Stochastic solutions for fractional wave equations

Abstract

A fractional wave equation replaces the second time derivative by a Caputo derivative of order between one and two. In this paper, we show that the fractional wave equation governs a stochastic model for wave propagation, with deterministic time replaced by the inverse of a stable subordinator whose index is one-half the order of the fractional time derivative.

1 Introduction

The traditional wave equation

$$\begin{aligned} \frac{\partial ^2}{\partial t^2} p(x,t)=\Delta _x p(x,t) \end{aligned}$$

(1.1)

models wave propagation in an ideal conducting medium. Assume a plane wave solution $p(x,t)=e^{-i\omega t + i k x}$ for some frequency $\omega >0$ and substitute into (1.1) to see that we must have $(-i \omega )^2=(ik)^2$ or in other words $k=\pm \omega$. This solution is a traveling wave at speed one (justified by a suitable choice of units), and the general solution to (1.1) can be written as a linear combination of plane waves.

In a complex inhomogeneous conducting medium, experimental evidence reveals that sound waves exhibit power-law attenuation, with an amplitude that falls off at an exponential rate $\alpha =\alpha (\omega )\approx \omega ^p$ for some power-law index $p$ (e.g., see Duck [12] for applications to medical ultrasound). A variety of modified wave equations have been proposed to model wave conduction in complex media [11, 14, 22, 23]. We note here, apparently for the first time, that a simple time-fractional wave equation

$$\begin{aligned} \frac{\partial ^\gamma }{\partial t^\gamma } p(x,t)=\Delta _x p(x,t),\qquad 1<\gamma <2, \end{aligned}$$

(1.2)

which replaces the second time derivative by a fractional derivative, also exhibits power-law attenuation. Assuming the same plane wave solution, and using the well-known formula $\frac{\mathrm{d}^\gamma }{\mathrm{d}t^\gamma } [\mathrm{e}^{at}]=t^\gamma \mathrm{e}^{at}$ (e.g., see [18, Example 2.6]), we now have $(-i \omega )^\gamma =-k^2$, and a little algebra yields $k=\beta (\omega )+i\alpha (\omega )$ with attenuation coefficient $\alpha (\omega )=\alpha _0 \omega ^{\gamma /2}$. Hence, solutions to the time-fractional wave equation (1.2) also exhibit power-law attenuation with power-law index $p=\gamma /2$.

The goal of this paper is to develop a new stochastic solution to the time-fractional wave equation (1.2). Our stochastic solution is based on limit theory for random walks and, therefore, provides a simple and illuminating statistical physics model for wave conduction in complex media.

2 Background

In one spatial dimension, the general solution to (1.1) with initial conditions $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t}p(x,0)=\psi (x)$ is given by the d’Alembert formula

$$\begin{aligned} p(x,t)=\frac{1}{2}\left[ \phi (x+t)+\phi (x-t)\right] +\int \limits _{x-t}^{x+t} \psi (y)\mathrm{d}y.\nonumber \\ \end{aligned}$$

(2.1)

In fact, for all continuous and exponentially bounded functions $\phi ,\psi$ the unique solution to the equivalent integral equation

$$\begin{aligned} p(x,t)=\phi (x)+t\psi (x)+\int \limits _0^t (t-s)\Delta _x p(x,s)\mathrm{d}s\nonumber \\ \end{aligned}$$

(2.2)

is given by the d’Alembert formula. In this paper, we consider the following integral form of the fractional wave equation:

$$\begin{aligned} p(x,t)&= \phi (x)+\frac{t^{\gamma /2}}{\Gamma (1+\gamma /2)}\psi (x)\nonumber \\&+\frac{1}{\Gamma (\gamma )}\int \limits _0^{t} (t-s)^{\gamma -1}\Delta _x p(x,s)\mathrm{d}s. \end{aligned}$$

(2.3)

The differential form of equation (2.3) employs the Riemann–Liouville fractional derivative. The Riemann–Liouville fractional integral of non-integer order $\gamma >0$ is defined by

$$\begin{aligned} \mathbb I_t^\gamma f(t)=\frac{1}{\Gamma (\gamma )}\int \limits _0^\infty (t-s)^{\gamma -1} f(s)\mathrm{d}s. \end{aligned}$$

(2.4)

The Riemann–Liouville fractional derivative of non-integer order $\gamma >0$ is defined by

$$\begin{aligned} \mathbb D_t^\gamma f(t)=\frac{1}{\Gamma (n-\gamma )}\frac{\mathrm{d}^n}{\mathrm{d}t^n}\int \limits _0^\infty (t-s)^{n-\gamma -1} f(s)\mathrm{d}s\nonumber \\ \end{aligned}$$

(2.5)

where $n$ is the smallest integer greater than $\gamma$.

Equation (2.3) corresponds to the following fractional differential equation:

$$\begin{aligned}&\mathbb D_t^{\gamma } p(x,t)-\phi (x)\frac{t^{-\gamma }}{\Gamma (1-\gamma )}\nonumber \\&\quad -\psi (x) \frac{t^{-\gamma /2}}{\Gamma (1-\gamma /2)}= \Delta _x p(x,t) \end{aligned}$$

(2.6)

with initial conditions

$$\begin{aligned} p(x,0)=\phi (x) \quad \text {and}\quad \frac{\partial ^{\gamma /2}}{\partial t^{\gamma /2}} p(x,0)=\psi (x). \end{aligned}$$

In this paper, we will show that the solution to (2.3) is

$$\begin{aligned} p(x,t)&= \frac{1}{2}\,\mathbb E\left[ \phi (x+E_t)+\phi (x-E_t)\right] \nonumber \\&+\,\frac{1}{2}\,\mathbb E\left[ \int \limits _{x-E_t}^{x+E_t} \psi (y)\mathrm{d}y \right] \end{aligned}$$

(2.7)

where $E_t$ is the inverse (hitting time or first passage time) of a standard stable subordinator with index $\gamma /2$. Then, using the general theory of second-order Cauchy problems, we will extend this result to a wide variety of fractional partial differential equations that model wave-like motions. Finally, we will develop random walk models that provide a physical explanation for these fractional wave equations.

3 Fractional wave equations

Let $D_u$ be a standard stable subordinator with $D_0=0$ a.s. and Laplace transform $\mathbb E[\mathrm{e}^{-sD_u}]=\mathrm{e}^{-us^\beta }$ for some $0<\beta <1$. The random variable $D_1$ has a smooth density function $g_\beta (u)$. Define the inverse subordinator (generalized inverse, first passage time, or hitting time)

$$\begin{aligned} E_t=\inf \{u\ge 0:D_u>t\} \end{aligned}$$

(3.1)

for $t\ge 0$. Then, a simple computation [16, Corollary 3.1] shows that $E_t$ has a smooth density

$$\begin{aligned} u\mapsto h(u,t)=\frac{t}{\beta }u^{-1-1/\beta }g_\beta (tu^{-1/\beta }),\quad u\!>\!0,\; t\!>\!0.\nonumber \\ \end{aligned}$$

(3.2)

Write ${\mathbb {R}}^+=[0,\infty )$, let ${\mathcal {B}} ({\mathbb R}\times {\mathbb R}^{+})$ denote the set of real-valued continuous functions $p(x,t)$ on ${\mathbb R}\times {\mathbb {R}}^+$ such that $|p(x,t)|\le A\mathrm{e}^{B(|x|+t)}$ for some constants $A,B>0$, and denote by ${\mathcal {B}}({\mathbb R})$ the set of real-valued continuous functions $\phi (x)$ on ${\mathbb R}$ such that $|\phi (x)|\le A\mathrm{e}^{B|x|}$ for some constants $A,B>0$. Denote by ${\mathcal {B}}^{m}({\mathbb R})$ the set of real-valued continuously differentiable functions $\phi (x)$ such that $\frac{\mathrm{d}^j}{\mathrm{d}x^j} \phi \in {\mathcal {B}}({\mathbb R})$ for all integers $0\le j\le m$ and by ${\mathcal {B}}^{m,0}({\mathbb R}\times {\mathbb R}_+)$ the set of real-valued functions $p(x,t)$ on ${\mathbb R}\times {\mathbb {R}}^+$ continuously differentiable in $x$ with $\frac{\mathrm{d}^j}{\mathrm{d}x^j} p(x,t)\in {\mathcal {B}}({\mathbb R}\times {\mathbb R}_+)$ for all integers $0\le j\le m$.

Theorem 3.1

Let $\Psi (x)=\int _0^x \psi (y)dy$. For any $\phi ,\,\psi$ such that $\phi , \,\Psi \in {\mathcal {B}}^2({\mathbb R})$, the unique solution to the fractional wave equation (2.3) in $\mathcal B^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,0)=\psi (x)$ is given by the formula (2.7), where $E_t$ is the inverse stable subordinator (3.1) with index $\beta =\gamma /2$.

Proof

The proof uses a result of Fujita [13] together with a duality result from [3]. Fujita considers a stable Lévy process $X_\gamma (t)$ with characteristic function

$$\begin{aligned} \mathbb E\left[ \mathrm{e}^{ikX_\gamma (t)}\right] =\exp \left[ -t|k|^{2/\gamma }\mathrm{e}^{-i(\pi /2)(2-2/\gamma )\mathrm{sgn }(k)} \right] .\nonumber \\ \end{aligned}$$

(3.3)

with index $1<2/\gamma <2$ along with its supremum process

$$\begin{aligned} Y_\gamma (t)=\sup _{0\le u\le t}X_\gamma (u). \end{aligned}$$

(3.4)

Fujita shows that for $\phi ,\,\psi$ such that $\phi , \,\Psi \in \mathcal B^2({\mathbb R})$, the unique solution to (2.3) in $\mathcal B^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ is

$$\begin{aligned} p(x,t)&= \frac{1}{2}\,\mathbb E\left[ \phi (x+Y_\gamma (t))+\phi (x-Y_\gamma (t))\right] \nonumber \\&+\,\frac{1}{2}\,\mathbb E\left[ \int \limits _{x-Y_\gamma (t)}^{x+Y_\gamma (t)} \psi (y)\,dy\right] \!. \end{aligned}$$

(3.5)

Using the parameterization of Samorodnitsky and Taqqu [21], the characteristic function of a generic stable process $\xi (t) = \xi _{\mu ,\alpha ,\sigma ,\theta }(t)$ is

$$\begin{aligned} \mathbb E\left[ \mathrm{e}^{ik \xi (t)}\right]&= \exp \left[ ik\mu -\sigma ^\alpha |k|^{\alpha } \right. \\&\times \left. \left\{ 1-i\theta \mathrm{sgn }(k)\tan (\pi \alpha /2)\right\} \right] . \end{aligned}$$

An elementary calculation (e.g., see [3, p. 1101]) shows that the process $X_\gamma (t)$ has stability index $\alpha =2/\gamma$, skewness $\theta =-1$, scale $\sigma ^\alpha =-t\cos (\pi \alpha /2)>0$, and centering constant $\mu =0$. Hence, $X_\gamma (t)$ is a spectrally negative stable Lévy process, with no positive jumps. Use the elementary formula (e.g., see [18, Eq. 5.5]) $(ik)^\alpha =|k|^\alpha \cos (\pi \alpha /2)(1+i\mathrm{sgn }(k)\tan (\pi \alpha /2))$ to write $\mathbb E\left[ \mathrm{e}^{ikX_\gamma (t)}\right] =\mathrm{e}^{t(ik)^\alpha }$ and then set $k=-is$ to see that

$$\begin{aligned} \mathbb E\left[ \mathrm{e}^{s X_\gamma (t)}\right] =\mathrm{e}^{ts^\alpha } \end{aligned}$$

for all $s\ge 0$ and $t\ge 0$. Now, it follows from [7, Theorem 1, p. 189] that the first-passage time process

$$\begin{aligned} D_u=\inf \{t\ge 0: X_\gamma (t)>u\} \end{aligned}$$

is a stable subordinator with Laplace transform $\mathbb E\big [\mathrm{e}^{-sD_u}\big ]=\mathrm{e}^{-us^\beta }$ for all $u\ge 0$ and $s\ge 0$, where the stability index $\beta =1/\alpha =\gamma /2$. Then, the inverse $\beta$-stable subordinator $E_t$ in (3.1) is the generalized inverse of $X_\gamma (t)$, which equals the supremum of $X_\gamma (t)$. Hence, we have $E_t=Y_\gamma (t)$ pathwise, see also Proposition 1 in [9]. Then, the form of the solution follows from (3.5).

The integral form of the fractional wave equation (2.3) corresponds to the differential form (1.2) with the initial conditions $p(x,0)=\phi (0)$ and $\psi = \frac{\partial ^\beta }{\partial x^\beta }p(x,t)$, $\beta =\gamma /2$. The first initial condition follows directly from (2.3). As for the second one, note that

$$\begin{aligned} \frac{1}{\Gamma (\gamma )}\int \limits _0^{t} (t-s)^{\gamma -1}\Delta _x p(x,s)\mathrm{d}s =\mathbb I_t^\gamma \Delta _xp(x,t) \end{aligned}$$

and apply the Caputo derivative to both sides of (2.3) to get

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,t)=\psi (x)+ \frac{\partial ^{\beta }}{\partial t^{\beta }} \mathbb I_t^\gamma \Delta _xp(x,t). \end{aligned}$$

Since Caputo and Riemann–Liouville derivatives of order $0<\beta <1$ are related by (e.g., see [18, p. 39])

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }} f(t)=\mathbb D_t^{\beta }f(x)-f(0)\frac{t^{-\beta }}{\Gamma (1-\beta )}, \end{aligned}$$

and $\mathbb I_t^\gamma \Delta _xp(x,t)$ evaluated at $t=0$ is zero, we have

$$\begin{aligned} \frac{\partial ^{\beta }}{\partial t^{\beta }} \mathbb I_t^\gamma \Delta _xp(x,t)=\mathbb D_t^{\beta }\mathbb I_t^\gamma \Delta _xp(x,t)=\mathbb I_t^{\beta }\Delta _xp(x,t). \end{aligned}$$

Since $p(x,t) \in \mathcal B^{2,0}({\mathbb R}\times {\mathbb R}^{+})$, $|\Delta _xp(x,s)|\le A\mathrm{e}^{B(|x|+t)}$ for some constants $A,\, B>0$ and $0\le s\le t$. Therefore,

$$\begin{aligned} \left| \mathbb I_t^{\beta }\Delta _xp(x,t)\right|&\le \frac{A\mathrm{e}^{B(|x|+t)}}{\Gamma (\beta )} \int \limits _0^t (t-s)^{\beta -1}\mathrm{d}s\nonumber \\&= \frac{A\mathrm{e}^{B(|x|+t)}}{\Gamma (\beta +1)} t^{\beta }\rightarrow 0 \,\, {\text {as}} \,\,\ t\rightarrow 0. \end{aligned}$$

Thus, the initial conditions corresponding to (2.3) are $p(x,0)=\phi (x)$ and $\frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,0)=\psi (x)$.$\square$

In the remainder of this section, we discuss related results in the literature and give some alternative stochastic representations of the solution.

Remark 3.2

Define the reflected stable process

$$\begin{aligned} Z_t = X_\gamma (t)-\inf \{X_\gamma (s): 0 \le s \le t\}, \end{aligned}$$

(3.6)

where $X_\gamma (t)$ is the spectrally negative stable process with index $1<\gamma \le 2$ and characteristic function (3.3). Apply [4, Lemma 4.5] to see that $Z_t$ has the same one-dimensional distributions as the inverse (3.1) of a standard $\beta$-stable subordinator with $\beta =\gamma /2$. Then, it follows from Theorem 3.1 that for any $\phi , \,\psi$ such that $\phi ,\Psi \in \mathcal B^2({\mathbb R})$, the unique solution to the fractional wave equation (1.2) in ${\mathcal {B}}^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,0)=\psi (x)$ is given by the formula

$$\begin{aligned} p(x,t)&= \frac{1}{2}\,\mathbb E\left[ \phi (x+Z_t)+\phi (x-Z_t)\right] \nonumber \\&+\,\frac{1}{2}\,\mathbb E\left[ \int \limits _{x-Z_t}^{x+Z_t} \psi (y)\,dy\right] \!. \end{aligned}$$

(3.7)

The advantage to this representation is that $Z_t$ is a Markov process.

Remark 3.3

Mainardi [15, Sect. 6.3] considers a version of (1.2) that employs the Caputo fractional derivative

$$\begin{aligned} \frac{\partial ^\gamma }{\partial t^\gamma } p(x,t)= \frac{1}{\Gamma (2-\gamma )}\int \limits _0^t\frac{\partial ^2}{\partial u^2} p(x,u)(t-u)^{1-\gamma }\mathrm{d}u\nonumber \\ \end{aligned}$$

(3.8)

of order $1<\gamma <2$. Mainardi [15, Sect. 6.4] derives the fractional wave equation (1.2) from a viscoelastic model with a power-law stress–strain relationship. He notes that the Green’s function solution to the fractional wave equation (1.2) can be also expressed in terms of stable densities. He considers the fractional wave equation (1.2) subject to the initial conditions $p(x,0)=\delta (x)$ and $\frac{\partial }{\partial t} p(x,0)=0$. Since the Caputo and Riemann-Liouville fractional derivatives of order $1<\gamma <2$ are related by (e.g., see [5, p. 11])

$$\begin{aligned} \frac{\partial ^\gamma }{\partial t^\gamma } f(t)&= \mathbb D_t^\gamma f(t)-f(0)\frac{t^{-\gamma }}{\Gamma (1-\gamma )}\nonumber \\&-f'(0)\frac{t^{1-\gamma }}{\Gamma (2-\gamma )}, \end{aligned}$$

(3.9)

when $\frac{\partial }{\partial t}p(x,0)=0$ Eq. (2.6) with initial conditions $p(x,0)=\phi (x)$ and $\frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,0)=0$ has the same integral form as Eq. (1.2) with $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t}p(x,0)=0$.

Letting $L_\alpha ^\eta (x)$ be the stable probability density function with characteristic function

$$\begin{aligned} \int \limits _{-\infty }^\infty \mathrm{e}^{ikx} L_\alpha ^\eta (x)\mathrm{d}x=\exp \left[ -|k|^\alpha \mathrm{e}^{i\mathrm{sgn }(k)\eta \pi /2} \right] \end{aligned}$$

Mainardi shows that

$$\begin{aligned} L_\alpha ^{\alpha -2}(x)=\frac{1}{\alpha }\Phi _{1/\alpha }(x)\quad \text {for any } x\in {\mathbb R}\text { and } 1<\alpha \le 2, \end{aligned}$$

where

$$\begin{aligned} \Phi _\beta (z)=\sum _{n=0}^\infty \frac{(-z)^n}{n!\Gamma (1-n\beta \!-\!\beta )}, \quad 0<\beta <1 \end{aligned}$$

(3.10)

is the Wright function. It follows [15, Eq. 6.37] that the solution to the fractional wave equation (1.2) with initial conditions $p(x,0)=\delta (x)$ and $\frac{\partial }{\partial t}p(x,0)=0$ can be written in the form

$$\begin{aligned} p(x,t)=\frac{1}{2\beta t^{\beta }}L_{1/\beta }^{(1/\beta )-2}\left( \frac{|x|}{t^\beta }\right) \end{aligned}$$

with $\beta =\gamma /2$. Hence, the solution to the fractional wave equation (1.2) with initial conditions $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t}p(x,0)=0$ is given by

$$\begin{aligned} p(x,t)&= \int \limits _0^\infty \frac{1}{2}\Big [\phi (x-u)+\,\phi (x+u)\Big ]\nonumber \\&\times \frac{1}{\beta t^{\beta }}L_{1/\beta }^{(1/\beta )-2}\left( \frac{u}{t^\beta }\right) \mathrm{d}u. \end{aligned}$$

(3.11)

The solution (3.11) to the fractional wave equation (1.2) involves a stable density with index $\alpha =1/\beta =2/\gamma$, whereas the solution in Theorem 3.1 uses the inverse of a stable law with index $\beta =\gamma /2$. This can be explained using the Zolotarev duality formula for stable densities [3, Theorem 2.1]. Baeumer et al. [3, Theorem 4.1] use Zolotarev duality to prove that $\beta h(u,t)=q(u,t)$ for all $t>0$ and $u\ge 0$, where $h(u,t)$ is the density (3.2) of the standard inverse $\beta$-stable subordinator on the set $u\ge 0$, $q(u,t)$ is the density of the spectrally negative stable process $X_\gamma (t)$ with index $1<\gamma \le 2$ on the set $-\infty <u<\infty$, and $\alpha =1/\beta$. For $u>0$, the self-similarity argument shows that the function

$$\begin{aligned} q(u,t)=\frac{1}{t^{\beta }}L_{1/\beta }^{(1/\beta )-2}\left( \frac{u}{t^\beta }.\right) \end{aligned}$$

Then, the solution (3.11) reduces to a special case of (2.7) with $\psi (x)=0$ since

$$\begin{aligned} p(x,t)&= \int \limits _0^\infty \frac{1}{2}\left[ \phi (x-u)\right. \\&\left. +\,\phi (x+u)\right] \frac{1}{\beta t^{\beta }}L_{1/\beta }^{(1/\beta )-2}\left( \frac{u}{t^\beta }\right) \mathrm{d}u \\&= \int \limits _0^\infty \frac{1}{2}\left[ \phi (x-u)+\phi (x+u)\right] \frac{q(u,t)}{\beta }\mathrm{d}u\\&= \int \limits _0^\infty \frac{1}{2}\left[ \phi (x-u)+\phi (x+u)\right] h(u,t)\mathrm{d}u. \end{aligned}$$

Theorem 4.1 in [3] also shows that the conditional distribution of $X_\gamma (t)$ given $X_\gamma (t)>0$ is identical to the distribution of $E_t$. Hence, for any $\phi ,\,\psi$ such that $\phi ,\Psi \in {\mathcal {B}}^2({\mathbb R})$, the unique solution to the fractional wave equation (2.6) in ${\mathcal {B}}^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{\partial ^{\beta }}{\partial t^{\beta }} p(x,0)=\psi (x)$ can be written as

$$\begin{aligned} p(x,t)&= \frac{1}{2}\,\mathbb E\left[ \phi (x+X_\gamma (t))+\phi (x-X_\gamma (t))\right. \nonumber \\&\left. +\int \limits _{x-X_\gamma (t)}^{x+X_\gamma (t)} \psi (y)\mathrm{d}y\,\bigg \vert \, X_\gamma (t)>0\right] . \end{aligned}$$

(3.12)

An extension of the well-known D. André reflection principle (see Appendix) shows that $\mathbb P[Y_\gamma (t)\ge x]=\mathbb P[X_\gamma (t)\ge x\,|\,X_\gamma (t)\ge 0]$, and this together with (3.5) gives another proof of (3.12).

4 General wave equations

Given a closed operator $L$ on a Banach space $X$ of functions, consider the second-order Cauchy problem

$$\begin{aligned}&\frac{\partial ^2}{\partial t^2}p(x,t)=L p(x,t); \nonumber \\&p(x,0)=\phi (x),\, \frac{\partial }{\partial t} p(x,0)=\psi (x). \end{aligned}$$

(4.1)

The traditional wave equation (1.1) is a special case where $L=\Delta _x$. Bajlekova [5, 6] developed the theory of fractional order Cauchy problems

$$\begin{aligned}&\frac{\partial ^\gamma }{\partial t^\gamma } p(x,t)=L p(x,t);\nonumber \\&p(x,0)=\phi (x),\ \frac{\partial }{\partial t}p(x,0)=0 \end{aligned}$$

(4.2)

using a Caputo fractional derivative of order $1<\gamma <2$.

The general theory of second-order Cauchy problems is laid out in [2, Sects. 3.14–3.16]. A strongly continuous (i.e., continuous in the Banach space norm) family of linear operators $(\mathrm {Cos}(t))_{t\ge 0}$ is called a cosine family if $\mathrm{Cos }(0)=I$ and $2\mathrm{Cos }(t)\mathrm{Cos }(s)=\mathrm{Cos }(t+s)+\mathrm{Cos }(t-s)$ for all $s,t\ge 0$. The generator $L$ of the cosine family is defined by

$$\begin{aligned} L f(x)=\lim _{t\downarrow 0}\frac{2}{t^2}\left[ \mathrm{Cos }(t)f(x)-f(x)\right] , \end{aligned}$$

and the domain $\mathrm{Dom }(L)$ is the set of functions $f\in X$ for which this limit exists strongly. If the operator $L$ in (4.1) is a generator of a cosine family $(\mathrm {Cos}(t))_{t\ge 0}$, then for any $\phi ,\psi \in X$, the unique mild solution to the second-order Cauchy problem (4.1) is given by

$$\begin{aligned} p(x,t)=\mathrm{Cos }(t)\phi (x)+\int \limits _0^t \mathrm{Cos }(s)\psi (x)\mathrm{d}s. \end{aligned}$$

(4.3)

That is, we have

$$\begin{aligned} p(x,t)=\phi (x)+t\psi (x)+L\int \limits _0^t (t-s)p(x,s)\mathrm{d}s \end{aligned}$$

for all $t\ge 0$, the integrated version of the second-order Cauchy problem. Furthermore, (4.3) is the unique (classical) solution to the second-order Cauchy problem (4.1) for any $\phi ,\psi \in \mathrm{Dom }(L)$ [2, Theorem 3.14.11].

Theorem 4.1

Suppose that the operator $L$ in (4.2) is a generator of a cosine family $(\mathrm {Cos}(t))_{t\ge 0}$. Then for any $\phi \in X$, the unique mild solution to the fractional Cauchy problem (4.2) is given by the formula

$$\begin{aligned} p(x,t)=\mathbb E\left[ \mathrm{Cos }(E_t)\phi (x)\right] , \end{aligned}$$

(4.4)

where $\mathrm{Cos }(t)\phi (x)$ is the unique mild solution to the second-order Cauchy problem (4.1) with $\psi =0$, and $E_t$ is the inverse (3.1) of the standard stable subordinator with index $\beta =\gamma /2$. Furthermore, equation (4.4) gives the unique classical solution to (4.2) for any $\phi \in \mathrm{Dom }(L)$.

Proof

Bajlekova [5, Theorem 3.1] proves that if $\mathrm{Cos }(t)\phi (x)=S_2(t)\phi (x)$ solves the second-order Cauchy problem (4.1) with $\psi = 0$, then the unique solution to the fractional Cauchy problem (4.2) is $p(x,t)=S_\gamma (t)\phi (x)$ where the family of solution operators $S_\gamma (t)$ is given by the subordination formula

$$\begin{aligned} S_\gamma (t)=\int \limits _0^\infty S_2(s)t^{-\gamma /2}\Phi _{\gamma /2}(st^{-\gamma /2})\mathrm{d}s , \end{aligned}$$

(4.5)

using the Wright function defined in (3.10). Recall the identity (e.g., see [5, Eq. 1.31])

$$\begin{aligned} \int \limits _0^\infty \mathrm{e}^{zt}\Phi _\beta (z)\mathrm{d}z=\mathcal E_\beta (t) \end{aligned}$$

(4.6)

where the Mittag-Leffler function

$$\begin{aligned} {\mathcal {E}}_\beta (z)=\sum _{n=0}^\infty \frac{z^n}{\Gamma (1+\beta n)} \end{aligned}$$

for $\beta >0$ and $z\in {\mathbb C}$. Bingham [8] and Bondesson et al. [10] show that the inverse $E_t$ of a $\beta$-stable subordinator has a Mittag-Leffler distribution with

$$\begin{aligned} \mathbb E(\mathrm{e}^{-sE_t})=\int \limits _0^\infty \mathrm{e}^{-su}h(u,t)\mathrm{d}u=\mathcal E_\beta (-st^\beta ). \end{aligned}$$

But, it follows from (4.6) along with a substitution $z=u/t^\beta$ that we also have

$$\begin{aligned} \int \limits _0^\infty \mathrm{e}^{-su}\frac{1}{t^{\beta }}\Phi _\beta \left( \frac{u}{t^\beta }\right) \mathrm{d}u&= \int \limits _0^\infty \mathrm{e}^{-szt^\beta }\Phi _\beta \left( z\right) \mathrm{d}z\\&= {\mathcal {E}}_\beta (-st^\beta ) \end{aligned}$$

where $\beta =\gamma /2$. Then, it follows from the uniqueness of the Laplace transform that the standard inverse $\beta$-stable density (3.2) is related to the Wright function by

$$\begin{aligned} h(u,t)=\frac{1}{t^{\beta }}\Phi _\beta \left( \frac{u}{t^\beta }\right) . \end{aligned}$$

(4.7)

Hence, Bajlekova’s solution (4.5) to the fractional wave equation is equivalent to the formula (4.4).$\square$

Remark 4.2

Mainardi [15, Sect. 6.3] shows that the solution to the fractional wave equation (1.2) with initial conditions $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t} p(x,0)=0$ is given by the convolution formula

$$\begin{aligned} p(x,t) =\int \limits _0^\infty \frac{1}{2}\left[ \phi (x\!-\!u)+\phi (x\!+\!u)\right] \frac{1}{t^{\beta }}\Phi _\beta \left( \frac{u}{t^\beta }\right) \mathrm{d}u. \end{aligned}$$

Using (4.7), this reduces to (2.7) with $\psi (x)\equiv 0$.

Example 4.3

Given an open subset $D$ of ${\mathbb R^d}$, consider the Laplace operator $L=\Delta _x$ on $L^2(D)$ with Dirichlet boundary conditions [2, Example 7.2.1]. For any $\phi \in \mathrm{Dom }(L)$ there exists a unique solution $p(x,t)$ to the wave equation

$$\begin{aligned} \frac{\partial ^2}{\partial t^2} p(x,t)&= \Delta _x p(x,t); \quad p(x,0)=\phi (x);\\ \frac{\partial }{\partial t}p(x,0)&= 0;\ p(x,t)=0\ \quad \forall x\notin D \end{aligned}$$

by [2, Theorem 7.2.2]. Then, it follows from Theorem 4.1 that the function $p_\gamma (x,t)=\mathbb E[p(x,E_t)]$ solves the corresponding fractional wave equation

$$\begin{aligned}&\frac{\partial ^\gamma }{\partial t^\gamma } p(x,t)=\Delta _x p(x,t);\quad p(x,0)=\phi (x);\\&\frac{\partial }{\partial t}p(x,0)=0;\ p(x,t)=0\ \quad \forall x\notin D \end{aligned}$$

on this bounded domain for $1<\gamma <2$, where $E_t$ is the inverse stable subordinator (3.1) with index $\beta =\gamma /2$.

Example 4.4

If $L=B^2$, where $B$ is a generator of a $C_0$-semigroup $(A(t))_{t\ge 0}$ on a Banach space of functions, then $L$ is a generator of a cosine family given by

$$\begin{aligned} \mathrm{Cos}(t)=\frac{1}{2} \left( A(t)+A(-t)\right) ,\quad t\in {\mathbb R}, \end{aligned}$$

see [2, Example 3.14.15]. When $B=\frac{\partial }{\partial x}$, $(A(t))_{t\ge 0}$ is a shift semigroup, and equation (4.1) becomes the traditional wave equation (1.1). Equation (4.3) giving the solution becomes the d’Alembert formula (2.1). Theorem 4.1 gives the solution to the fractional wave equation (1.2) with the initial conditions $p(x,0)=0$, $\frac{ \partial }{\partial t} p(x,0)=0$.

Example 4.5

If $L$ is a self-adjoint linear operator on some Hilbert space such that $(Lx,x)_H\le \omega \Vert x\Vert _H^2$ for some $\omega >0$ and all $x\in \mathrm{Dom }(L)$, then $L$ generates a cosine family [2, Example 3.14.16], and hence, (4.3) is the unique classical solution to the wave equation

$$\begin{aligned}&\frac{\partial ^2}{\partial t^2} p(x,t)=L p(x,t); \\&p(x,0)=\phi (x);\ \frac{ \partial }{\partial t}p(x,t)=0 \end{aligned}$$

for any $\phi \in \mathrm{Dom }(L)$. Then, Theorem 4.1 implies that the function $p_\gamma (x,t)=\mathbb E[p(x,E_t)]$ solves the corresponding fractional wave equation

$$\begin{aligned}&\frac{\partial ^\gamma }{\partial t^\gamma } p(x,t)=L p(x,t); \\&p(x,0)=\phi (x);\ \frac{ \partial }{\partial t} p(x,t)=0 , \end{aligned}$$

where $E_t$ is the inverse stable subordinator (3.1) with index $\beta =\gamma /2$.

5 Random walk models

In this section, we will develop a random walk model for the fractional wave equation (1.2). First, we decompose the fractional wave equation into simpler parts. Using the notation (2.4) for the Riemann–Liouville fractional integral, the integral form (2.3) of the fractional wave equation with $\psi \equiv 0$ can be written as

$$\begin{aligned} p(x,t)=\phi (x)+\mathbb I_t^{\gamma }\Delta _x p(x,t). \end{aligned}$$

(5.1)

Using the property $\mathbb D_t^\gamma =\mathbb D_t^n\mathbb I_t^{n-\gamma }$ for the Riemann–Liouville factional derivative and integral, and the semigroup property $\mathbb I_t^\alpha \mathbb I_t^\beta =\mathbb I_t^{\alpha +\beta }$, it follows easily that $\mathbb D_t^\gamma \mathbb I_t^\gamma f(t)=f(t)$ [5, Theorem 1.5]. Apply the operator $\mathbb D_t^\gamma$ to both sides of (5.1) to get the equivalent form

$$\begin{aligned} \mathbb D_t^\gamma p(x,t)=\Delta _x p(x,t)+\mathbb D_t^\gamma \phi (x). \end{aligned}$$

(5.2)

An easy computation [18, Example 2.8] shows that $\mathbb D_t^\gamma 1=t^{-\gamma }/\Gamma (1-\gamma )$, and then, (5.2) becomes

$$\begin{aligned} \mathbb D_t^\gamma p(x,t)-\phi (x)\frac{t^{-\gamma }}{\Gamma (1-\gamma )} =\Delta _x p(x,t). \end{aligned}$$

(5.3)

Since the Caputo and Riemann–Liouville fractional derivatives of order $1<\gamma <2$ are related by (3.9), Eq. (5.3) is equivalent to the fractional wave equation (1.2) with initial conditions $p(x,0)=\phi (x)$ and $\frac{ \partial }{\partial t} p(x,0)=0$.

Now, consider the one way fractional wave equations

$$\begin{aligned} \mathbb D_t^{\gamma /2} p^{+}(x,t)&= -\nabla _x p^{+}(x,t) +\frac{1}{2}\phi (x)\frac{t^{-\gamma /2}}{\Gamma (1-\gamma /2)}\nonumber \\ \mathbb D_t^{\gamma /2} p^{-}(x,t)&= \nabla _x p^{-}(x,t) +\frac{1}{2}\phi (x)\frac{t^{-\gamma /2}}{\Gamma (1-\gamma /2)}\nonumber \\ \end{aligned}$$

(5.4)

where again $1<\gamma <2$. Apply the operator $\mathbb I_t^{\gamma /2}$ to both sides to obtain the integral forms

$$\begin{aligned} (I+\mathbb I_t^{\gamma /2}\nabla _x )p^{+}(x,t)&= \frac{1}{2}\phi (x)\nonumber \\ (I-\mathbb I_t^{\gamma /2}\nabla _x )p^{-}(x,t)&= \frac{1}{2}\phi (x) \end{aligned}$$

(5.5)

where $I$ is the identity operator.

Theorem 5.1

For any $\phi \in \mathcal B^1({\mathbb R})$, the unique solutions to the one way fractional wave equations (5.5) in $\mathcal B^{1,0}({\mathbb R}\times {\mathbb R}^{+})$ are given by the formulae

$$\begin{aligned} p^{+}(x,t)&= \frac{1}{2}\mathbb E\left[ \phi (x-E_t)\right] \nonumber \\ p^{-}(x,t)&= \frac{1}{2}\mathbb E\left[ \phi (x+E_t)\right] \end{aligned}$$

(5.6)

where $E_t$ is the generalized inverse (3.1) of the standard stable subordinator with index $\beta =\gamma /2$. Furthermore, the unique solution to the fractional wave equation (2.3) in ${\mathcal {B}}^{1,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{ \partial }{\partial {t}} p(x,0)=\psi (x)\equiv 0$ is then given by $p(x,t)=p^{+}(x,t)+p^{-}(x,t)$.

Proof

Fujita [13] proves the same result with $E_t$ replaced by the supremum process (3.4). As noted in the proof of Theorem 3.1, these two processes have the same one-dimensional distributions. Then, the result follows.$\square$

Remark 5.2

A direct proof of Theorem 5.1 uses an idea from Fujita [13]. Apply [17, Theorem 4.1] to see that the density (3.2) of the inverse stable subordinator with index $\beta =\gamma /2$ solves equation

$$\begin{aligned} \mathbb D_t^{\gamma /2} h(x,t) =-\nabla _x h(x,t) +\delta (x)\frac{t^{-\gamma /2}}{\Gamma (1-\gamma /2)}.\nonumber \\ \end{aligned}$$

(5.7)

It follows using the principle of superposition that

$$\begin{aligned} p^{+}(x,t)=\frac{1}{2}\mathbb E\left[ \phi (x\!-\!E_t)\right] =\frac{1}{2}\int \limits _0^\infty \phi (x\!-\!u)h(u,t)\mathrm{d}y \end{aligned}$$

solves the positive one way fractional wave equation. Then, a simple change of coordinates shows that $p^{-}(x,t)=\mathbb E\left[ \phi (x+E_t)\right] /2$ solves the negative one way fractional wave equation. Now, write

$$\begin{aligned}&(I\!-\!\mathbb I^\gamma _t\Delta _x)(p^{+}\!+\!p^{-})=(I\!-\!\mathbb I^{\gamma /2}_t\nabla _x)(I+\mathbb I^{\gamma /2}_t\nabla _x)p^{+}\\&\qquad +\,(I+\mathbb I^{\gamma /2}_t\nabla _x)(I-\mathbb I^{\gamma /2}_t\nabla _x)p^{-}\\&\quad =(I-\mathbb I^{\gamma /2}_t\nabla _x)\frac{1}{2}\phi +(I+\mathbb I^{\gamma /2}_t\nabla _x)\frac{1}{2}\phi =\phi \end{aligned}$$

which is equivalent to the fractional wave equation (2.3) with $p(x,0)=\phi (x)$ and $\frac{ \partial }{\partial t}p(x,0)=0$. One can also prove Theorem 3.1 in the same manner. Just apply the same argument again with $\phi (x)$ replaced by the function $\Psi (x)=\int _0^x \psi (y)\mathrm{d}y$, and then add the two solutions.

Remark 5.3

Here, we indicate an alternative proof of Theorem 4.1 using Riemann–Liouville fractional derivatives and an idea from [19]. Suppose that $p(x,t)$ solves the second-order Cauchy problem (4.1) with initial conditions $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t} p(x,0)=0$. Let $h(u,t)$ be the density (3.2) of the inverse stable subordinator with index $\beta =\gamma /2$. From (5.7), it follows that $\mathbb D_t^\beta h(u,t)=-\partial _u h(u,t)$ on $t>0$ and $u>0$. It follows from (3.2) and the asymptotic behavior of stable densities that $h(0+,t)=t^{-\beta }/\Gamma (1-\beta )$ and $h(0-,t)=0$ for all $t>0$ (e.g., see [20, Eq. 20]). Write

$$\begin{aligned} p_\gamma (x,t)=\int \limits _0^\infty p(x,u)h(u,t)\mathrm{d}u \end{aligned}$$

and integrate by parts to get

$$\begin{aligned} \mathbb D_t^\beta p_\gamma (x,t)&= \int \limits _0^\infty p(x,u) \mathbb D_t^\beta h(u,t)\mathrm{d}u\\&= \int \limits _0^\infty p(x,u) \left[ -\partial _u h(u,t) \right] \mathrm{d}u\\&=p(x,0)h(0+,t)\\&\quad +\int \limits _0^\infty \partial _{u} p(x,u)h(u,t)\mathrm{d}u. \end{aligned}$$

Integrate by parts again and use (4.1) to get

$$\begin{aligned} \mathbb D_t^{2\beta } p_\gamma (x,t)&=p(x,0)\mathbb D_t^{\beta }h(0+,t) \\&\quad + \int \limits _0^\infty \partial _{u} p(x,u) \left[ -\partial _{u} h(u,t) \right] \mathrm{d}u\\&=p(x,0)\mathbb D_t^{\beta }h(0+,t)\\&\quad +\int \limits _0^\infty \partial _{u}^2 p(x,u) h(u,t)\mathrm{d}u \\&=L p_\gamma (x,t)+ \phi (x)\frac{t^{-{2\beta }}}{\Gamma (1-2\beta )} \end{aligned}$$

using the general formula $\mathbb D_t^{\beta }[t^\alpha ]=\Gamma (1+\alpha )t^{\alpha -\beta }/\Gamma (1+\alpha -\beta )$ [18, Example 2.7]. This equation for $p_\gamma (x,t)$ is equivalent to the second-order Cauchy problem (4.1) with initial conditions $p_\gamma (x,0)=\phi (x)$ and $\frac{\partial }{\partial t} p_\gamma (x,0)=0$, since the Caputo and Riemann–Liouville fractional derivatives of order $\gamma =2\beta \in (1,2)$ are related by (3.9).

Finally, we develop a simple particle tracking method for solving the fractional wave equation (2.3), using a continuous-time random walk [16, 17] that converges to the stochastic solution of the fractional wave equation. The main idea is to construct a random walk model that converges to the inverse stable subordinator $E_t$ and use the fact that the density (3.2) of $E_t$ solves the positive one way fractional wave equation.

Theorem 5.4

Given a continuous probability density function $\phi (x)$ on ${\mathbb R}$, let $X_0$ be a random variable with density $\phi (x)$. Let $X_1$ be a Bernoulli random variable independent of $X_0$ such that $\mathbb P[X_1=1]=\mathbb P[X_1=-1]=1/2$, set $X_n=X_1$ for $n>1$, and let $S(n)=X_1+\cdots +X_n=nX_1$. Let $W_n$ be iid random variables independent of $X_0,X_1$ with $\mathbb P[W_n>t]= Ct^{-\beta }$ for $t>C^{1/\beta }$, where $0<\beta <1$ and $C=1/\Gamma (1-\beta )$. Let $T_0=0$, $T_n=W_1+\cdots +W_n$ for $n\ge 1$, and $N_t=\max \{n\ge 0:T_n\le t\}$ for $t\ge 0$. Then,

$$\begin{aligned} X_0+c^{-\beta }S(N_{ct})\Rightarrow U_t \quad {\text{ as } } c\rightarrow \infty \end{aligned}$$

(5.8)

in $\mathcal D[0,\infty )$ with the Skorokhod $J_1$ topology, where the random variable $U_t$ has density

$$\begin{aligned} p(x,t)=\frac{1}{2}\,\mathbb E\left[ \phi (x+E_t)+\phi (x-E_t)\right] , \end{aligned}$$

(5.9)

and $E_t$ is the inverse stable subordinator (3.1) with index $\beta =\gamma /2$. Hence, $p(x,t)$ is the unique solution to the fractional wave equation (2.3) in $\mathcal B^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t}p(x,0)=\psi (x)\equiv 0$.

Proof

For any $c>0$, it follows from [18, Theorem 3.41 and Eq. 4.29] that $c^{-1/\beta }T_{[ct]}\Rightarrow D_t$ as $c\rightarrow \infty$ in $J_1$ topology in $\mathcal D[0, \infty )$, where $D_t$ is a $\beta$-stable subordinator with characteristic function $\mathbb E[\mathrm{e}^{ikD_t}]=\exp [-tC\Gamma (1-\beta )(-ik)^\beta ]$. Taking $C=1/\Gamma (1-\beta )$, the limit is a standard stable subordinator with Laplace transform $\mathbb E[\mathrm{e}^{-sD_t}]=\mathrm{e}^{-ts^\beta }$, and then, [16, Theorem 3.2 and Corollary 3.4] implies that $c^{-\beta }N_{ct}\Rightarrow E_t$ as $c\rightarrow \infty$, where $E_t$ is the inverse (3.1) of the standard stable subordinator $D_t$. Since $S(n)=nX_1$ it follows easily that

$$\begin{aligned} X_0+c^{-\beta }S(N_{ct})=X_0+c^{-\beta }N_{ct}X_1\Rightarrow X_0+E_tX_1 \end{aligned}$$

as $c\rightarrow \infty$. Then, (5.8) holds with $U_t=X_0+E_tX_1$, and a simple conditioning argument yields (5.9). Then, Theorem 3.1 shows that $p(x,t)$ is the unique solution to the fractional wave equation (2.3) in $\mathcal B^{2,0}({\mathbb R}\times {\mathbb R}^{+})$ with $p(x,0)=\phi (x)$ and $\frac{\partial }{\partial t} p(x,0)=\psi (x)\equiv 0$.$\square$

Theorem 5.4 provides a physical model for the fractional wave equation. Each sample path represents a packet of wave energy moving out from its initial position $X_0$ at unit speed, represented by the process $S(n)$. For the traditional wave equation, this is the correct particle model. In the fractional case, time delays with a power-law probability distribution occur between movements, and this retards the progress of the wave outward from the starting point. These delays are related to the heterogeneous structure of the conducting medium, see Mainardi [15, Sect. 6.4].

Remark 5.5

Theorem 5.4 implies that the histogram of a large number $M$ of identical continuous-time random-walk processes $X_0+c^{-\beta }S(N_{ct})$ gives an approximate solution to the fractional wave equation, which gains accuracy at $M\rightarrow \infty$ and $c\rightarrow \infty$. It is a simple matter to simulate the waiting times using the formula $W_n=(U_n/C)^{-1/\beta }$ where $U_n$ are iid uniform random variables on $(0,1)$. Theorem 5.4 remains true for any iid waiting times $W_n>0$ in the domain of attraction of the $\beta$-stable subordinator, except that the norming constants $c^{-\beta }$ need to be adjusted as in [16, Theorem  3.2].

Appendix

1.1 Reflection principle

The goal of this appendix is to establish the following variation of the D. André reflection principle for Brownian motion, which may also be useful in other contexts. Since this extension is not completely standard, we include its simple proof.

Theorem 6.1

(Reflection principle) Suppose that $Y_t$ is a Lévy process started at the origin, with no positive jumps, and let $S_t=\sup \{Y_u:0\le u\le t\}$. Assume that $\mathbb {P}(Y_t>0) = \mathbb {P}(Y_1 \ge 0)$, for all $t>0$. Then,

$$\begin{aligned} \mathbb {P}(S_t\ge x)&= \mathbb {P}(Y_t\ge x \,|\, Y_t\ge 0)\nonumber \\&= \frac{\mathbb {P}(Y_t\ge x)}{\mathbb {P}(Y_t\ge 0)} \end{aligned}$$

(6.1)

for all $t,x>0$.

Proof

Let $\tau _x := \inf \{u>0\,:\, Y_u > x\}$ denote the first-passage time process. Since $(Y_t)_{t\ge 0}$ has stationary independent increments, it follows that $(Y_{t+\tau _x} - Y_{\tau _x})_{t\ge 0}$ is a Lévy process, which is independent of the $\sigma$-algebra generated by $(Y_t)_{t\le \tau _x}$, and it has the same finite-dimensional distributions as $(Y_t)_{t\ge 0}$. Consequently,

$$\begin{aligned}&\mathbb {P}\left( \tau _x\le t, Y_t < Y_{\tau _x} \right) =\mathbb {P}\left( \tau _x\le t, Y_t - Y_{\tau _x} < 0\right) \nonumber \\&\quad =\mathbb {P}\left( \tau _x\le t\right) \mathbb {P}\left( Y_1<0\right) \nonumber \\&\quad =\frac{\mathbb {P}\left( Y_1<0\right) }{\mathbb {P}\left( Y_1\ge 0\right) } \mathbb {P}\left( \tau _x\le t\right) \mathbb {P}\left( Y_1\ge 0\right) \nonumber \\&\quad =\frac{\mathbb {P}\left( Y_1<0\right) }{\mathbb {P}\left( Y_1\ge 0\right) } \mathbb {P}\left( \tau _x\le t, Y_t \ge Y_{\tau _x} \right) . \end{aligned}$$

(6.2)

Observe that we have $\{\tau _x < t\} \subset \{S_t > x\} \subset \{\tau _x \le t\} \subset \{S_t\ge x\}$ for all $t$ and $x>0$. Therefore,

$$\begin{aligned}&\mathbb {P}(S_t>x) \le \mathbb {P}(\tau _x\le t) = \mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x}) \\&\qquad + \mathbb {P}(\tau _x\le t, Y_t < Y_{\tau _x})\\&\quad = \left( 1+ \frac{\mathbb {P}\left( Y_1<0\right) }{\mathbb {P}\left( Y_1\ge 0\right) }\right) \mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x})\\&\quad = \frac{1}{\mathbb {P}\left( Y_1\ge 0\right) }\,\mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x})\\&\quad = \frac{1}{\mathbb {P}\left( Y_t\ge 0\right) }\,\mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x},Y_t\ge 0)\\&\quad = \mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x}\,|\,Y_t\ge 0). \end{aligned}$$

On the other hand, and with similar arguments,

$$\begin{aligned}&\mathbb {P}(S_t>x) \ge \mathbb {P}(\tau _x< t) = \mathbb {P}(\tau _x < t, Y_t\ge Y_{\tau _x}) \\&\qquad + \mathbb {P}(\tau _x< t, Y_t < Y_{\tau _x})\\&\quad = \mathbb {P}(\tau _x< t, Y_t\ge Y_{\tau _x}\,|\,Y_t\ge 0)\\&\quad \ge \mathbb {P}(\tau _x< t, Y_t > Y_{\tau _x}\,|\,Y_t\ge 0). \end{aligned}$$

Therefore,

$$\begin{aligned}&\mathbb {P}(\tau _x< t, Y_t > Y_{\tau _x}\,|\,Y_t\ge 0) \le \mathbb {P}(S_t>x) \\&\quad \le \mathbb {P}(\tau _x\le t, Y_t\ge Y_{\tau _x}\,|\,Y_t\ge 0). \end{aligned}$$

Since $Y_t$ has no upward jumps, $Y_{\tau _x} = x$ and $\{\tau _x < t\}\cap \{Y_t > x\} = \{Y_t > x\}$. Therefore,

$$\begin{aligned}&\mathbb {P}(Y_t > x\,|\,Y_t\ge 0) \le \mathbb {P}(S_t>x) \\&\quad \le \mathbb {P}(Y_t\ge x\,|\,Y_t\ge 0). \end{aligned}$$

We can now use standard approximation techniques:

$$\begin{aligned} \{X \ge x\}&= \bigcap _n\big \{X> x-1/n\big \} \\&= \bigcap _n\big \{X\ge x-1/n\big \} \end{aligned}$$

and

$$\begin{aligned} \{X > x\} = \bigcup _n\{X > x+1/n\} \end{aligned}$$

to get

$$\begin{aligned} \mathbb {P}(Y_t\ge x \,|\,Y_t\ge 0)&= \lim _{n\rightarrow \infty } \mathbb {P}(Y_t > x - 1/n \,|\,Y_t\ge 0)\\&\le \lim _{n\rightarrow \infty } \mathbb {P}(S_t > x - 1/n)\\&= \mathbb {P}(S_t\ge x) \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}(S_t\ge x)&= \lim _{n\rightarrow \infty }\mathbb {P}(S_t > x-1/n)\\&\le \lim _{n\rightarrow \infty }\mathbb {P}(Y_t \ge x-1/n\,|\, Y_t\ge 0)\\&= \mathbb {P}(Y_t \ge x \,|\, Y_t\ge 0) \end{aligned}$$

which proves $\mathbb {P}(S_t\ge x) = \mathbb {P}(Y_t\ge x\,|\, Y_t\ge 0)$.$\square$

Remark 6.2

If $(Y_t)_{t\ge 0}$ is a Brownian motion, then (6.1) becomes the classical reflection principle: $\mathbb {P}(Y_t\ge 0)=1/2$, so that (6.1) is equivalent to $\mathbb {P}(S_t\ge x) = 2\mathbb {P}(Y_t\ge ~x)$.

The proof of Theorem 6.1 relies essentially on local symmetry and the strong Markov property. Let $Y_t$ be a strong Markov process with càdlàg paths and transition function $p_t(z,\mathrm{d}y)=\mathbb {P}^z(Y_t\in \mathrm{d}y)$. Write $\tau _x^z = \inf \{u>0\,:\, Y_u-z > x\}$ for the first passage time above the level $x+z$ for the process $Y_t$ started at $z$; observe that, in general, $Y_{\tau _x^z} \ge x+z$. We can use the strong Markov property in (6.2) to get for any starting point $z$

$$\begin{aligned}&\mathbb {P}^z\left( \tau _x^z\le t, Y_t < Y_{\tau _x^z} \right) =\mathbb {P}^z\left( \tau _x^z\le t, Y_t - Y_{\tau _x^z} < 0\right) \\&\quad = \int \limits _{\{ \tau _x^z\le t\}} \mathbb {P}^{Y_{\tau _x^z}(\omega )}\big (Y_{t- \tau _x^z(\omega )} - Y_0 < 0\big )\, \mathbb {P}^z(\mathrm{d}\omega ). \end{aligned}$$

If we assume, in addition, some local “symmetry,” i.e., that for some constant $c\in (0,\infty )$ we have

$$\begin{aligned} \frac{\mathbb {P}^z(Y_t - z < 0)}{\mathbb {P}^z(Y_t - z \ge 0)} = c \quad \text {for all } t>0,\;z\in \mathbb {R}, \end{aligned}$$

(6.3)

then we get $\mathbb {P}^{Y_{\tau _x^z}(\omega )}\big (Y_{t-\tau _x^z(\omega )} - Y_0 < 0\big ) = c \,\mathbb {P}^{Y_{\tau _x^z}(\omega )}\big (Y_{t-\tau _x^z(\omega )} - Y_0 \ge 0\big )$ and, with a similar argument,

$$\begin{aligned} \mathbb {P}^z\left( \tau _x^z\le t, Y_t < Y_{\tau _x^z} \right) = c\,\mathbb {P}^z\left( \tau _x^z\le t, Y_t\ge Y_{\tau _x^z}\right) . \end{aligned}$$

This means that we can follow the lines of the proof of Theorem 6.1 to derive the following general result.

Theorem 6.3

(Markov reflection principle) Suppose $(Y_t,\mathbb {P}^z)$ is a strong Markov process satisfying the local symmetry condition (6.3). Set $S_t = \sup \{Y_u-Y_0 \,:\, 0\le u\le t\}$. Then, we have for all $t,x>0$ and $z\in \mathbb {R}$

$$\begin{aligned} \mathbb {P}^z(S_t>x)&\le \mathbb {P}^z (S_t\ge x, Y_t\ge Y_{ \tau _x^z}\,|\,Y_t\ge z)\\&\le \mathbb {P}^z (Y_t-z\ge x\,|\,Y_t\ge z). \end{aligned}$$

(6.4)

If $Y_t$ has only non-positive jumps, then $Y_{ \tau _x^z}=x+z$ a.s., and we get for all $t,x>0$ and $z\in \mathbb {R}$

$$\begin{aligned} \mathbb {P}^z(S_t\ge x) = \mathbb {P}^z(Y_t-z\ge x\,|\,Y_t\ge z). \end{aligned}$$

(6.5)

Remark 6.4

It is also possible to prove (6.1) using relation (3.6) in Alili and Chaumont [1].