Brownian subordinators and fractional Cauchy problems. (English) Zbl 1186.60079
The authors establish a connection between two seemingly disparate classes of subordinated stochastic processes and their govering equations.
More precisely, let \(X_0(t)\) be continuous Markov process and \(L_x\) its generator. Let be \(Y\) an independent Brownian motion and set \(Z_t= X(Y_t)\), where \(X\) is the two-sided pocess defined by \(X(t)= X_0(t)\), for \(t\geq 0\), and \(X(t)= X_1(-1)\), for \(t< 0\), with \(X_1\) an independent copy of \(X_0\).
It is known since [H. Allouba and W. Zheng, Ann. Probab. 29, No .4, 1780–1795 (2001; Zbl 1018.60066)] that \(u(t,x)= E[f(Z_t)|Z_0= x]\) solves the following initial value problem \[ {\partial\over\partial t} u(t,x)= {L_xf(x)\over \sqrt{\pi t}}+ L^2_x u(t,x);\;u(0,x)= f(x).\tag{1} \] Further, for \(\beta\in (0,1)\), let \(E_t= \text{inf}\{x> 0; D(x)> t\}\), where \(D\) is a stable subordinate independent of \(X_0\), with \(E[e^{-sD_t}]= e^{-ts^\beta}\). The subordinated process \(Z_t= x+ X_0(E_t)\) occurs as the scaling limit of a continuous time random walk [M. M. Meerschaert and H.-P. Scheffler, J. Appl. Probab. 41, No. 3, 623–638 (2004; Zbl 1065.60042)]. Furthermore \(u(t,x)= E[f(Z_t)|Z_0= x]\) satisfies the fractional Cauchy problem \[ {\partial^\beta\over\partial t^\beta} u(t,x)= L_x u(t,x);\;u(0,x)= f(x).\tag{2} \] The main result of this paper states that, for \(\beta={1\over 2}\), the two equations (1) and (2) have the same solution.
More precisely, let \(X_0(t)\) be continuous Markov process and \(L_x\) its generator. Let be \(Y\) an independent Brownian motion and set \(Z_t= X(Y_t)\), where \(X\) is the two-sided pocess defined by \(X(t)= X_0(t)\), for \(t\geq 0\), and \(X(t)= X_1(-1)\), for \(t< 0\), with \(X_1\) an independent copy of \(X_0\).
It is known since [H. Allouba and W. Zheng, Ann. Probab. 29, No .4, 1780–1795 (2001; Zbl 1018.60066)] that \(u(t,x)= E[f(Z_t)|Z_0= x]\) solves the following initial value problem \[ {\partial\over\partial t} u(t,x)= {L_xf(x)\over \sqrt{\pi t}}+ L^2_x u(t,x);\;u(0,x)= f(x).\tag{1} \] Further, for \(\beta\in (0,1)\), let \(E_t= \text{inf}\{x> 0; D(x)> t\}\), where \(D\) is a stable subordinate independent of \(X_0\), with \(E[e^{-sD_t}]= e^{-ts^\beta}\). The subordinated process \(Z_t= x+ X_0(E_t)\) occurs as the scaling limit of a continuous time random walk [M. M. Meerschaert and H.-P. Scheffler, J. Appl. Probab. 41, No. 3, 623–638 (2004; Zbl 1065.60042)]. Furthermore \(u(t,x)= E[f(Z_t)|Z_0= x]\) satisfies the fractional Cauchy problem \[ {\partial^\beta\over\partial t^\beta} u(t,x)= L_x u(t,x);\;u(0,x)= f(x).\tag{2} \] The main result of this paper states that, for \(\beta={1\over 2}\), the two equations (1) and (2) have the same solution.
Reviewer: Catherine Rainer (Brest)
Keywords:
fractional diffusion; Lévy process; Cauchy problem; iterated Brownian motion; Brownian subordinator; Caputo derivativeReferences:
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