On Diffusion Processes, Lévy Noises, and Confusion C é cile Penland NOAA/Earth Systems Research Laboratory

How do we model unresolved variability? Diffusion approximation? Lévy-driven processes? Better representation of tails? Does our experience with diffusion processes transfer to (white) Lévy-driven processes? Diffusion approximation? Lévy-driven processes? Better representation of tails? Does our experience with diffusion processes transfer to (white) Lévy-driven processes?

The diffusion approximation: dx/dt =    G(x,t) +  F(x,t)    G(x,t) is slow  F(x,t) is fast The diffusion approximation: dx/dt =    G(x,t) +  F(x,t)    G(x,t) is slow  F(x,t) is fast

Choose a scaling s =   t:. (*) For simplicity, say and Lim (*) -> t->∞  ->0 (W is a Brownian motion; dW  N (  dt 

The Good: Have a systematic way of handling a lot of multiscale processes. Rigorous connection between dynamics (PDEs) and probabilistic description. implies a Stratonovich Fokker-Planck eqn. The Good: Have a systematic way of handling a lot of multiscale processes. Rigorous connection between dynamics (PDEs) and probabilistic description. implies a Stratonovich Fokker-Planck eqn.

The Bad: Existence of multiple calculi can make numerical generation difficult. The difference between Ito and Stratonovich integrals is physically-based; a thermometer will not perform the Ito correction for you. Mathematically, it’s an issue of who speaks first: the continuum limit or the white-noise limit. The Bad: Existence of multiple calculi can make numerical generation difficult. The difference between Ito and Stratonovich integrals is physically-based; a thermometer will not perform the Ito correction for you. Mathematically, it’s an issue of who speaks first: the continuum limit or the white-noise limit.

NWS operational GCM (1993 version)

The Bad: Existence of multiple calculi can make numerical generation difficult. The difference between Ito and Stratonovich integrals is physically-based; a thermometer will not perform the Ito correction for you. Mathematically, it’s an issue of who speaks first: the continuum limit or the white-noise limit. The Bad: Existence of multiple calculi can make numerical generation difficult. The difference between Ito and Stratonovich integrals is physically-based; a thermometer will not perform the Ito correction for you. Mathematically, it’s an issue of who speaks first: the continuum limit or the white-noise limit.

The Ugly: Sometimes, the approximation just doesn’t hold. The white-noise limit and Gaussian limit in the CLT are more or less taken simultaneously when that doesn’t always happen in nature. The Ugly: Sometimes, the approximation just doesn’t hold. The white-noise limit and Gaussian limit in the CLT are more or less taken simultaneously when that doesn’t always happen in nature.

The new fashion: Lévy noises P(X>x) ~ x -    x s  for s>  Also get Langevin equations: dX = f(X)dt+  (X)  dL 

Why do we care?  Anomalous diffusion in hydrology (Hurst 1951)  Paleoclimate models, particularly as concerns intermittency in the ice core record (e.g., Ditlevsen 1999)  Atmospheric turbulence (Viecelli 1998)  Anomalous diffusion in hydrology (Hurst 1951)  Paleoclimate models, particularly as concerns intermittency in the ice core record (e.g., Ditlevsen 1999)  Atmospheric turbulence (Viecelli 1998) Even if we object to some of these models, we still have to know how to analyze them.

 dX = f(X)dt+  (X)  dL  Fractional Fokker-Planck equation is derived in spectral form as the continuum limit of a finite jump process.

Questions for the Mathematicians  Theoretical justification  Implementation  Theoretical justification  Implementation

 When can we get away with treating a Lévy process as a system driven by multiplicative Brownian noise?  Are there limit theorems for continuous systems which converge to continuous Lévy- driven processes?  If there are limit theorems converging to Lévy- driven processes, what are the requirements for convergence?  How forgiving are these limit theorems?  When can we get away with treating a Lévy process as a system driven by multiplicative Brownian noise?  Are there limit theorems for continuous systems which converge to continuous Lévy- driven processes?  If there are limit theorems converging to Lévy- driven processes, what are the requirements for convergence?  How forgiving are these limit theorems?

 Do Lévy processes ever approximate the physical system when the white-noise limit is pretty good but the Gaussian limit is not? Under what circumstances?  Are there classes of Lévy noises for which the stochastic integral is not unique?  If so, what does the Lévy equivalent of a noise- induced drift look like? Does it involve a fractional derivative? Are there transformations between calculi?  Do Lévy processes ever approximate the physical system when the white-noise limit is pretty good but the Gaussian limit is not? Under what circumstances?  Are there classes of Lévy noises for which the stochastic integral is not unique?  If so, what does the Lévy equivalent of a noise- induced drift look like? Does it involve a fractional derivative? Are there transformations between calculi?

 Are there really limit theorems that would give combinations of Lévy and Brownian noises in the same dynamical equation?  Is there a recipe like the CLT that scientists can use to get an approximate stochastic equation?  Is there such a thing as a Lévy-Taylor expansion?  Are there really limit theorems that would give combinations of Lévy and Brownian noises in the same dynamical equation?  Is there a recipe like the CLT that scientists can use to get an approximate stochastic equation?  Is there such a thing as a Lévy-Taylor expansion?

Choose a scaling s =   t:. (*) For simplicity, say and Lim (*) -> t->∞  ->0 (W is a Brownian motion; dW  N (  dt 

 Are there limit theorems that would give combinations of Lévy and Brownian noises in the same dynamical equation?  Is there a recipe like the CLT that scientists can use to get an approximate stochastic equation?  Is there such a thing as a Lévy-Taylor expansion?  Are there limit theorems that would give combinations of Lévy and Brownian noises in the same dynamical equation?  Is there a recipe like the CLT that scientists can use to get an approximate stochastic equation?  Is there such a thing as a Lévy-Taylor expansion?

 Is there such a thing as an Ito-Lévy-Taylor expansion?  What about the numerical generation of Lévy noises? Do we use (  ) 1/  R to update the noisy increment?  Since some models use combinations of Wiener and Lévy noises, can we use the same numerical scheme to update both terms?  Is there such a thing as an Ito-Lévy-Taylor expansion?  What about the numerical generation of Lévy noises? Do we use (  ) 1/  R to update the noisy increment?  Since some models use combinations of Wiener and Lévy noises, can we use the same numerical scheme to update both terms?

 How do we handle multiple stochastic integrals when some of the noises are Brownian and others are Lévy?  If we only care about the distribution of the solution, are there weak and strong numerical schemes for Lévy-driven processes like there are for Wiener-driven processes?  Does the accuracy of these schemes depend on any particular calculus like it does for Wiener- driven schemes?  How do we handle multiple stochastic integrals when some of the noises are Brownian and others are Lévy?  If we only care about the distribution of the solution, are there weak and strong numerical schemes for Lévy-driven processes like there are for Wiener-driven processes?  Does the accuracy of these schemes depend on any particular calculus like it does for Wiener- driven schemes?

 What else is out there?