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International Journal for Uncertainty Quantification
DOI: 10.1615/Int.J.UncertaintyQuantification.2011003485
pages 195213 DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATIONAbstract By using functional integral methods we determine new types of differential constraints satisfied by the joint probability density function of stochastic solutions to the wave equation subject to uncertain boundary and initial conditions. These differential constraints involve unusual limit partial differential operators and, in general, they can be grouped into two main classes: the first one depends on the specific field equation under consideration (i.e., on the stochastic wave equation), the second class includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. Preliminary results we have obtained for stochastic dynamical systems and firstorder nonlinear stochastic particle differential equations (PDEs) suggest that the set of differential constraints is complete and, therefore, it allows determining uniquely the probability density function of the solution to the stochastic problem. The proposed new approach can be extended to arbitrary nonlinear stochastic PDEs and it could be an effective way to overcome the curse of dimensionality for random boundary and initial conditions. An application of the theory developed is presented and discussed for a simple random wave in one spatial dimension. KEYWORDS: stochastic partial differential equations, highdimensional methods, random fields 1. INTRODUCTIONMany physical phenomena such as sound propagation, electromagnetic scattering at random surfaces, and random vibrations in solid mechanics can be described in terms of random waves satisfying the standard wave equation
for random boundary conditions, random initial conditions, or random group velocity. The purpose of this paper is to introduce a new method to compute the statistical properties of the stochastic solutions to Eq. (1). This method is based on a set of differential constraints for the joint probability density function of the wave A and its derivatives and it may present an advantage with respect to more classical stochastic approaches such as polynomial chaos [1– 4], probabilistic collocation [5, 6], and generalized spectral decompositions [7–11]. In fact, it seems that it does not suffer from the curse of dimensionality problem, at least when randomness comes from boundary or initial conditions. Indeed, since we are solving for probability density function of the system, we can actually prescribe these conditions in terms of the probability distributions and this is obviously not dependent on the number of random variables characterizing the underlying probability space. This observation immediately leads us to the question: Is it possible to determine a closed evolution equation for the probability density function of the solution to Eq. (1) at a specific spacetime location? Unfortunately, the answer is negative. In fact, the selfinteracting nature of the wave equation is associated with nonlocal solutions in space and time that, in turn, yield the impossibility of determining a pointwise equation for the probability density function. However, an equation for the probability density functional of the wave always can be obtained. This very general approach relies on the use of functional integral techniques [12–16]; in particular, those involving the Hopf characteristic functional (see Appendix B). These functional methods aim to cope with the global probabilistic structure of the solution and they have been extensively studied in the past as a possible tool to tackle many fundamental problems in physics such as turbulence [17, 18]. Their usage grew very rapidly in the 1970s, when it became clear that diagrammatic functional techniques [12] could be applied, at least formally, to many different problems in classical statistical physics. However, functional differential equations involving the Hopf characteristic functional unfortunately are not amenable to numerical simulation. In addition, the amount of statistical information carried on by the Hopf characteristic functional is often far beyond the needs of practical uncertainty quantification, which usually reduces only to the computation of a few statistical moments of the solution at specific spacetime locations. Thus, we are led to look for alternative ways to determine the evolution of the probability density function associated with the solution to Eq. (1). To this end, by using a functional integral technique we have introduced recently in the context of stochastic dynamical systems, in this paper we will show that it is possible to formulate a set of differential constraints that have to be satisfied locally by the probability density function of every random wave process governed by Eq. (1). These constraints involve, in general, unusual limit partial differential operators and they can be grouped into two main classes: the first one depends on the specific field equation under consideration; i.e., on the stochastic wave equation; the second class is represented by a set of intrinsic relations arising from the structure of the joint probability density function of the wave and its derivatives. A fundamental question we address in this paper is whether the set of these differential constraints allows us to determine uniquely the probability density function associated with the stochastic solution to the wave equation. This paper is organized as follows. In Section 2 we consider random waves in one spatial dimension and we show how to determine a closed and exact differential constraint that has to be satisfied by the probability density function of every random wave process for random boundary conditions and random initial conditions. All the calculations are based on a functional integral technique that is presented in detail in Appendix A. Section 3 deals with the formulation of additional differential constraints depending on the structure of the joint probability density function of the wave and its derivatives. The completeness of the set of differential constraints is discussed in Section 4 for a prototype problem involving a firstorder nonlinear stochastic particle differential equation (PDE). In Section 5 we generalize the theory developed for two and threedimensional random wave processes. An example of the application of the theory is presented in Section 6. Finally, the main findings and their implications are summarized in Section 7. We also include two additional appendices. The first one (i.e., Appendix B), deals with the application of the Hopf characteristic functional approach to onedimensional random waves. The second one (i.e., Appendix C), deals with the perturbation expansion of differential constraints in the neighborhood of a selected spacetime location. 2. ONEDIMENSIONAL RANDOM WAVESIn one spatial dimension Eq. (1) can be written as
We supplement this equation with appropriate random boundary conditions and with a random initial condition in a suitable spacetime domain. The solution to this boundary value problem is clearly a random wave, which we assume to admit a probability density function. At this point a fundamental question is whether we can actually determine an evolution equation for such a probability density based on Eq. (2). To this end, let us first look for a representation of the joint probability of the wave and its firstorder derivatives with respect to space and time calculated at different spacetime locations. The reason why we need such joint probability density will become clear in a while. For notational convenience, let us set
This allows us to write the joint probability density of ψ, ψ′_{t}, and ψ″_{x} as (see Appendix A for further details) Differential Constraints for the Probability Density Function
where δ denotes the Dirac delta function and the average operator ⟨⋅⟩ is defined as a functional integral with respect to the joint measure of the random initial condition and the random boundary conditions. The definition we give of “constructor field” relies on the fact that we will extract from that Dirac delta function, also called the “indicator function” by Klyatskin [19, p. 42]; all the derivatives we need to build up the wave equation. As we will see, this procedure also generates additional terms that can be treated in a limit process by employing the “absorbing fields”. By using the differentiation rules for the probability density function discussed in Appendix A we obtain
and
Similarly,
If we subtract Eq. (7) from Eq. (6) and we take Eq. (2) into account we obtain
The average appearing in Eq. (8) can be easily represented if we take the limit for (x′; x″) and (t′; t″) going to x and t, respectively. In fact, by using the absorbing fields appearing in Eq. (4) we see that
This yields the final result
Note that this identity involves unusual partial differential operators that we shall call limit partial derivatives. For subsequent mathematical developments it is convenient to reserve a special symbol for these operators; i.e., we shall define
This allows us to write Eq. (10) in a compact form as
Equation (12) is a differential constraint that must be satisfied by the probability density function of every onedimensional random wave governed by Eq. (2) for the random initial condition or random boundary conditions. These types of differential constraints made their first appearance in [20] in the context of stochastic dynamical systems. Note that Eq. (12) always involves only three variables (x, t, a) and two parameters (b and c) independently of the dimensionality of the random boundary conditions and the random initial condition. This very attractive feature immediately leads us to the question: Is it possible to solve a boundary value problem for Eq. (12) and, therefore, obtain the joint probability density function of the random wave and its derivatives at a specific spacetime location uniquely? Certainly, if we would have this possibility then we would be able to overcome the curse of dimensionality problem. However, as we will see in Section 6, there exist an infinite number of probability densities satisfying the differential constraint (12) for the same boundary and initial conditions. This suggests that a boundary value problem for Eq. (12) is, in general, ill posed. A possible way to get rid of the limit partial derivatives is to integrate Eq. (12) with respect to the variables b and c. This procedure yields the following equation for the response probability, i.e., for the probability density of the wave ψ:
where
denotes the marginal density of p^{(a,b,c)}_{ψ ψtψx}. The integrals appearing in Eq. (14) are formally written from ∞ to ∞ although the probability density function we integrate out may be compactly supported. Note that Eq. (13) is not closed. In fact, the last term on the righthand side can be written as
The last two averages of this equation involve the calculation of the correlations between two functionals of the random initial conditions and the random boundary conditions; e.g., δ(a − ψ) and ψ^{2}_{t}. Such a correlation structure can be disentangled, for example, by using a functional power series [21, 22] (see also [23], p. 311, and [19], p. 70). In other words, the correlations 〈δ(a − ψ)ψ^{2}_{t}〉 and 〈δ(a − ψ)ψ^{2}_{x}〉 can be represented in terms of cumulants of the random initial conditions and the random boundary conditions. Alternatively, other types of closures can be considered [24–27]. We conclude this section by emphasizing that Eq. (13) can be obtained more directly from the representation of the response probability of the wave; i.e.,
In fact, if we differentiate Eq. (16) twice with respect to t and x
and sum up the derivatives, we immediately obtain
This shows that the differential constraint (12) includes the evolution equation of the response probability of the wave as a subcase. 2.1 Some Remarks on Evolution Equations Involving Limit Partial Differential OperatorsEquation (12) involves unusual limit partial differential operators in space and time. These operators arise very often when the functional integral method or, equivalently, the Hopf characteristic functional approach (see Appendix B) are applied to a stochastic PDE in order to determine a pointwise equation for the probability density function of its solution. In this section we would like to provide some heuristic justification of why this happens. To this end, let us first rewrite the onedimensional wave Eq. (2) as a firstorder system
This formulation clearly shows that the temporal evolution of the wave ψ is driven by the random field ∂η/∂x while the temporal evolution of η is governed by ∂ψ/∂x. In other words, the stochastic dynamics of random waves is, in general, selfinteracting. A similar situation arises in the simpler context of timedependent stochastic dynamical systems of order greater than 1, where the dynamics of the stochastic solution is influenced by the dynamics of higherorder time derivatives. A selfinteracting system can yield to nonlocal solutions in space and time. Perhaps, the simplest example in this context is the diffusion equation. In this case, a pointwise equation for the probability density function cannot be derived in a closed form. However, there are a few exceptions where we can actually obtain a closed equation for the probability density function. Among them, we recall firstorder nonlinear and quasilinear stochastic PDEs. In these cases the limit partial derivatives do not appear and the evolution problem for the probability density function is well defined in a classical sense. A prime example is the inviscid Burgers equation
with random boundary or initial conditions. In this case it can be shown that the probability density function of the solution satisfies
3. INTRINSIC RELATIONS DEPENDING ON THE STRUCTURE OF THE JOINT PROBABILITY DENSITY FUNCTIONThe fields appearing in joint probability density (4) are related locally to each other and, therefore, we expect that there exists a certain number of relations between the limit partial derivatives of the joint probability density with respect to different arguments. These relations are independent of the particular stochastic field equation describing the physical phenomenon [e.g., Eq. (2)], but they are defined intrinsically by the structure of the joint probability density function. For instance, there is clearly a local deterministic relation between the random field A and its spatial derivative
This relation, and similar ones for other derivatives, can be translated into an intrinsic relation involving joint probability density function (4). To this end, we first observe that the limit derivatives of Eq. (4) with respect to different arguments (e.g., x, x′, and x″) are different. Indeed,
Similarly,
From Eqs. (25) and (29) it follows that
Next, let us consider the average of the secondorder spatial derivative of A. A comparison between Eq. (26) and Eq. (7) (in the limit for x′, x″ → x and t′, t″ → t) immediately gives
Similarly, a comparison between Eqs. (28) and (6) yields
Equations (24), (27), and (30)–(32) are simple examples of intrinsic relations involving the probability density function. These relations are in the form of differential constraints and they have to be satisfied independently of the particular stochastic problem under consideration; e.g., Eq. (2). In principle, if the wave field ψ is analytic, by repeatedly applying the arguments above we can construct an infinite set of differential constraints to be satisfied at every spacetime location by the joint probability density function of the wave and its derivatives. Note that from Eqs. (28), (26), and (2) we easily obtain that
which is another, more compact, way to write the differential constraint for the probability density function associated with wave Eq. (2). Indeed, a substitution of Eqs. (31) and (32) into Eq. (33) consistently gives Eq. (12). 4. ON THE COMPLETENESS OF THE SET OF DIFFERENTIAL CONSTRAINTSIn some cases, it can be proven that the set of differential constraints involving the firstorder limit partial derivatives is equivalent to a standard evolution equation for the joint probability density function of the system. This circumstance implies that the set of differential constraints is complete; i.e., that it allows for the computation of the joint density. This happens, for example, in the context of highorder stochastic dynamical systems subject to a random initial state. The set of differential constraints, in this case, is equivalent to the wellknown Liouville equation. In this section we would like to prove a similar result for firstorder nonlinear stochastic PDEs. To this end, let us consider the equation
subject to random boundary and initial conditions of arbitrary dimensionality. The solution to this problem can be represented in probability space by using the joint density
The full set of firstorder differential constraints associated with Eqs. (34) and (35) is easily obtained as
From Eqs. (37) and (39) it follows that
At this point, let us recall the following identities between standard partial derivatives and limit partial derivatives
Note that at the left hand side we first set x = x′ and t = t′ and then we differentiate while at the right hand side we first differentiate and then we set x = x′ and t = t′. By substituting Eqs. (36) and (40) into Eq. (41) and using Eqs. (42) and (38), we obtain
This is the correct evolution equation for the joint probability density function associated with the stochastic solution to Eq. (34). We can show this by considering Eq. (35), with both ψ and ψ_{x} set at the same spacetime location. A differentiation with respect to time yields, in this case,
where the last term can be written in terms of the probability density function by inverting the relation
i.e.,
A substitution of Eq. (46) into Eq. (44) gives exactly Eq. (43). Thus, we have shown that the set of firstorder differential constraints (36)–(39) is equivalent to a standard PDE for joint probability density function (35) on the hyperplane x′ = x, t′ = t. This implies that the set of differential constraints, in this case, is complete. However, it is still an open question if such a set is complete for more general stochastic problems; e.g., for the stochastic wave Eq. (2). The dynamics of the joint probability density function in these cases does not satisfy a standard PDE. This is due to the fact that such dynamics, in general, arises from the projection of a functional differential equation on the hyperplane x = x′ = x″ = …, t = t′ = t″ = …. In some sense, the set of differential constraints we have obtained represents the components of such a projection. 5. TWO AND THREEDIMENSIONAL RANDOM WAVESThe theoretical apparatus developed in the previous sections can be easily extended to two and threedimensional wave equations. To this end, let us first consider two spatial dimensions and the equation
The joint probability density function of the random wave and its derivatives has the functional integral representation
As before, we now differentiate the constructor field with respect to time and space variables and then we use the absorbing fields in a limit procedure in order to get rid of the additional terms arising from the differentiation process. We obtain,
A summation of Eqs. (49)(51) and a following limit procedure for (x′, x″, x′′′) and (t′, t″, t′′′) going to x and t, respectively, yields
that is the differential constraint for the joint probability density function of every twodimensional random wave satisfying wave Eq. (47) for random boundary or random initial conditions. Straightforward extensions of the arguments above yield the following differential constraint for the joint probability density function of threedimensional random waves:
Differential constraints (52) and (53) can be supplemented with additional intrinsic relations involving the joint probability density function. These relations can be constructed by following the same mathematical steps presented in Section 3. 6. AN EXAMPLELet us consider a random wave in one spatial dimension satisfying the following boundary value problem:
where ξ_{k}(ω) (k = 1, ..., 3) are random variables with known joint probability density functions and h_{k}(x) are prescribed deterministic functions on the real line. The analytical solution to problem (54) is the wellknown d′Alambert wave [28, p. 41]
For subsequent mathematical developments, it is convenient to define
This allows us to write solution (55) and its derivatives with respect to x and t in a compact form as
The joint probability density of ψ, ψ′_{t}, and ψ″_{x} (i.e., the joint probability of the wave and its firstorder derivatives at different spacetime locations) can be obtained by using the classical mapping approach. To this end, let us notice that Eqs. (59) define a threedimensional linear transformation from {ξ_{1},ξ_{2},ξ_{3}} to {ψ,ψ′_{x},ψ″_{t}}. In matrixvector form this transformation can be represented as
The determinant of the system′s matrix is
while the transpose of its algebraic complement has entries
A classical result of probability theory [29, p. 183] states that if p^{(α1,α2,α3)}_{ξ1ξ2ξ3} is the joint probability density of ξ_{1}, ξ_{2}, and ξ_{3}, then the joint probability of ψ, ψ′_{t}, and ψ″_{x} is given by
where
Note that every is also a function of all the spacetime variables because of the matrix components H_{ij}. Therefore, Eq. (63) is a rather complex function of nine variables. Under the assumption that ξ_{i}(ϖ) are mutually independent normal random variables, we can apply the convolution theorem and obtain the probability density function of the random wave ψ, as
where G_{1i} are given in Eq. (56). This probability density is shown in Fig. 1 at different time instants for U = 2 and an initial condition defined in terms of the functions FIG. 1: Probability density function p^{(a)}_{ψ(x,t)} of the random wave at different time instants.
According to Eq. (65) the mean wave is zero, while the variance σ(x, t)^{2} evolves in time as shown in Fig. 2. FIG. 2: Temporal evolution of the wave variance. 6.1 Analytical Verification of Some Differential ConstraintsIt is easy to prove that probability density function (63) verifies simple differential constraints (24) and (27); hereafter, rewritten for convenience as To this end, let us first notice that the limit partial derivatives of Jacobian determinant (61) with respect to x and t are identically zero; i.e., we have
In fact, by using definitions (57) and (58) we obtain
At this point, we can calculate the limit derivatives of probability density (63) with respect to x and t:
where we have used Eqs. (67). From Eqs. (64) and (67) and definitions (57) and (58) it follows that
Substituting these results back into Eqs. (70) and (71) we obtain
At the same time we notice that
and, therefore, differential constraints (24) and (27) are satisfied identically [simply compare Eqs. (74) and (75) with Eqs. (73)]. Similarly, it is possible to show analytically that higherorder constraints such as Eqs. (33), (12), or (31) are satisfied identically by joint probability density function (63). We conclude this section by emphasizing that there exist many joint densities satisfying a single differential constraint [e.g., Eq. (12)], for the same boundary and initial conditions. As an example, simply consider the family of probability densities in the form of Eq. (63) where functions are translated by rather arbitrary functions χ_{i}(x′, x″, t′, t″) defined for x′, x″ ∈ ℝ, t′, t″ ≥ t_{0} and satisfying χ_{i} = 0 for t′ = t_{0} or t″ = t_{0}. This clearly shows that a boundary value problem for Eq. (12) alone is ill posed; i.e., it admits an infinite number of solutions. 7. SUMMARYWe have determined and discussed new differential constraints satisfied by the joint probability density function of arbitrary random waves governed by Eq. (1) for random boundary conditions and random initial conditions. These differential constraints involve unusual limit partial differential operators, and they can be classified into two main groups: the first one depends on the stochastic wave equation while the second one includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. We have shown that in those cases where an evolution equation for the probability density function exists (e.g., for firstorder nonlinear stochastic PDEs) the set of differential constraints is complete; i.e., it allows us to determine the probability density function of the solution. We have applied the new theory to a random wave process in a onedimensional spatial domain and we have shown analytically that the differential constraints for the joint probability density function are satisfied identically. In addition, we have argued that a boundary value problem involving only one differential constraint is, in general, not well posed; i.e., it admits an infinite number of solutions. This brings us back to the fundamental question we have addressed at the beginning of this paper: How many differential constraints do we need in order to have enough information to compute the probability density function of the system uniquely? Most likely the full set of firstorder differential constraints, which has been proven to be complete for firstorder stochastic PDEs, is not sufficient and the full hierarchy of equations have to be considered (see, e.g., [40]).We conclude by emphasizing that the sequence of steps that led us to formulate the set of differential constraints presented in this paper can be repeated with some modifications for other linear and nonlinear stochastic PDEs. In other words, we have provided a general methodology that allows us to obtain an ensemble of conditions, in the form of partial differential equations, for the probability density function of the solution to an arbitrary stochastic PDE. ACKNOWLEDGMENTSWe acknowledge financial support from OSDMURI Grant Number FA95500910613, DOE Grant Number DEFG02 07ER25818, and NSF Grant Number DMS0915077. REFERENCES1. Ghanem, R. G. and Spanos, P. D., Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1998. 2. 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P. L., Generalized spectral decomposition for stochastic nonlinear problems, J. Comput. Phys., 228:202–235, 2009. 9. Venturi, D., On proper orthogonal decomposition of randomly perturbed fields with applications to flow past a cylinder and natural convection over a horizontal plate, J. Fluid Mech., 559:215–254, 2006. 10. Venturi, D., Wan, X., and Karniadakis, G. E., Stochastic lowdimensional modelling of a random laminar wake past a circular cylinder, J. Fluid Mech., 606:339–367, 2008. 11. Venturi, D., A fully symmetric nonlinear biorthogonal decomposition theory for random fields, Physica D, 240(45):415–425, 2010. 12. Martin, P. C., Siggia, E. D., and Rose, H. A., Statistical dynamics of classical systems, Phys. Rev. A, 8:423–437, 1973. 13. Jouvet, B. and Phythian, R., Quantum aspects of classical and statistical fields, Phys. Rev. A, 19:1350–1355, 1979. 14. Jensen, R. V., Functional integral approach to classical statistical dynamics, J. Stat. Phys., 25(2):183–210, 1981. 15. Phythian, R., The functional formalism of classical statistical dynamics, J. Phys. A: Math. Gen., 10(5):777–788, 1977. 16. Phythian, R., The operator formalism of classical statistical dynamics, J. Phys A: Math. Gen., 8(9):1423–1432, 1975. 17. Tennekes, H. and Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge, MA, 1972. 18. Monin, A. S. and Yaglom, A. M., Statistical Fluid Mechanics, vol. 1, Dover, New York, 2007. 19. Klyatskin, V. I., Dynamics of Stochastic Systems, Elsevier, Amsterdam, The Netherlands, 2005. 20. Sapsis, T. P. and Athanassoulis, G., New partial differential equations governing the responseexcitation joint probability distributions of nonlinear systems under general stochastic excitation, Prob. Eng. Mech., 23:289–306, 2008. 21. Bochkov, G. N. and Dubkov, A. A., Concerning the correlation analysis of nonlinear stochastic functionals, Radiophys. Quantum Electron., 17(3):288–292, 1974. 22. Bochkov, G. N., Dubkov, A. A., and Malakhov, A. N., Structure of the correlation dependence of nonlinear stochastic functionals, Radiophys. Quantum Electron., 20(3):276–280, 1977. 23. Hänggi, P., On derivations and solutions of master equations and asymptotic represenations, Z. Phys. B, 30:85–95, 1978. 24. Chen, H., Chen, S., and Kraichnan, R. H., Probability distribution of a stochastically advected scalar field, Phys. Rev. Lett., 63:2657–2660, 1989. 25. Pope, S. B., Mapping closures for turbulent mixing and reaction, Theor. Comput. Fluid Dyn., 2:255–270, 1991. 26. Kim, S. H., On the conditional variance and covariance equations for secondorder conditional moment closure, Phys. Fluids, 14(6):2011–2014, 2002. 27. Tartakovsky, D. M. and Broyda, S., PDF equations for advectivereactive transport in heterogeneous porous media with uncertain properties, J. Contam. Hydrol., 120121:129–140, 2011. 28. Carrier, G. F. and Pearson, C. E., Partial Differential Equations: Theory and Technique, Academic, New York, 1976. 29. Papoulis, A., Probability, Random Variables and Stochastic Processes, 3rd ed., McGrawHill, New York, 1991. 30. Khuri, A. I., Applications of Dirac′s delta function in statistics, Int. J. Math. Educ. Sci. Technol., 35(2):185–195, 2004. 31. Kanwal, R. P., Generalized Functions: Theory and Technique, 2nd ed., Birkhäuser, Boston, 1998. 32. Rosen, G., Dynamics of probability distributions over classical fields, Int. J. Theor. Phys., 4(3):189–195, 1971. 33. Lewis, R. M. and Kraichnan, R. H., A spacetime functional formalism for turbulence, Commun. Pure Appl. Math., 15:397– 411, 1962. 34. Rosen, G., Formulations of Classical and Quantum Dynamical Theory, vol. 60, Academic, New York, 1969. 35. Volterra, V., Theory of Functionals and of Integral and IntegroDifferential Equations, Dover, Mineola, NY, 1959. 36. Volterra, V., Leçons sur les Fonctions de Ligne, Gauthier Villas, Paris, 1913. 37. Vainberg, M. M., Variational Methods for the Study of Nonlinear Operators, HoldenDay, San Francisco, 1964. 38. Nashed, M. Z., Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, Nonlinear Functional Analysis and Applications, Rall, L. B., ed., Academic, New York, 1971. 39. Tuckerman, M. E., Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press, Oxford, UK, 2010. 40. Lundgren, T. S., Distribution functions in the statistical theory of turbulence, Phys. Fluids, 10(5):969–975, 1967. APPENDIX A. FUNCTIONAL REPRESENTATION OF THE PROBABILITY DENSITY FUNCTION OF THE SOLUTION TO STOCHASTIC PDESLet us consider a physical system described in terms of PDEs subject to uncertain initial conditions, boundary conditions, physical parameters, or external forcing terms. The solution to these types of boundary value problems is a random field whose regularity properties in space and time are related strongly to the type of nonlinearities appearing in the equations as well as to the statistical properties of the random input processes. In this paper we assume that the probability density function of the solution field exists. In order to fix ideas, let us consider the advectiondiffusion equation
with deterministic boundary and initial conditions. The parameter ξ(ω) is assumed to be a random variable with known probability density function. The solution to Eq. (A.1) is a random field depending on the random variable ξ(ω) in a possibly nonlinear way. We shall denote such a functional dependence as A(x, t; [ξ]). The joint probability density of A(x, t; [ξ]) and ξ [i.e., the solution field at the spacetime location (x, t) and the random variable ξ] admits the following integral representation (see Eq. (16) in [30]):
The average operator 〈·〉, in this particular case, is defined as a simple integral with respect to the probability density of ξ(ω); i.e.,
where p^{(z)}_{ξ} denotes the probability density of ξ which could be compactly supported (e.g., a uniform distribution). The support of the probability density function p^{(a,b)}_{ ψ(x, t)ξ} is determined by the nonlinear transformation ξ → ψ(x, t, [ξ]) appearing within the delta function δ[a − ψ(x, t, [ξ])] (see, e.g., Chap. 3 in [31]). Simple representation (A.3) can be generalized easily to infinite dimensional random input processes. To this end, let us examine the case where the scalar field ψ is advected by a random velocity field U according to the equation
for some deterministic initial condition and boundary conditions. Disregarding the particular form of the random field U(x, t; ω), let us consider its collocation representation for a given discretization of the spacetime domain. This gives us a certain number of random variables {U(x_{i}, t_{j}; ω)} (i = 1, ...,N, j = 1, ..., M). The random field ψ solving Eq. (A.4) at each one of these locations is, in general, a nonlinear function of all the variables {U(x_{i}, t_{j}; ω)}. In order to see this, it is sufficient to write an explicit finitedifference numerical scheme of Eq. (A.4). The joint probability density of the solution field ψ at (x_{i}, t_{j}) and the driving field U at (x_{n}, t_{m}), admits the following integral representation:
where the average, in this case, is with respect to the joint probability of all the random variables {U(x_{n}, t_{m}; ω)} (n = 1, ..., N, m = 1, ..., M). The notation ψ(x_{i}, t_{j}; [U(x_{1}, t_{1}), ..., U(x_{N}, t_{M})]) emphasizes that the solution field ψ(x_{i}, t_{j}) is, in general, a nonlinear function of all the random variables {U(x_{n}, t_{m}; ω)}. If we send the number of these variables to infinity (i.e., we refine the spacetime mesh to the continuum level), we obtain a functional integral representation of the joint probability density
where W[U] is the probability density functional of random field U(x, t; ω) and 𝒟[U] is the usual functional integral measure [13–15]. Depending on the particular equation of motion, we will need to consider different joint probability density functions; e.g., the joint probability of a field and its derivatives with respect to space variables. The functional representation described above allows us to deal with these different situations in a very practical way. For instance, the joint probability density of a field ψ(x, y, t) and its firstorder spatial derivatives at different spacetime locations can be represented as
where, for notational convenience, we have denoted by ψ_{x} ∂ψ/∂x, ψ_{y} ∂ψ/∂y and we have omitted the functional dependence on the random input variables within each field. Similarly, the joint probability of ψ(x, y, t) at two different spatial locations is
In order to lighten the notation further, sometimes we will drop the subscripts indicating the spacetime variables and write, for instance
or even more compactly
A.1 Representation of DerivativesThe differentiation of the probability density function with respect to space and time variables involves generalized derivatives of the Dirac delta function and it can be carried out in a systematic manner. To this end, let us consider Eq. (A.2) and define the following linear functional:
where ρ(a) is a continuously differentiable and compactly supported function. A differentiation of Eq. (A.11) with respect to t gives
This equation holds for an arbitrary ρ(a) and therefore we have the identity
Similarly,
Straightforward extensions of these results allow us to compute derivatives of joint probability density functions involving more fields; e.g., ψ(x, t) and its firstorder spatial derivative ψ_{x}(x, t). For instance, we have
where ψ_{tx} ∂^{2}ψ/∂t∂x. A.2 Representation of AveragesIn this section we determine important formulas to compute the average of a product between Dirac delta functions and various fields. To this end, let us first consider the average 〈δ(a − ψ)ψ〉. By applying wellknown properties of Dirac delta functions it can be shown that
This result is a multidimensional extension of the following trivial identity that holds for one random variable ξ (with probability density p^{(z)}_{ξ}) and a nonlinear function g(ξ) (see, e.g., Ch. 3 of [31] or [30]):
where = g^{−1}(a) are roots of g(z) = a. Since g() = g(g^{−1}(a)) = a, from Eq. (A.17) it follows that
which is the onedimensional version of Eq. (A.16). Similarly, one can show that
and, more generally, that
where, for the purposes of the present paper, h(ψ, ψ_{x}, x, t) is any continuous function of ψ, ψ_{x}, x, and t. The result [Eq. (A.21)] can be generalized even further to averages involving a product of functions in the form
In short, the general rule is: we are allowed to take out of the average all those functions involving fields for which we have available a Dirac delta. As an example, if A is a timedependent random field in a twodimensional spatial domain we have
APPENDIX B. HOPF CHARACTERISTIC FUNCTIONAL APPROACHIn this appendix we employ a Hopf characteristic functional approach [19, 32–34] to derive the differential constraint (12) for the probability density function of onedimensional random waves satisfying Eq. (2). This allows us to show how the differential constraints for the probability density function can be obtained from general principles. The joint Hopf characteristic functional of the wave ψ and its derivatives with respect to space and time is defined as
where
and the average operator is a functional integral with respect to the joint probability functional of the random initial condition and the random boundary conditions. The Volterra functional derivative [35, 36] of Eq. (B.1) with respect to ψ(x, t); i.e., the Gâteaux differential [37, 38] of F[α, β, γ] with respect to α evaluated at z(X, t) = δ(t − τ)δ(x − X), is
Similarly, the functional derivatives of Eq. (B.1) with respect to ψ_{t}(x, t) and ψ_{x}(x, t) are
Note that, by construction,
If we differentiate Eq. (B.3) twice with respect to t and x
and we take into account Eq. (2), we easily determine the following functional differential equation governing for the dynamics of the Hopf characteristic functional:
We notice that up to this point we have made no use of the fields β and γ appearing in Eq. (B.1). Indeed, we can safely set β = 0 and γ = 0 in Eqs. (B.2) and (B.1) and determine exactly the same evolution [Eq. (B.9)]. The need for β and γ clearly appears when one tries to extract a pointwise equation for the probability density function of the random wave from Eq. (B.9). To this end, let us first remark that functional differential equation (B.9) holds for arbitrary test functions α, β, and γ. In particular, it holds for
An evaluation of Eq. (B.9) for α = α^{+}, β = β^{+}, and γ = γ^{+} yields the condition
This condition is equivalent to a differential constraint involving the joint characteristic function of the wave and its derivatives
In order to see this, let us notice that
If we take the limit of this expression for (t′, t″) → t and (x′, x″) → x we obtain
where 𝔡 denotes the limit partial derivative operator. Similarly,
At this point we can combine the results above to obtain the following differential constraint for the joint characteristic function:
The inverse Fourier transform of this relation with respect to a, b, and c gives us exactly Eq. (12). In order to prove this, we simply recall the definition of p^{(a,b,c)} _{ψψxψt} as the inverse Fourier transform of the characteristic function ψ^{(a,b,c)} _{ψψxψt} :
and two simple relations arising from Fourier transformation theory of a one dimensional function f(x):
APPENDIX C. PERTURBATION EXPANSIONS OF DIFFERENTIAL CONSTRAINTSIn this appendix we show that differential constraint (12) can be expanded in a perturbation series leading to an equation governing the local behavior of the probability density function in the neighborhood of the hyperplane x″ = x′ = x, t″ = t′ = t. To this end, let us consider again Eq. (8):
and look for an approximation to both averages appearing on the righthand side. The simplest one consists in expanding ψ^{2}_{t} and ψ^{2}_{x} in a firstorder Taylor series around (x′, t′) and (x″, t″), respectively; i.e.,
A substitution of Eqs. (C.2) and (C.3) into the averages in Eq. (C.1) yields
At this point we can use property (A.22) and write Eqs. (C.4) and (C.5) as
A substitution of Eqs. (C.6) and (C.7) into Eq. (C.1) yields a fourthorder (ninedimensional) partial differential equation that characterizes locally; i.e., in the neighborhood of the hyperplane x = x′ = x″, t = t′ = t″ the evolution of the probability density function of the random wave. Equivalently, we can say that we have determined a local (nonpointwise) realization of functional differential Eq. (B.9). An alternative method to obtain this result consists in evaluating functional differential Eq. (B.9) for test functions α, β, and γ that are very concentrated near specific spacetime locations. These types of test functions can be any of those belonging to a delta sequence (see, e.g., Appendix A in [39]). 
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