Space-time fractional equations
European Congress of Mathematics, Sevilla
Co-authors
Juan Luis Vázquez
U. Autónoma de Madrid, Real Academia de Ciencias
Nicola Abatangelo
U. Bologna
Hardy Chan
U. Basel
\[ \newcommand{\dRL}{{\color{blue} \prescript{RL}{}\partial_\alpha}} \newcommand{\dC}{{\color{blue} \prescript{C}{}\partial_\alpha}} \newcommand{\iRL}{{\color{blue} I_\alpha}} \newcommand{\Rd}{{\mathbb R^d}} \]
Aim
\[ \text{``}{\color{blue}\partial_t^\alpha}\text{''} u + \text{``}{\color{red} (-\Delta)^s} \text{''} u = f \]
Fractional in time: “\({\color{blue} \partial_t}\)”
Fractional in time
Let \({\color{blue} I}: C([0,T]) \to C([0,T])\) the Riemann integral \[ {\color{blue} I} f(t) = \int_0^t f(s) ds. \] Then \[\begin{align*} {\color{blue} I}^2 f (t) &= \int_0^t I_1f (s) ds = \int_0^t \int_0^s f(\sigma) d\sigma ds = \int_0^t f(\sigma) \int_s^t ds d \sigma \\ &= \int_0^t f(\sigma) (t-s) d \sigma, \\ {\color{blue} I}^n f (t) &= \int_0^t f(\sigma) K_n(t-s) d \sigma, \qquad K_n(t) = \frac{t^{n-1}}{(n-1)!}. \end{align*}\]
We define the Riemann-Liouville integral for \(\alpha > 0\) \[ \iRL f (t) = \int_0^t f(\sigma) K_\alpha(t-s) d \sigma, \qquad {\color{blue} K_\alpha}(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}. \]
Key properties
\[ \iRL f (t) = \int_0^t f(\sigma) K_\alpha(t-s) d \sigma, \qquad K_\alpha(t) = \frac{t^{\alpha-1}}{\Gamma(\alpha)}. \]
Semigroup: \(\iRL {\color{blue} I_\beta} = {\color{blue} I_{\alpha + \beta}}\)
Monomial, for \(\beta > -1\) \[ \iRL t^\beta =\frac{\Gamma(\beta+1)}{\Gamma(\beta+1+\alpha)} t^{\beta+\alpha}. \]
Laplace transform symbol \[ {\mathfrak L}[\iRL f](s) = s^{-\alpha} {\mathfrak L}[f](s). \]
Fractional derivatives
A fractional derivative should have symbol \(s^\alpha\). Thus we get two choices:
Riemann-Liouville derivative \[ \dRL u = \partial_t {\color{blue} I_{1-\alpha}} u \]
Caputo derivative \[ \dC u = {\color{blue} I_{1-\alpha}} \partial_t u \]
Using the properties of \(I_\alpha\) and \(\partial_t\) it follows directly that \[\begin{align*} \dC t^\beta &= \dRL t^\beta = C(\beta, \alpha) t^{\beta - \alpha} \qquad \text{if } \beta > 0. \\ \dC 1 &= 0 \\ \dRL 1 &= \frac{t^{\alpha - 1}}{\Gamma(1-\alpha)} \ne 0 \\ \dRL t^{\alpha - 1} &= 0. \end{align*}\]
Integral formulation
\[\begin{align*} \dC u = f &\iff u = u(0) + \iRL f \\ \dRL v = f &\iff v = C_\alpha {\color{blue} I_{1-\alpha}} v(0) t^{\alpha-1} + \iRL f \end{align*}\]
Who cares? Stochastic volatility model for asset pricing.
Let \(S_t\) be the discounted spot price of an asset. We model \[ dS_t = S_t \sqrt{V_t} dW_t, % \qquad \text{i.e. } S_t = S_0 + \int_0^t S_s \sqrt{V_s} dW_s, \] where \(W_t\) be a Brownian motion in risk-neutral probability measure.
Consider \(B_t\) a second Brownian motion with \(\langle dW_t, dB_t \rangle = \rho\).
- Black-Scholes (1973): \(V_t = \sigma^2\) constant.
Heston (1993): \[ V_t = V_0 + \int_0^t \kappa( \theta - V_s ) ds + \int_0^t \sigma \sqrt{V_t} d B_s, \]
Rough Heston (2014) model: \[ V_t = V_0 + \int_0^t \kappa( \theta - V_s ) {\color{blue} K_\alpha (t-s)} ds + \int_0^t \sigma \sqrt{V_t}{\color{blue} K_\alpha (t-s)} d B_s. \] Several problem related to RH involve fractional ODEs.
Does \(\alpha\) matter?
Arbitrage-free price of European call option: \(e^{-rT} \mathbb E[(S_T - K)_+]\)
Fractional in space: “\({\color{red} (-\Delta)^s}\)”
Fractional Laplacian
\[ {\color{red} (-\Delta)^s} u (x) = {C(d,s)} \,\, \underbrace{\lim_{\varepsilon \to 0} \int_{\mathbb R^d \setminus B_\varepsilon (x) }}_{\text{ P.V. }\int _{\mathbb R^d}} \frac{u(x) - u(y)}{|x-y|^{d+2s}} d y. \]
Equivalent definitions (up-to ten [Kwasnicki, 2017])
The unique operator such that
\[ \mathfrak F[ {\color{red} (-\Delta)^s} u ] = |\xi|^{2s} \mathfrak F[u] \]
i.e. the spectral fractional power of \(-\Delta\).
The infinitesimal generator of a Lévy process \(X_h\), characterised by “long jumps”
\[ {\color{red} (-\Delta)^s} u (x) = \lim_{h \to 0} \frac{ \mathbb E[ f(x) - f(x - X_h) ] }{ h } \]
Fractional Elliptic/Heat/Porous Medium in \(\mathbb R^d\)
Analysis of the operator: [Caffarelli & Silvestre, 2007] extension
Obstacle: [Silvestre, 2007], [Barrios, Figalli & Ros-Oton, 2018]
Heat: We can compute estimates for the heat kernel \(K(t,\cdot) = \mathfrak F^{-1} [e^{-t|\xi|^{2s}}]\).
Porous Medium: [De Pablo, Quirós, Rodríguez & Vázquez, 2011], [Vázquez & Volzone, 2014]
Connection to geometry: [Chang & González, 2011]
Semilinear elliptic: [Nguyen & Véron, 2018],
[Chan, González, Huang, Mainini & Volzone, 2020]
There is a different Porous-Medium-type fractional equation \[ \partial_t u = \mathrm{div} (u^{m-1} \nabla (-\Delta)^{-s} u) \]
We will not discuss it:
\(m = 2\) [Caffarelli & Vazquez, 2011], \(m \ne 2\) [Stan, Del Teso & Vázquez, 2016]
Fractional LaplacianS
in bounded domains
Restricted fractional Laplacian (RFL). Singular integral operator: \[ {\color{red} (-\Delta)^s_{\mathrm {RFL}}} u (x)= C(d,s) \, \mathrm{P.V.}\int_{ \mathbb R^d } \frac{u(x) - u(y)}{|x-y|^{d+2s}} \; dy. \] If we work only in \(\Omega\), we must prescribe \(u\) in \(\Omega^c = \mathbb R^d \setminus \Omega\).
Censored fractional Laplacian (CFL). For \(s > \frac 1 2\) \[\begin{equation} {\color{red} (-\Delta)^s_{\mathrm{CFL}}} u (x) = C(d,s) \, \mathrm{P.V.} \int_{ \Omega } \frac{u(x) - u(y)}{|x-y|^{n+2s}} \; dy, \end{equation}\]
Spectral fractional Laplacian (SFL). Operational power. \(-\Delta \varphi_m = \lambda_m \varphi_m \textrm{ in } \Omega\), \(\varphi_m = 0 \textrm{ on } \partial \Omega.\) one defines \[ u(x) = \sum_{m=1}^{+\infty} {u_m} \varphi_m(x) \qquad \longmapsto \qquad {\color{red} (-\Delta)_{\mathrm{SFL}}^s} u (x) = \sum_{m=1}^{+\infty} \lambda_m^s u_m \varphi_m (x). \] The “boundary condition” is \(u = 0\) on \(\partial \Omega\).
Green kernels
The solution of the Laplace equation \[ \begin{dcases} {\color{red} \mathcal L} u = f & \Omega \\ u = 0 & \partial \Omega \text{ or } \mathbb R^d \setminus \overline \Omega \\ \end{dcases} \]
can be written via the Green function \[ u(x) = {\color{red} \mathrm G}[f] (x) = \int_\Omega {\color{red} \mathbb G}(x,y) f(y) dy. \]
The probabilistic approach provides in each of our cases a similar shape \[ {\color{red} \mathbb G}(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma \]
Since we deal only with self-adjoint \({\color{red} \mathcal L}\), then \({\color{red} \mathbb G}(x,y) = {\color{red} \mathbb G}(y,x)\).
The probabilistic approach provides in each of our cases a similar shape \[ {\color{red} \mathbb G}(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma \]
And \(\gamma \in (0,1]\) depends on the setting
RFL: \(\gamma = s\).
SFL: \(\gamma = 1\)
CFL: \(\gamma = 2s-1\), only \(s \in (\frac 1 2, 1)\).
Parabolic problems with homogeneous Dirichlet data
Linear fractional ODEs
Let \(E_{\alpha, \beta}\) be the Mittag-Leffler function
We define
\(E_\alpha(z) = E_{\alpha,\alpha}(z)\) and
\(P_{\alpha}(t;\lambda) = t^{\alpha-1} E_{\alpha,\alpha}(-\lambda t^\alpha)\).
Using Laplace transform \[\begin{align*} &\dC u = -\lambda u + f \\ &\qquad \implies u (t) = E_\alpha (-\lambda t^\alpha) u(0) + \int_0^t P_{\alpha}(t-\tau;\lambda) f(\tau) d \tau.\\ &\dRL v = -\lambda v + g \\ &\qquad \implies v(t) = P_{\alpha}(t;\lambda) I_{1-\alpha}v (0) + \int_0^t P_{\alpha} (t-\tau;\lambda) g(\tau) d \tau \end{align*}\]
Duhamel’s formulas for space-time problems
(homogeneous Dirichlet conditions)
Using the spectral decomposition, we write \[\begin{align*} &\dC u = -{\color{red} \mathcal L} u + f \\ &\qquad \implies u (t) = E_\alpha (-t^\alpha {\color{red} \mathcal L}) u(0) + \int_0^t P_{\alpha}(t-\tau;{\color{red} \mathcal L}) f(\tau) d \tau.\\ &\dRL v = -{\color{red} \mathcal L} v + g \\ &\qquad \implies v(t) = P_{\alpha}(t;{\color{red} \mathcal L}) I_{1-\alpha}v (0) + \int_0^t P_{\alpha} (t-\tau;{\color{red} \mathcal L}) g(\tau) d \tau \end{align*}\]
Here we are using the notation for \(L\varphi_i = \lambda_i \varphi_i\) and \(F:\mathbb R_+\to \mathbb R\) \[ F(L) \sum_i a_i \varphi_i = \sum_i a_i F(\lambda_i) \varphi_i. \]
Singular boundary data for
Laplace equation
Estimates for interior data for \((-\Delta)^s u = f\)
Given that \({\color{red} \mathbb G}(x,y) \asymp \frac{1}{|x-y|^{d-2s}} \left(1 \wedge \frac{\delta(x)}{|x-y|} \right)^\gamma\left(1 \wedge \frac{\delta(y)}{|x-y|} \right)^\gamma\)
For \(K \Subset \Omega\) \[ \int_K |{\color{red} \mathrm G}(f)| \le C_K \int_\Omega |f| \delta^\gamma. \]
So \({\color{red} \mathrm G}: L^1 (\Omega, \delta^\gamma) \to L^1_{loc} (\Omega)\)
If \(f \ge 0\) then, using the kernel estimates \[ \tag{Hopf} {\color{red} \mathrm G}[f] (x) \ge c \delta^\gamma(x) \int_\Omega f(y) \delta(y)^\gamma dy. \]
If \(0\le f \notin L^1 (\Omega, \delta^\gamma)\), then \({\color{red} \mathrm G}[f] = +\infty\).
Large solutions with boundary blow-up for \({\color{red} \mathcal L}\)
[Abatangelo, GC & Vázquez, 2022] denoting \(\delta(x) = \mathrm{dist}(x, \partial \Omega)\)
We can compute \({\color{red} \mathrm G}[\delta^\beta] \asymp \delta^\alpha\) with
\[ {\color{red} \mathrm G}[\delta^\beta] \asymp \begin{dcases} \delta^\gamma & \beta +2s > \gamma \\ \delta^{\gamma} \log|\delta| & \beta +2s = \gamma \\ \delta^{\beta + 2s} & \beta +2s < \gamma \\ &\text{ and } \beta > -1-\gamma \\ \infty & \beta \le -1-\gamma \end{dcases} \]
\(\infty \not \equiv {\color{red} \mathrm G}[\delta^\beta] \notin L^\infty \iff -1-\gamma < \beta < -2s\).
For \(-\Delta\) we have \(\gamma = 1 = s\). Either \({\color{red} \mathrm G}[\delta^\beta] \in L^\infty\) or \({\color{red} \mathrm G}[\delta^\beta] \equiv + \infty\)
Singular solutions of the elliptic “Dirichlet” problem
The harmonic singular solution
Take \(s \in (0,1)\)
\[ u(x) = \begin{cases} (1-|x|^2)^{s-1} & \text{if } |x|<1 \\ 0 & \text{if } |x| \ge 1 \\ \end{cases} \]
satisfies
\[ (-\Delta)_{\mathrm{RFL}}^s u(x) = 0 \qquad \text{if } |x| < 1. \]
No right-hand side? There is an extra “boundary” condition
Martin problem
Because of the scaling we consider \[ f_m (x) = \frac{|\partial \Omega|}{ |A_m| } \frac{\chi_{A_m} (x)}{\delta (x)^\gamma} \qquad \text{ where } A_m = \left\{ x: \frac 1 m \delta(x) < \frac 2 m \right\} \]
[Abatangelo, GC & Vázquez, 2022] We show that \(\mathrm{supp}(f_m) \to \partial \Omega\) and \({\color{red} \mathrm G}[f_m] \to u^\star \ne 0\).
In fact, for any \(h \in L^1 (\partial \Omega)\) the problem
\[ \begin{dcases} {\color{red} \mathcal L} u = 0 & \Omega \\ u = 0 & \mathbb R^d \setminus \overline \Omega \\ \lim_{x \to z} \frac{u(x)}{u^\star (x)} = h(z) & \text{for all } z \in \partial \Omega \end{dcases} \]
admits a solution
(constructed in similar fashion).
We show that \(u^\star \asymp \delta^{2s-\gamma -1 }.\)
RFL: \(s-1\)
SFL: \(2(s-1)\)
CFL: \(0\), i.e. bounded.