A Kolmogorov operator with time dependent coefficients on \({{\mathbb{R}}^{d}}\) is an elliptic second order differential operator of the form

$$L\varphi (x,t) = \sum\limits_{i,j \leqslant d} {{a}^{{ij}}}(x,t){{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}\varphi (x) + \sum\limits_{i \leqslant d} {{b}^{i}}(x,t){{\partial }_{{{{x}_{i}}}}}\varphi (x),$$

where for a fixed interval \([\tau ,T]\) we are given on \({{\mathbb{R}}^{d}} \times [\tau ,T]\) a mapping \(b = ({{b}^{i}})\) with values in \({{\mathbb{R}}^{d}}\) and a mapping \(A = ({{a}^{{ij}}})\) with values in the space of nonnegative definite symmetric d × d-matrices such that the functions \({{b}^{i}}\) and \({{a}^{{ij}}}\) are Borel measurable. For such an operator we pose the Cauchy problem for the Fokker–Planck–Kolmogorov equation

$${{\partial }_{t}}{{\mu }_{t}} = \sum\limits_{i,j \leqslant d} {{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}({{a}^{{ij}}}{{\mu }_{t}}) - \sum\limits_{i \leqslant d} {{\partial }_{{{{x}_{i}}}}}({{b}^{i}}{{\mu }_{t}})$$
(1)

with initial condition \({{\mu }_{\tau }} = \nu \), where \(\nu \) is a Borel probability measure on \({{\mathbb{R}}^{d}}\). A probability solution to this Cauchy problem is defined as a family \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) of Borel probability measures on \({{\mathbb{R}}^{d}}\) for which the function \(t \mapsto {{\mu }_{t}}(E)\) is Borel measurable for every Borel set E in \({{\mathbb{R}}^{d}}\), the integrals

$$\int\limits_0^T {\int\limits_K^{} {{\text{|}}{{a}^{{ij}}}(x,s){\text{|}}\, + \,{\text{|}}{{b}^{i}}(x,s){\text{|}}{\kern 1pt} {{\mu }_{s}}(dx){\kern 1pt} ds} } ,$$

are finite for all compact sets \(K \subset {{\mathbb{R}}^{d}}\) and, for every function \(\varphi \in C_{0}^{\infty }({{\mathbb{R}}^{d}})\), the identity

$$\int\limits_{{{\mathbb{R}}^{d}}}^{} {\varphi {\kern 1pt} d{{\mu }_{t}} - \int\limits_{{{\mathbb{R}}^{d}}}^{} {\varphi {\kern 1pt} d\nu = \,} \int\limits_\tau ^t {\int\limits_{{{\mathbb{R}}^{d}}}^{} {L\varphi {\kern 1pt} d{{\mu }_{s}}{\kern 1pt} ds} } } $$

holds. The required condition of local integrability of the coefficients with respect to the solution is automatically fulfilled in the case of locally bounded coefficients. By a solution we also mean the measure \(\mu = {{\mu }_{t}}(dx)dt\) defined by the equality

$$\int\limits_{{{\mathbb{R}}^{d}} \times [\tau ,T]}^{} {f{\kern 1pt} d\mu = } \,\int\limits_\tau ^T {\int\limits_{{{\mathbb{R}}^{d}}}^{} {f(x,y){\kern 1pt} {{\mu }_{t}}(dx){\kern 1pt} dt.} } $$

The indicated condition on the coefficients is the local integrability with respect to \(\mu \).

Let \(y \in {{\mathbb{R}}^{d}}\) and let \({{(\mu _{t}^{y})}_{{t \in [\tau ,T]}}}\) be a family of Borel probability measures on \({{\mathbb{R}}^{d}}\) satisfying Eq. (1) with initial condition \({{\mu }_{\tau }} = {{\delta }_{y}}\) that is the Dirac measure at the point \(y\). It was shown by Kolmogorov [1] that in the case of sufficiently regular coefficients they can be calculated by means of such solutions by the formulas

$$\mathop {\lim }\limits_{h \to 0 + } \frac{1}{h}\int\limits_{{{\mathbb{R}}^{d}}}^{} {({{x}_{i}} - {{y}_{i}})\mu _{{\tau + h}}^{y}(dx) = {{b}^{i}}(y,\tau ),} $$
(2)
$$\mathop {\lim }\limits_{h \to 0 + } \frac{1}{{2h}}\int\limits_{{{\mathbb{R}}^{d}}}^{} {({{x}_{i}} - {{y}_{i}})({{x}_{j}} - {{y}_{j}})\mu _{{\tau + h}}^{y}(dx) = {{a}^{{ij}}}(y,\tau ).} $$
(3)

The question naturally arises about the possibility of reconstructing the coefficients by solutions in the general case. It has been shown in the recent paper [2] that if the coefficients are continuous and satisfy the estimates

$${\text{|}}{{a}^{{ij}}}(x,t){\text{|}} \leqslant C(1 + \,{\text{|}}x{{{\text{|}}}^{2}}),\quad {\text{|}}{{b}^{i}}(x,t){\text{|}} \leqslant C(1 + \,{\text{|}}x{\text{|}})$$

with some constant \(C\) (under these estimates the integrals of |x|2 against the measures \({{\mu }_{t}}\) are bounded on compact intervals), then the equalities indicated by Kolmogorov are valid.

If the coefficients \({{a}^{{ij}}}\), \({{b}^{i}}\) have no continuous versions, then these equalities must be clarified, since they can fail to hold pointwise simply because changing the coefficients on a set of \({{\mu }_{t}}{\kern 1pt} dt\)-measure zero does not change the equation. In addition, it would be interesting to obtain analogs of these equalities in the case where the initial condition is not a delta-measure. In Theorem 2 below we present a result of this kind. It gives broad sufficient conditions for reconstructing the coefficients by a solution, covering the case of continuous coefficients and Dirac initial distributions and also the case of discontinuous coefficients and the initial distribution given by a density with certain properties. The proofs will be published in a detailed paper.

Definition 1. We shall say that a Borel function f on \({{\mathbb{R}}^{d}} \times [\tau ,T]\) is approximable with respect to a family of probability measures \({{({{\mu }_{t}})}_{{[\tau ,T]}}}\) if there exists a sequence of bounded Borel functions fn on \({{\mathbb{R}}^{d}} \times [\tau ,T]\) such that these functions are continuous in x, the equality

$$\mathop {\lim }\limits_{t \to \tau } \mathop {\sup }\limits_{x \in U} {\text{|}}{{f}_{n}}(x,t) - {{f}_{n}}(x,\tau ){\text{|}} = 0$$
(4)

is fulfilled for every ball U, and also the equality

$$\mathop {{\text{lim}}}\limits_{n \to \infty } {\text{esssu}}{{{\text{p}}}_{{s \in [\tau ,T]}}}{\text{||}}f( \cdot ,s) - {{f}_{n}}( \cdot ,s){\text{|}}{{{\text{|}}}_{{{{L}^{1}}({{\mu }_{s}})}}} = 0$$
(5)

holds.

According to this definition, for every \(\varepsilon > 0\), there exists a number N such that for all \(m,n \geqslant N\) and almost all \(s \in [\tau ,T]\) the following estimate is valid:

$${\text{||}}{{f}_{m}}( \cdot ,s) - {{f}_{n}}( \cdot ,s){\text{|}}{{{\text{|}}}_{{{{L}^{1}}({{\mu }_{s}})}}} \leqslant \varepsilon .$$

Hence there exists a set \(S \subset [\tau ,T]\) with complement of measure zero such that for all \(s \in S\) the sequence \({{f}_{n}}({\kern 1pt} \cdot {\kern 1pt} ,s)\) converges in \({{L}^{1}}({{\mu }_{s}})\) to some limit \(\widetilde f({\kern 1pt} \cdot ,s)\). It is readily verified that convergence also holds for \(s = \tau \), so that we can assume that \(\tau \in S\) and hence there is a function \(\widetilde f({\kern 1pt} \cdot ,\tau )\), which will play an important role in Theorem 2.

The new function \(\widetilde f\) is a version of the original one in the sense that \(\widetilde f(x,s) = f(x,s)\) almost everywhere with respect to the measure \(\mu \) on \({{\mathbb{R}}^{d}} \times [\tau ,T]\). This property depends, of course, on the measure \(\mu \), but if it is known that every probability solution to the given Fokker–Planck–Kolmogorov equation possesses an almost everywhere positive density with respect to Lebesgue measure on \({{\mathbb{R}}^{d}} \times [\tau ,T]\), then the constructed version is universal for all measures with densities. Different sufficient conditions for positivity of densities of all probability solutions are known, in particular, it suffices that the diffusion matrix be locally Lipschitz continuous and invertible and the drift coefficient be locally bounded (see [3], Chapter 8).

It is clear that every continuous and bounded function f on \([\tau ,T] \times {{\mathbb{R}}^{d}}\) is approximable with respect to \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\), moreover, we can assume that \(\widetilde f = f\).

Let us give a sufficient condition for the approximability of the function f with respect to the family \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) fulfilled without assumptions about its continuity.

Theorem 1. Suppose that for almost all \(t \in [\tau ,T]\) the measure \({{\mu }_{t}}\) has a density \(\varrho ({\kern 1pt} \cdot {\kern 1pt} ,t)\) with respect to Lebesgue measure and the functions \(\varrho ({\kern 1pt} \cdot {\kern 1pt} ,t)\) are uniformly integrable on every ball U (for example, are uniformly bounded in \({{L}^{p}}(U)\), where \(p > 1\)). Assume that a locally bounded measurable function f satisfies the following conditions:

$$\mathop {\lim }\limits_{R \to \infty } \mathop {\sup }\limits_{t \in [\tau ,T]} \int\limits_{|x| \geqslant R}^{} {{\text{|}}f(x,t{\text{)|}}{\kern 1pt} {{\mu }_{t}}(dx) = 0} $$
(6)

for every ball U, the family of functions \(x \mapsto f(x,t)\) has compact closure in \({{L}^{1}}(U)\) and

$$\mathop {\lim }\limits_{t \to \tau } \mathop {\sup }\limits_{x \in U} {\text{|}}f(x,t) - f(x,\tau ){\text{|}} = 0.$$
(7)

Then the function f in approximable with respect to the family \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\).

Example 1. (i) Any bounded measurable function f independent of \(t\) is approximable with respect to the family of measures \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) that is continuous in \(t\) in the weak topology and satisfies the condition on the density \(\varrho \) indicated in the proposition. Indeed, the precompactness condition and (7) are trivially fulfilled, and for verification of (6) it suffices to observe that the given family of measures is uniformly tight by the Prohorov theorem, since it is compact in the weak topology by the continuity of \({{\mu }_{t}}\) in \(t\).

(ii) Any continuous function f independent of t is approximable with respect to any family of measures \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) with respect to which it is uniformly integrable. Here for \({{g}_{n}}\) one can take the cut-off functions \({{\psi }_{n}}(f)\), where \({{\psi }_{n}}(u) = u\) if \({\text{|}}u{\text{|}} \leqslant n\), \({{\psi }_{n}}(u) = n\) if \(u > n\), and \({{\psi }_{n}}(u) = - n\) if \(u < - n\). For example, this is true if the function f is continuous, satisfies the estimate \({\text{|}}f(x){\text{|}} \leqslant C + C{\text{|}}x{{{\text{|}}}^{k}}\) and the measures \({{\mu }_{t}}\) possess uniformly bounded moments of some order \(m > k\).

(iii) More generally, a function f continuous on \({{\mathbb{R}}^{d}} \times [\tau ,T]\) is approximable with respect to the family of measures \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) if the functions \(f({\kern 1pt} \cdot {\kern 1pt} ,t)\) are uniformly integrable with respect to the measures \({{\mu }_{t}}\). For example, this is true under the same estimate \({\text{|}}f(x,t){\text{|}} \leqslant C + C{\text{|}}x{{{\text{|}}}^{k}}\) and the condition of uniform boundedness of moments of the measures \({{\mu }_{t}}\) of order \(m > k\). In particular, if \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) is a solution to prob-lem (1), where \({\text{|}}{{a}^{{ij}}}(x,t{\text{)|}} \leqslant C(1 + \,{\text{|}}x{{{\text{|}}}^{2}})\), \({\text{|}}{{b}^{i}}(x,t){\text{|}} \leqslant C(1\, + \,{\text{|}}x{\text{|}})\) and the initial condition has finite second moment, then any continuous function f satisfying the estimate \({\text{|}}f(x,t){\text{|}} \leqslant {{C}_{1}}(1\, + \,{\text{|}}x{{{\text{|}}}^{2}})\) is approximable with respect to \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\).

In applications to solutions to problem (1) the indicated condition on the density excludes singular initial distributions at time \(\tau \), but it is known that it is fulfilled if the matrices \(A{{(x,t)}^{{ - 1}}}\) are uniformly bounded, the matrices \(A(x,t)\) are uniformly Lipschitz in x, and the density \(\varrho ({\kern 1pt} \cdot {\kern 1pt} ,\tau )\) is locally bounded (see [3], Sect. 7.3).

Let us present our main result on reconstruction of the coefficients \(A\) and \(b\) by a probability solution \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\) to Eq. (1) under the following condition:

$$\begin{gathered} \frac{{{{a}^{{ij}}}(x,t)}}{{{\text{|}}x{\text{|}} + 1}},\;{{b}^{i}}(x,t)\;{\kern 1pt} {\text{are}}\;{\text{integrable}}\;{\text{against}} \\ {\text{the}}\;{\text{measure}}\;\mu = {{\mu }_{t}}(dx)dt.{\kern 1pt} \\ \end{gathered} $$
(8)

The formulation makes use of some family of measures \(\mu _{t}^{y}\) generated by the solution to problem (1) with initial distribution \({{\mu }_{\tau }}\) and giving solutions to problem (1) with Dirac initial distributions \({{\delta }_{y}}\). This family is constructed in the following way. According to the known superposition principle (see [6–8]), under condition (8), there a exists a Borel probability measure \(P\) on the space of continuous paths Ω := \(C([\tau ,T],{{\mathbb{R}}^{d}})\) with is standard sup-norm such that

$${{\mu }_{t}} = P \circ e_{t}^{{ - 1}},\quad {\kern 1pt} {\text{where}}\quad {{e}_{t}}(\omega ) = \omega (t),$$

and for every function \(\varphi \in C_{0}^{\infty }({{\mathbb{R}}^{d}})\) the mapping

$$(\omega ,t) \mapsto \varphi (\omega (t)) - \varphi (\omega (\tau )) - \int\limits_\tau ^t {L\varphi (\omega (s),s){\kern 1pt} ds} $$

is a martingale with respect to P and the filtration \({{\mathcal{F}}_{t}}\) generated by the mappings \({{e}_{s}}\), \(\tau \leqslant s \leqslant t\). Such a measure P is called a solution to the martingale problem (see [4, 5]). Let \({{P}^{y}}\) be the conditional measures obtained by disintegration of the measure P with respect to the measure \({{\mu }_{\tau }}\). This means that the measure P is written in the form \(P(d\omega ) = {{P}^{y}}(d\omega ){{\mu }_{\tau }}(dy)\), i.e., for every Borel set \(B \subset \Omega \) we have the equality

$$P(B) = \int\limits_{{{\mathbb{R}}^{d}}}^{} {{{P}^{y}}(B){\kern 1pt} {{\mu }_{\tau }}(dy),} $$

where every measure \({{P}^{y}}\) is concentrated on the set \(e_{\tau }^{{ - 1}}(y)\) and depends on \(y\) Borel measurably. It is straightforward to verify that for \({{\mu }_{\tau }}\)-almost every y the family of measures \(\mu _{t}^{y} = {{P}^{y}} \circ e_{t}^{{ - 1}}\) is a solution to Eq. (1) with initial condition \(\mu _{\tau }^{y} = {{\delta }_{y}}\).

Note that solutions \(\mu _{t}^{y}\) need not exist for all y. In addition, we do not assume uniqueness of solutions to the Cauchy problem. In case of non-uniqueness, it is important to use the indicated solutions generated by the solution to the martingale problem (its uniqueness is not assume as well).

Theorem 2. Suppose that

$$\mathop {\sup }\limits_{t \in [\tau ,T]} \int\limits_{{{\mathbb{R}}^{d}}}^{} {{\text{|}}x{{{\text{|}}}^{2}}{\kern 1pt} {{\mu }_{t}}(dx) < \infty } ,$$

and the coefficients \({{b}^{i}}\) are approximable with respect to the family \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\). Then the equalities

$$\mathop {\lim }\limits_{h \to 0 + } \int\limits_{{{\mathbb{R}}^{d}}}^{} {\left| {\frac{1}{h}\int\limits_{{{\mathbb{R}}^{d}}}^{} {({{x}^{i}} - {{y}^{i}})\mu _{{\tau + h}}^{y}(dx) - {{{\widetilde b}}^{i}}(y,\tau )} } \right|} {{\mu }_{\tau }}(dy) = 0$$

hold. If, in addition, the functions \({{a}^{{ij}}}\) are integrable with respect to the measure \(\mu \), the functions \({{x}_{j}}{{b}^{i}}\) are approximable with respect to the family \({{({{\mu }_{t}})}_{{t \in [\tau ,T]}}}\), and for some \(p > 2\) the integrals of \({\text{|}}x{{{\text{|}}}^{p}}\) against the measures \({{\mu }_{t}}\) are uniformly bounded and \({{\sup }_{t}}{\text{||}}b({\kern 1pt} \cdot {\kern 1pt} ,t){\text{|}}{{{\text{|}}}_{{{{L}^{2}}({{\mu }_{t}})}}}\, < \,\infty \), then the equalities

$$\mathop {\lim }\limits_{h \to 0 + } \int\limits_{{{\mathbb{R}}^{d}}}^{} {\left| {\frac{1}{{2h}}\int\limits_{{{\mathbb{R}}^{d}}}^{} {({{x}^{i}}\, - \,{{y}^{i}})({{x}^{j}}\, - \,{{y}^{j}})\mu _{{\tau + h}}^{y}(dx)\, - \,{{{\tilde {a}}}^{{ij}}}(y,\tau )} } \right|} {{\mu }_{\tau }}(dy)\, = \,0$$

hold. Thus, equalities (2) and (3) are fulfilled in the sense of convergence in \({{L}^{1}}({{\mu }_{\tau }})\).