STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝ^d

Abstract

Abstract. We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

Probing the quantum nature of spacetime by diffusion

“…In this case, it is an important observation [45,46,[50][51][52][53] that the probability density (31) can also be obtained as a solution of an ordinary partial differential equation including higher-order spatial derivatives, once suitable source terms are included. In this section we generalize these ideas in order to obtain positive-definite solutions for the UV limit of HL gravity in three and four spacetime dimensions.…”

Diffusion in multiscale spacetimes

“…An example of particular importance for us is iterated Brownian motion (IBM) [86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104]. To illustrate iterated Brownian motion, we set α = 1 and consider the ordinary D-dimensional higher-order operator ∇ n , Eq.…”

From Brownian-Time Brownian Sheet to a Fourth Order and a Kuramoto–Sivashinsky-Variant Interacting PDEs Systems

“…then u is the solution to the PDE (1.1) for an appropriate class of initial functions f (see also [4,16,21]). The novelty in the PDE (1.1) is the x f x / √ 8 t term which expresses the BTP memory-preserving (non-Markovian) property.…”

“…(1) Probabilistically (1.1) is the PDE solved by "running" the intrinsicallyinteresting BTP, giving a dynamic description of the average of the modulated process f x B ; (2) The term x f x / √ 8 t results in an interesting unconventional interaction between the initial data f and the solution u (this feature is under investigation for its potential applications to financial models); (3) Along with Zheng, we also initiated in [5] the link between Brownian-time processes and fractional time derivatives by way of the BTP half-derivative generator-an implicit equivalence of (1.1) to temporally-fractional PDEs that was later made explicit and extended in several directions in the nice works of Nane, Meerschaert, Baeumer, Vellaisamy, Orsingher, and Beghin [12,13,19,21,22]; and (4) In [3], we adapted our BTPs construction in [4,5] and their PDEs connections and gave an explicit representation of the solution to a linearization of the celebrated Kuramoto-Sivashinsky (KS) PDE in all spatial dimensions. Besides giving a probabilistic Brownian-time-motivated and explicit representation of this important equation, this achievement is noteworthy since the existence of the KS semigroup is still unsettled for dimensions larger than 1 via traditional analytical methods, precluding the possibility to have definitive results using semigroup methods in higher dimensions.…”

“…This connection opens up another new Allouba fundamental front in the rapidly growing field of Brownian-time and iterated processes and their connections to both (1) new and important higher order deterministic and stochastic PDEs and (2) some equivalent fractional Cauchy problems (e.g., Allouba and Allouba et al [1][2][3][4][5][6][8][9][10] and followup papers; as well as the recent articles by Nane [20,21], Baeumer et al [12], Meerschaert et al [19], Orsingher et al [13,22], and DeBlassie [16]). In this new Brownian-time Brownian sheet PDEs direction, the BTBS-PDEs connections (1.5) and (1.4) introduce a way for a random field to shed light on nonlinear PDEs-and their corresponding system of n linear PDEs-that are fourth order and interacting; and these PDEs, in turn, could be used to gain insight into the underlying Brownian-time random field.…”