Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond

We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of L\'evy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also L\'evy processes, of which Brownian Motion is a special case, have become increasingly popular. L\'evy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include L\'evy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.Comment: 22 pages, including 4 figures and 8 pages of source code for the generation of sample paths of Feller processe

Epidemiology, comorbidities, and healthcare utilization of patients with chronic urticaria in Germany

Background: Comprehensive data on the epidemiology and comorbidities of chronic urticaria (CU) in Germany are either limited, or not contemporary. Objectives: To investigate the epidemiology of CU, overall comorbidities and healthcare resource utilized by patients with CU in Germany, using an anonymized statutory health insurance (SHI) database. Methods: Anonymized SHI claims research database of the Institute for Applied Health Research, Berlin [InGef] (01 January 2015-30 September 2018) was used to analyse insured individuals with a confirmed diagnosis of CU (ICD-10-GM codes). Twelve-month diagnosed prevalence and incidence, comorbidities (vs. atopic dermatitis and psoriasis), and healthcare utilization by patients with CU were investigated. Results: Of 4 693 772 individuals of all ages listed in the database, 3 538 540 were observable during 2017. Overall, 17 524 patients (˜0.5%) were diagnosed with CU; chronic spontaneous urticaria (CSU: 71.2%), chronic inducible urticaria (CIndU: 19.7%), CSU+CIndU (9.1%). Females, vs. males, had higher diagnosed prevalence (0.62% vs. 0.37%) and diagnosed incidence (0.18% vs. 0.11%) of CU among all patients. Patients most frequently visited general practitioners (41.3% of total visits). Hypertensive diseases (43.5%), lipoprotein metabolism disorders (32.1%) and affective disorders (26.0%) were the most frequently reported comorbidities of special interest. Rates of most comorbidities of special interests were similar to atopic dermatitis and psoriasis patients, and all higher vs. overall population. More than half (54.1%) of all CU patients were not prescribed any treatment. Second-generation H1 -antihistamines were the most commonly prescribed medication for adult (17.9%) and paediatric (27.9%) patients. Patients with CIndU (paediatric, 15.5%; adult, 7.8%) were more often hospitalized versus patients with CSU (paediatric, 9.9%; adult, 4.6%). Conclusions: In Germany, prevalence of CU along with multiple comorbidities may pose increased burden on the healthcare system. Awareness of adhering to treatment guidelines, and aiming for complete control of urticaria, needs to be driven and may improve outcomes

Subdiffusive transport in intergranular lanes on the Sun. The Leighton model revisited

In this paper we consider a random motion of magnetic bright points (MBP) associated with magnetic fields at the solar photosphere. The MBP transport in the short time range [0-20 minutes] has a subdiffusive character as the magnetic flux tends to accumulate at sinks of the flow field. Such a behavior can be rigorously described in the framework of a continuous time random walk leading to the fractional Fokker-Planck dynamics. This formalism, applied for the analysis of the solar subdiffusion of magnetic fields, generalizes the Leighton's model.Comment: 7 page

Epidemiology, comorbidities, and healthcare utilization of patients with chronic urticaria in Germany

Abstract Background Comprehensive data on the epidemiology and comorbidities of chronic urticaria (CU) in Germany are either limited, or not contemporary. Objectives To investigate the epidemiology of CU, overall comorbidities and healthcare resource utilized by patients with CU in Germany, using an anonymized statutory health insurance (SHI) database. Methods Anonymized SHI claims research database of the Institute for Applied Health Research, Berlin [InGef] (01 January 2015–30 September 2018) was used to analyse insured individuals with a confirmed diagnosis of CU (ICD‐10‐GM codes). Twelve‐month diagnosed prevalence and incidence, comorbidities (vs. atopic dermatitis and psoriasis), and healthcare utilization by patients with CU were investigated. Results Of 4 693 772 individuals of all ages listed in the database, 3 538 540 were observable during 2017. Overall, 17 524 patients (˜0.5%) were diagnosed with CU; chronic spontaneous urticaria (CSU: 71.2%), chronic inducible urticaria (CIndU: 19.7%), CSU+CIndU (9.1%). Females, vs. males, had higher diagnosed prevalence (0.62% vs. 0.37%) and diagnosed incidence (0.18% vs. 0.11%) of CU among all patients. Patients most frequently visited general practitioners (41.3% of total visits). Hypertensive diseases (43.5%), lipoprotein metabolism disorders (32.1%) and affective disorders (26.0%) were the most frequently reported comorbidities of special interest. Rates of most comorbidities of special interests were similar to atopic dermatitis and psoriasis patients, and all higher vs. overall population. More than half (54.1%) of all CU patients were not prescribed any treatment. Second‐generation H 1 ‐antihistamines were the most commonly prescribed medication for adult (17.9%) and paediatric (27.9%) patients. Patients with CIndU (paediatric, 15.5%; adult, 7.8%) were more often hospitalized versus patients with CSU (paediatric, 9.9%; adult, 4.6%). Conclusions In Germany, prevalence of CU along with multiple comorbidities may pose increased burden on the healthcare system. Awareness of adhering to treatment guidelines, and aiming for complete control of urticaria, needs to be driven and may improve outcomes

Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $\nu$) of the Fresnel equation is also performed and, in detail, some specific cases, like $\nu=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure

Convolution-type derivatives, hitting-times of subordinators and time-changed C0C_0-semigroups

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.Comment: Final version, Potential analysis, 201

Fractional Cauchy problems on bounded domains: survey of recent results

In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \citet{nigmatullin}, and \citet{zaslavsky} in $\mathbb R^d$ for modeling some physical phenomena. The fractional derivative models time delays in a diffusion process. We will give a survey of the recent results on the fractional Cauchy problem and its generalizations on bounded domains D\subset \rd obtained in \citet{m-n-v-aop, mnv-2}. We also study the solutions of fractional Cauchy problem where the first time derivative is replaced with an infinite sum of fractional derivatives. We point out a connection to eigenvalue problems for the fractional time operators considered. The solutions to the eigenvalue problems are expressed by Mittag-Leffler functions and its generalized versions. The stochastic solution of the eigenvalue problems for the fractional derivatives are given by inverse subordinators

Spectral fiber dosimetry with beryllium oxide for quality assurance in hadron radiation therapy

Using the radioluminescence light of solid state probes coupled to long and flexible fibers for dosimetry in radiotherapy offers many advantages in terms of probe size, robustness and cost efficiency. However, especially in hadron fields, radioluminophores exhibit quenching effects dependent on the linear energy transfer. This work describes the discovery of a spectral shift in the radioluminescence light of beryllium oxide in dependence on the residual range at therapeutic proton energies. A spectrally resolving measurement setup has been developed and tested in scanned proton fields. It is shown that such a system can not only quantitatively reconstruct the dose, but might also give information on the residual proton range at the point of measurement

Numerical approximations for the tempered fractional Laplacian: Error analysis and applications

In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of $O(h^\epsilon)$, for $u \in C^{0, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{1, \alpha-1+\epsilon} (\bar{\Omega})$ if $\alpha \ge 1$) with $\epsilon > 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in C^{2, \alpha+\epsilon} (\bar{\Omega})$ if $\alpha < 1$ (or $u \in C^{3, \alpha - 1 + \epsilon} (\bar{\Omega})$ if $\alpha \ge 1$). Numerical experiments confirm our analytical results and provide insights in solving the tempered fractional Poisson problem. It suggests that to achieve the second order of accuracy, our method only requires the solution $u \in C^{1,1}(\bar{\Omega})$ for any $0<\alpha<2$. Moreover, if the solution of tempered fractional Poisson problems satisfies $u \in C^{p, s}(\bar{\Omega})$ for $p = 0, 1$ and $0<s \le 1$, our method has the accuracy of $O(h^{p+s})$. Since our method yields a (multilevel) Toeplitz stiffness matrix, one can design fast algorithms via the fast Fourier transform for efficient simulations. Finally, we apply it together with fast algorithms to study the tempered effects on the solutions of various tempered fractional PDEs, including the Allen-Cahn equation and Gray-Scott equations.Comment: 21 pages, 11 figures, 3 table