Exact and limit results for the CTRW in presence of drift and position dependent noise intensity

Abstract

Continuous-time random walks (CTRWs) with drift and position-dependent jumps provide a highly general framework for describing a wide range of natural and engineered systems. We analyze the stochastic differential equation (SDE) associated with this class of models, in which the driving noise $ξ(t)$ consists of spike (shot) events, and we derive two exact analytical results. First, we obtain a closed-form expression for the $n$-time correlation functions of $ξ(t)$, expressed as a sum over all $2^{\,n-1}$ ordered partitions of the observation times (Proposition~2). Second, using the $G$-cumulant formalism, we derive an \emph{exact} non-local master equation (ME) for the probability density function of the CTRW variable $x(t)$, valid without invoking diffusive limits, fractional scaling assumptions, or closure hypotheses (Proposition~3). In interaction representation, this ME retains the same structural form as that of the standard CTRW without drift or position-dependent jumps. Our main result is the emergence of a \textbf{universal local master equation}: at long times, the exact non-local ME is universally and accurately approximated by a time-local ME whose only coefficient is the instantaneous renewal rate $R(t)$. This approximation reproduces the exact Poissonian ME when $R$ is constant, and numerical experiments confirm its remarkable accuracy even far beyond regimes where a naive time-scale separation would justify it.

Figures (12)

• Figure 1: A trajectory realization $\xi(t)$ for the case of Lévy flight-CTRW, with $t_0=0$. We have $\xi( t)=\sum_{q=0}^{\infty} \xi_q \; \delta\left(t-\sum_{k=0}^q \theta_k\right)$, (see text for details).
• Figure 2: $P(x\hbox{;}\, t)$ at times $t = 0.5$ for the SDE \ref{['SDE']} in the multiplicative case, i.e., with $C(x) =\gamma\,x$ and $I(x)=1+\beta \,x$, where $\gamma=1.0$ and $\beta =0.5$. The jump PDF is both dichotomous (black color) and Gaussian (green color), the WT is $\psi(t)=(\mu -1){T}^{-1}\left(1+t/T\right)^{-\mu }$ with $T=1$ and $\mu=1.5$. The figure shows the results of numerical simulations of the SDE together with the solution of the exact theoretical PDE in Eq. \ref{['ME_fin']} (dashed lines, barely visible). The insert shows the same plot but in log scale. As expected, the agreement between theory and simulation is perfect.
• Figure 3: The same as fig. \ref{['fig:def_comp_t0.2_pde_dico-gauss_mu1.50_tz1.00']}, but for $\mu=3.5$ (from which $\tau=T/(\mu-2)=2/3$) and for the sole dichotomous jump PDF case. Because $\mu>2$ from the result 3, point \ref{['en:1']}, we can exploit the analytical findings of \ref{['app:ME_resuls']}. For example, the tail of the PDF goes as $x^{-(\alpha+1)}$, where $\alpha\approx3.9$ is obtained from the transcendental implicit Eq. \ref{['power2']} (dashed brown line).
• Figure 4: Log-plot (left) and linear plot (right) of $P(x,t)$ at $t=5$ (top), $t=30$ (center), and $t=100$ (bottom) for the SDE \ref{['SDE']} in the additive case (i.e., with $I(x)=1$) with strongly nonlinear drift, $C(x) = -x(1.2 - x^2)$. The system is driven by renewal noise with dichotomous jumps PDF and WT PDF $\psi(t)=(\mu -1)T^{-1}(1+t/T)^{-\mu}$, with $T=2$ and $\mu=1.5$ (infinite aging). Circles correspond to direct numerical simulations of the SDE \ref{['SDE']}, while the dotted curves show the solution of the approximate local-in-time ME given in Eq. \ref{['ME_universal_intro']}. The vertical scale on the right hand plates decreases as time increases, to show the vanishing of the PDF as predicted by Eq. \ref{['ME_universal_intro']}, and it is clipped: at the two equilibrium points the PDF reaches a stationary value outside the range shown (see the left hand plates) The agreement between theory and simulations is very good.
• Figure 5: Same as Fig. \ref{['fig:Def_pdedico_t5-30-100_mu1.5_tz2.00_ga1.2_Cubic_appr']}, but with $\mu=2.5$ and shown only for $t=30$ and $t=100$. The two curves are almost perfectly superimposed, indicating that the Poissonian equilibrium has effectively been reached.

Theorems & Definitions (11)

• Proposition 1: Universal Local ME — informal statement
• Proposition 2
• Proposition 3
• Theorem 1: Universal Local ME
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 1
• Proposition 8