Global in Time Asymptotic Solutions to Kolmogorov--Feller-Type Parabolic Pseudodifferential Equations with a Small Parameter. Forward and Backward in Time Motion

The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics.


Forward in time motion
The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics. This approach allows one to construct global in time solutions not only for the usual Cauchy problems but also for the inverse problems. We will work with Kolmogorov-Feller-type equations with diffusion, potential, and jump terms. The equation under study has the form: where P (x, ξ) is the symbol of the Kolmogorov-Feller operator, ε → +0 is a small parameter characterizing the frequency and the amplitude of jumps of the Markov stochastic process with transition probability given by P (x, ξ).
To be more precise, we bear in the mind the following form of P (x, ξ): P (x, ξ) = (A(x)ξ, ξ) + V (x) + (B(x), ξ) +  [4,19,23], for a version of this construction, see also in [20,22,21]. Maslov's approach is based on ideas similar to those used in his famous canonical operator construction (or in Fourier integral operators theory). This construction is based on some integral representation and is not suitable for constructing backward in time solutions. Another approach to the global asymptotic solution construction was suggested in [2] and is based on the construction of generalized solutions to continuity equation in a discontinuous velocity field [1]. We assume that the class of solutions under study admits the following limits: (1) -logarithmic pointwise limit lim ε→0 (−εlnu). We denote this limit by S = S(x, t) and assume that it is a piecewise smooth function with bounded first-order derivatives and a singular support in the form of a stratified manifold M .
(2) -weak limit of the expression exp (2S/ε)u 2 . We denoted it by ρ and assume that ρ is the sum of the function (ρ reg ) smooth outside M and the Dirac δ-function on M . Note that here we deal with the limit in the weighted weak sense! If S and ρ are smooth function, then the following representation is true (in the usual sense) Example 1: where S 0 ≥ 0 is a smooth function, ϕ 0 ∈ C ∞ 0 .
Here a WKB-like approach can be used (Yu. Kifer, [3]; V. Maslov, [4,19,23]). It gives an asymptotic (approximating) solution in the form (cf. [7]) for arbitrary k. Here S(x, t) is the solution to the Cauchy problem for the Hamilton-Jacobi equation and ϕ 0 = ϕ 0 (x, t) is the solution to the transport equation Both of the solutions S and ϕ 0 are defined via solutions of the Hamilton systeṁ They are smooth while Dx Dα = 0.
There are symplectic geometry objects corresponding to this construction: (1) the phase space R 2n x,p = R n x × R n p ; (2) the Lagrangian manifold Λ t n ∈ R 2n x,p , Λ 0 n = (x = α, p = ∇S 0 (α)), Λ t n = g t P Λ 0 n , where g t P is a shift mapping along the Hamiltonian system trajectories; (3) the projection mapping π : Λ t n → R n x with Jacobi matrix ∂x ∂α .
The main assumption that is required is the following one. The trajectories of the Hamilton system form a manifold of the phase space (at least in the area of the phase space under study).
Let Dx/Dα = 0 for t ∈ [0, T ], then we have the following statement (V. Maslov, [4,19]; V. Danilov, [20,22,21]): η ∈ R n is necessary and sufficient for the following estimation to hold: where M = M (k) → ∞ as k → ∞ and the function S is a solution to the Hamilton-Jacobi equation with Hamiltonian P (x, p).
It is easy to verify that the function P = P (x, ξ) -the symbol introduced above -satisfies the inequality mentioned in the theorem.
Example 2: The corresponding system is: Its solution has the form and For t > t * in case (ii), we get Λ 1 t of the shape plotted in Fig. 3.   One can see that in the case under study there are three values of S at the pointx. This means that we can present an asymptotic solution near this point in the form of a linear combination: where each of the functions u j = exp(−S j /ε)ϕ j , j = 1, 2, 3, satisfies the equation with the same accuracy. But the functions themselves are not equivalent in contrast to the hyperbolic case. For example, it is clear that if the inequality holds at a certain pointx, then the "WKB" solutions u 1 and u 2 at the point x satisfy the relation where N > 0 is an arbitrary number. This follows from the fact that the difference (S 1 − S 2 )|x in parentheses in the exponent is positive. Thus, at each point in formula (1.5), it is necessary to choose the term where the function S j is minimal. Such a choice leads to an expression of the form is the leading term of the approximate solution.
The corresponding Lagrangian manifold is the following one: Figure 4: The vertical line position is such that the above-mentioned squares are equal.
It is interesting to note that, in the one-dimensional case, there is a direct connection between the Hamilton-Jacobi equation and the conservation law of the form where u = S x and the velocity of the vertical line is defined by Rankine-Hugoniot condition corresponding to the conservation law. If the Lagrangian manifold has a jump, then the corresponding value function has a jump in the first derivative. Assumption of the actual analyticity for all objects provides that there is no "concentration of singularities", -each one can be considered separately in a sense and one can construct a value function as a solution of the Hamilton-Jacobi-Bellman equation.
What about the amplitude function? As was mentioned above, it is a solution of the transport equation Generally, the velocity field calculated from the Hamilton system has a jump simultaneously with a jump in p = ∇S (and then inẋ). Thus, the problem (still open!) is to solve the transport equation in a discontinuous velocity field. We avoid this problem considering the squared solution of the transport equation ρ = ϕ 2 0 . Madelung, [8] (about 100 years ago!) observed that it satisfies the continuity equation in the smooth case. Our case is more complicated: we again have a discontinuity velocity field. There are few approaches to this problem solution: the theory of measure solutions (F. Murat, P. LeFloh, B. Hayes, T. Zang, Y. Zheng et al., [10,11,16,12,13,14]), the box approximation (M. Oberguggenberger, M. Nedel'kov, [9]), the weak asymptotics method (V. Danilov, V. Shelkovich, [29,30,31,32,33]), the generalized characteristics method (V. Danilov, D.Mitrovich and V. Danilov, [24,25,28,26]). The last allows one to construct a solution to the continuity equation in the case where a singular support of the velocity field is a stratified manifold with smooth strata which are transversal to the (incoming!) trajectories of the velocity field.
If for some time interval [t 1 , t 2 ], the singular support of the velocity field preserves its structure (the mapping of the singular support induced by the shift along the Hamilton flow is a diffeomorphism), then it is possible to show that the singular support of the velocity has the required structure. If the structure is changing (e.g., a jump appears, see Fig.5 and Fig.4-the last step of evolution in time), then one can use the weak asymptotics method to construct a global solution to the Hamilton-Jacobi and continuity equations. This approach is based on a "new (generalized) characteristics" constructed by V. Danilov and D. Mitrovic, [24,25,26] in the case where the strata of the singular support are of codimension 1. The main idea of this approach is to consider the singularity origination as a result of nonlinear solitary wave interaction.
A simple example is the Hamilton flow corresponding to the heat equation from the previous example. The Hamilton-Jacobi equation in this case is equivalent to the Hopf equation for the momentum p: The solution in this case has the form and is plotted below. To consider p 2 , we have to calculate the product Here the following equality holds: for each ψ, which is a test function. The time evolution of the function p is such that the slanting intercept of a straight line preserves its shape until it takes the vertical position and then a jump begins to propagate. This means that, at every time instant, the solution anzatz can be presented in the form of a linear combination of Heaviside functions. This allows one to use a formula which express the product of Heaviside functions as their linear combination, and hence, uniformly in time, we see that the functions p and p 2 (the last up to a small quantity) belong to the same linear space, for detail, see [33,24,25,26]. Thus, we can prove the following theorem.
Theorem 2. Assume that the following conditions are satisfied for t ∈ [0, T ], T > 0: (1) there exists a smooth solution of the Hamiltonian system, (2) the singularities of the velocity field form a stratified manifold with smooth strata and Hess ξ P (x, ξ) > 0. Then there exists a generalized solution ρ of the Cauchy problem for continuity equation in the sense of the integral identity introduced in [31,32] and at the points where the projection π is bijective, the asymptotic solution of the Cauchy problem for Kolmogorov-Feller type equation has the form u = exp(−S(x, t)/ε)( √ ρ reg + O(ε)).

Backward in time motion
As was shown above, all that we need to go forward in time is the Hamilton system:ẋ = ∇ ξ P (x, p), x| t=0 = α, Let us change the time direction as t → −t, then We want to solve the inverse problem: The right-hand sides are considered as given data, and we are looking for X(α, t), Ξ(α, t) for 0 ≤ t ≤ T . Obviously, in our case the solutions have the form Conclusion: we can use the "same" trajectories to move forward and backward in time. But the incoming trajectories become outcoming and vice versa.

Corollary 1. Stable jumps become unstable.
But if there are no jumps (singularities of the projection mapping π : Λ t n → R n x ), then our geometry (and the asymptotic solution!) is invertible in time. This means that if we take the Cauchy problem solution u for parabolic PDE such that then the asymptotic solution for t = T has the "WKB" form Then taking the last function as the initial data for parabolic PDE in inverse time (let v as (x, t) be its asymptotic solution), we get: v as (x, T ) = exp (−S 0 (x)/ε)ϕ 0 (x)(1 + 0(ε)) It can be easily verified in the case P (x, ξ) = ξ 2 i.e. for simplest heat equation. If one constructs the solution of inverse heat equation with initial data at t = T of the form using the Green function and calculates the integral be saddle point method at t = 0then the following result will be obtained We want to stress once again that this statement is true if there is no singularities of projection mapping π : Λ t n → R n x . But a jump brings problems: There is no unique reconstruction of the part of Lagrangian manifold coming to the vertical line as t increases (see Fig.9)! But, fortunately, we can move ahead using the sense considerations. The main point is that the function S cannot attain its minimum (maximum) inside the "terra incognita". This allows one to calculate the integrals containing the reconstructed solution without taking "terra incognita" into account in the case where the integrand support contains this "terra incognita", and we can formulate the following statement.
Theorem 3. Let the symbol P (x, ξ) defined by (1.1) be such that the function B and measure µ do not depend on x and V = 0. Let u ε (x, t) be a solution of the Cauchy problem to a Kolmogorov-Feller-type equation, and assume that, for some t ∈ (0, T ), there exists a logarithmic limit S(x, t) = lim ε→0 − ε ln u ε , namely, the action function and the generalized amplitude ρ = w − lim ε→0 exp(2S/ε)u 2 ε .
Let the singular support of S(x, t) be a stratified manifold and ρ reg be the regular part of the generalized solution to the continuity equation in the distribution sense.
Then, for an arbitrary test function φ = φ(x) > 0 from the Schwartz space and for all t ∈ [0, T ], the limit as ε → 0 of the integral This result can be extended to C ∞ 0 -test functions and to the set of smooth functions integrable with weight ρ 1/2 reg exp(−S/ε) under the same assumption as in the theorem above.
Let Ω(t) be a subset in R n x , where we cannot define the functions S and ϕ uniquely by the characteristics. Here we have two possible statements: (i) let Ω(t) ∈ supp φ, then relation (2.3) is true; (ii) assume that x ∈ R 1 , Ω(t) is a union of segments I k , and the intersection between some Ik and supp φ is not empty but Ik does not belong to Ω(t), then the limit of the integral as ε → 0 exp(Ψ/ε) u ε − ρ 1/2 reg exp(−S/ε) φdx equals 0, where Ψ is the minimal value of S at the ends of Ik.
These statements actually mean that, from the viewpoint of the weak sense (momentum), the density reconstructed arbitrarily inside the "terra incognita" and according to the characteristics outside it can be used in the same manner as the leading term of the asymptotic solution constructed earlier by Maslov's tunnel canonical operator and its modifications, [4,19,22,23,21,20]. Now I will briefly speak about the proofs of the statements about the invertibility in time that was formulated above. Each of them can be divided into two parts. First, it is to prove that, for all smooth reconstructions of Lagrangian manifold in a "terra incognita" domain, the corresponding function S 0 = S 0 (x) (for some fixed t) cannot attain its minimum value inside this domain, see the lemma below. This allows one to apply the Laplace method for calculating the integrals mentioned in those statements taking into account that, due to this method and Lemma, the results of these calculations do not depend on the values of the integrands inside the "terra incognita" domain. The proof is finished by taking account of the estimation which is true outside the singular support of the function S, see Theorem above. Now I formulate the Lemma.
Lemma 1. Let the symbol P (x, ξ) defined by (1.1) be such that the function B and the measure µ do not depend on x and V = 0. Then the function S, i.e., the action function corresponding to the Lagrangian manifold, cannot attain the minimal value inside the "terra incognita" domains.
We begin the consideration with a particular case when operator symbol P (x, ξ) does not depend on x and restrict ourselves by one dimensional case studying. Let (a,b) is an interval inside the "terra incognita" domain, and letx 0 ∈ (a, b). We will proceed by contradiction. Assume that the function S attain its minimal value at the pointx 0 ∈ (a, b). We prove that, along the trajectory of Hamilton system whose projection starts atx 0 ∈ (a, b), the following inequality is true: This inequality leads to a contradiction because of the assumption that x 0 belongs to the "terra incognita" domain, and hence it belongs to the projection of the image of the singular (vertical) part of the Lagrangian manifold under backward in time shift along the trajectories of the Hamilton system. In turn, this means that the projections of all trajectories whose starting points are projected to the "terra incognita" must intersect at a point for the forward in time motion. So the above-mentioned inequality leads to a contradiction. To prove this inequality, we write the projection of the Hamilton system trajectory starting atx 0 . It has the form It is clear that p 0 = S 0x 0 (x 0 ) = 0. Thus, Taking into account that that because of the convexity of P and due to the assumption thatx 0 is a point of minimal value, we get the needed inequality (2.4). A multidimensional case differs from the case considered by changing the scalar values P ξ and S 0x 0 by vectors (gradients). In turn, this gives matrix inequalities in (2.7) and (2.8). The problem is to prove that the eigenvalues of the matrix P ξξ (0)S 0x 0 x 0 (x 0 ) are nonnegative by using (2.7) and (2.8). For this, we can make a change of variables reducing the matrix S 0x 0 x 0 (x 0 ) to diagonal form. This transformation induces the corresponding transformation in the p-plane that transforms the matrix P pp (0) to a new symmetric positive matrix. Now we note that the principal minors of the new matrix product are products of matrix-factor principal minors (because the second one is of diagonal form). The determinants of the matrix-factor principal minors are nonnegative, so the spectrum of the matrix P ξξ (0)S 0x 0 x 0 (x 0 ) is also nonnegative and we again get (2.4). To finish our consideration, we have to investigate the case where the symbol P = P (x, ξ) depends on x.
We again stay at the pointx 0 , where the function S 0 attains its minimum. If so, then and, by assumptions, p = 0 for t > 0.
The system for the matrices ∂x ∂x 0 and ∂p ∂x 0 follows from the Hamilton system and has the form Because of our assumptions (see the Lemma formulation), we have P | ξ=0 = 0, P xx | ξ=0 = 0 and P ξx | ξ=0 = 0. This means that, along the Hamilton system trajectory starting from the point x =x 0 , p = 0, we have p = 0 and Here we used the fact that the matrix t 0 exp t ′ ∂B ∂x dt ′ is in diagonal form with positive eigenvalues. Thus we came to the relation with the same properties as (2.6). One can proceed further and generalize the statement to the case of an arbitrary drift. But up to now the presence of the potential destroys our picture and I will think about it.