3.5 , b = 1 t=0.3,a=3.5,b=1 .
Figure 2

The shape of metric ball B t ψ ( 0 ) when we fix t = 0.3 , b = 1 and a takes the values from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 ; (b) t = 0.3 , a = 1.5 , b = 1 ; (c) t = 0.3 , a = 2 , b = 1 ; (d) t = 0.3 , a = 2.5 , b = 1 ; (e) t = 0.3 , a = 3 , b = 1 ; (f) t = 0.3 , a = 3.5 , b = 1 .

Figure 3 The shape of metric ball δ t ( x 1 , x 2 ) {\delta }_{t}\left({x}_{1},{x}_{2}) when we fix t = 0.3 t=0.3 , a = 1 a=1 and b b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 t=0.3,a=1,b=1 ; (b) t = 0.3 , a = 1 , b = 1.5 t=0.3,a=1,b=1.5 ; (c) t = 0.3 , a = 1 , b = 2 t=0.3,a=1,b=2 ; (d) t = 0.3 , a = 1 , b = 2.5 t=0.3,a=1,b=2.5 ; (e) t = 0.3 , a = 1 , b = 3 t=0.3,a=1,b=3 ; (f) t = 0.3 , a = 1 , b = 3.5 t=0.3,a=1,b=3.5 .
Figure 3

The shape of metric ball δ t ( x 1 , x 2 ) when we fix t = 0.3 , a = 1 and b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 ; (b) t = 0.3 , a = 1 , b = 1.5 ; (c) t = 0.3 , a = 1 , b = 2 ; (d) t = 0.3 , a = 1 , b = 2.5 ; (e) t = 0.3 , a = 1 , b = 3 ; (f) t = 0.3 , a = 1 , b = 3.5 .

Figure 4 The shape of metric ball δ t ( x 1 , x 2 ) {\delta }_{t}\left({x}_{1},{x}_{2}) when we fix t = 0.5 t=0.5 , a = 1 a=1 and b b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.5 , a = 1 , b = 1 t=0.5,a=1,b=1 ; (b) t = 0.5 , a = 1 , b = 1.5 t=0.5,a=1,b=1.5 ; (c) t = 0.5 , a = 1 , b = 2 t=0.5,a=1,b=2 ; (d) t = 0.5 , a = 1 , b = 2.5 t=0.5,a=1,b=2.5 ; (e) t = 0.5 , a = 1 , b = 3 t=0.5,a=1,b=3 ; (f) t = 0.5 , a = 1 , b = 3.5 t=0.5,a=1,b=3.5 .
Figure 4

The shape of metric ball δ t ( x 1 , x 2 ) when we fix t = 0.5 , a = 1 and b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.5 , a = 1 , b = 1 ; (b) t = 0.5 , a = 1 , b = 1.5 ; (c) t = 0.5 , a = 1 , b = 2 ; (d) t = 0.5 , a = 1 , b = 2.5 ; (e) t = 0.5 , a = 1 , b = 3 ; (f) t = 0.5 , a = 1 , b = 3.5 .

Figure 5 The shape of metric ball δ t ( x 1 , x 2 ) {\delta }_{t}\left({x}_{1},{x}_{2}) when we fix t = 0.75 t=0.75 , a = 1 a=1 and b b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.75 , a = 1 , b = 1 t=0.75,a=1,b=1 ; (b) t = 0.75 , a = 1 , b = 1.5 t=0.75,a=1,b=1.5 ; (c) t = 0.75 , a = 1 , b = 2 t=0.75,a=1,b=2 ; (d) t = 0.75 , a = 1 , b = 2.5 t=0.75,a=1,b=2.5 ; (e) t = 0.75 , a = 1 , b = 3 t=0.75,a=1,b=3 ; (f) t = 0.75 , a = 1 , b = 3.5 t=0.75,a=1,b=3.5 .
Figure 5

The shape of metric ball δ t ( x 1 , x 2 ) when we fix t = 0.75 , a = 1 and b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.75 , a = 1 , b = 1 ; (b) t = 0.75 , a = 1 , b = 1.5 ; (c) t = 0.75 , a = 1 , b = 2 ; (d) t = 0.75 , a = 1 , b = 2.5 ; (e) t = 0.75 , a = 1 , b = 3 ; (f) t = 0.75 , a = 1 , b = 3.5 .

Finally, we come to the stage of graphic study of p ( x 1 , x 2 ) . Through calculations we can get

(4.5) p ( x 1 , x 2 ) = 1 2 π t b π a t exp − x 1 2 4 t a + ln ( t 2 b 2 ) − ln ( t 2 b 2 + x 2 2 ) .

So we need to consider the impact of the transition function p ( x 1 , x 2 ) created by the weight of a and b in two different directions, which are shown in Figures 6 and 7. The experiments and the drawing of graphs have been carried out using Matlab.

Figure 6 The shape of transition function p ( x 1 , x 2 ) p\left({x}_{1},{x}_{2}) when we fix t = 0.3 t=0.3 , a = 1 a=1 and b b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 t=0.3,a=1,b=1 ; (b) t = 0.3 , a = 1 , b = 1.5 t=0.3,a=1,b=1.5 ; (c) t = 0.3 , a = 1 , b = 2 t=0.3,a=1,b=2 ; (d) t = 0.3 , a = 1 , b = 2.5 t=0.3,a=1,b=2.5 ; (e) t = 0.3 , a = 1 , b = 3 t=0.3,a=1,b=3 ; (f) t = 0.3 , a = 1 , b = 3.5 t=0.3,a=1,b=3.5 .
Figure 6

The shape of transition function p ( x 1 , x 2 ) when we fix t = 0.3 , a = 1 and b changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 ; (b) t = 0.3 , a = 1 , b = 1.5 ; (c) t = 0.3 , a = 1 , b = 2 ; (d) t = 0.3 , a = 1 , b = 2.5 ; (e) t = 0.3 , a = 1 , b = 3 ; (f) t = 0.3 , a = 1 , b = 3.5 .

Figure 7 The shape of transition function p ( x 1 , x 2 ) p\left({x}_{1},{x}_{2}) when we fix t = 0.3 t=0.3 , b = 1 b=1 and a a changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 t=0.3,a=1,b=1 ; (b) t = 0.3 , a = 1.5 , b = 1 t=0.3,a=1.5,b=1 ; (c) t = 0.3 , a = 2 , b = 1 t=0.3,a=2,b=1 ; (d) t = 0.3 , a = 2.5 , b = 1 t=0.3,a=2.5,b=1 ; (e) t = 0.3 , a = 3 , b = 1 t=0.3,a=3,b=1 ; (f) t = 0.3 , a = 3.5 , b = 1 t=0.3,a=3.5,b=1 .
Figure 7

The shape of transition function p ( x 1 , x 2 ) when we fix t = 0.3 , b = 1 and a changes from 1 to 3.5 in the interval of 0.5. (a) t = 0.3 , a = 1 , b = 1 ; (b) t = 0.3 , a = 1.5 , b = 1 ; (c) t = 0.3 , a = 2 , b = 1 ; (d) t = 0.3 , a = 2.5 , b = 1 ; (e) t = 0.3 , a = 3 , b = 1 ; (f) t = 0.3 , a = 3.5 , b = 1 .

From graphs above, we can draw the following conclusions:

  1. For the metric ball B t ψ ( 0 ) , as shown in Figures 1 and 2, we can see that the greater deformation as a increases than b . The main reason can be seen from the definition of B t ψ ( 0 ) in (4.3) that the square term ∣ ξ 1 ∣ 2 in the negative definite function has more impact on the shape of the ball than the absolute term ∣ ξ 2 ∣ .

  2. For the metric ball δ t ( x 1 , x 2 ) , as shown in Figures 3, 4, and 5, we can see that as t increases, the size of the metric ball increases exponentially as we change b values. The main reason can been seen from the square term of b ( x ) in (4.4).

  3. For the transition function p ( x 1 , x 2 ) , as shown in Figures 6 and 7, we can see that when we fix the t value and a value, the transition function behaves like a Cauchy process and changes more in the direction of x 2 and when we fix the t values and b values, the transition function behaves like a Brownian motion and changes more in the direction of x 1 . The main reason can be seen in the definition of p ( x 1 , x 2 ) in (4.5). When we fix b , the transition functions are of Gaussian type in the direction of x 1 . When we fix a , the transition functions are of Cauchy type in the direction of x 2 .

To summarize the graphical experiments of this section, from drawing the graphs of the metric ball of B t ψ ( 0 ) and δ t ( x 1 , x 2 ) and transition function p ( x 1 , x 2 ) , we attempt to depict the shape changes of the variable coefficient version of the characteristic functions related to specific Levy-type process in different directions. The geometric intuitive understanding of transition densities with variable coefficients of concrete jump-type Markov processes will shed light on the research of the general jump-type Markov processes in future.

5 Concluding remarks

In this paper, we have studied the estimation of transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. Different from the previous methods which only focus on the theoretical analysis, we have discussed the comparison results of the general transition probability of certain Levy processes and given the detailed calculations of decoupled situation. Moreover, the graphical experiments of the metric balls and probability transition functions of concrete Levy processes related to Brownian motion and Cauchy process have been shown to gain insight of the general jump-type Markov processes.

In recent years, the Levy-type processes have played an important role in option pricing model building [27,28], the stability analysis of stochastic differential equations [29,30] and simulation in Bioinformatics [31]. Therefore, our work could shed some light on the inner mechanism of Levy-type processes and good supplement to the existing literature.

Acknowledgements

The authors are grateful to the editor and anonymous reviewers for their suggestions in improving the quality of the paper.

  1. Funding information: Scientific research fund for high-level talents, Jimei University.

  2. Author contributions: Yuanying Zhuang is in charge of the ideas of the paper and Xiao Song is in charge of the graphs and typesetting.

  3. Conflict of interest: Authors state no conflict of interest.

  4. Data availability statement: Not applicable.

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Received: 2021-03-13
Revised: 2021-09-24
Accepted: 2021-09-27
Published Online: 2021-12-31

© 2021 Yuanying Zhuang and Xiao Song, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2021-0120
Keywords for this article
Levy-type processes; pseudo-differential operator; heat kernels
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  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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