Applications of Zvonkin’s Transform to Stationary Kolmogorov Equations

Abstract—

In this note we develop a new analytic version of Zvonkin’s transform of the drift coefficient of a stationary Kolmogorov equation and apply this transform to derive the Harnack inequality for nonnegative solutions in the case where the diffusion matrix is not locally Sobolev. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation.

In this note we develop a new analytic version of Zvonkin’s transform [1] of the drift coefficient of a stationary Kolmogorov equation on ${{\mathbb{R}}^{d}}$ and apply this transform to derive the Harnack inequality for nonnegative solutions provided that the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to some power $p > d$. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini’s condition or belongs to the class VMO.

Let us consider the stationary Kolmogorov equation

$${{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}({{a}^{{ij}}}\varrho ) - {{\partial }_{{{{x}_{i}}}}}({{b}^{i}}\varrho ) = 0$$

(1)

on an open set $\Omega \subset {{\mathbb{R}}^{d}}$, where the coefficients ${{a}^{{ij}}}$ and ${{b}^{i}}$ are Borel functions, the matrix $A = ({{a}^{{ij}}})$ is symmetric and positive definite. Let

$$L\varphi \, = \,{{a}^{{ij}}}{{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}\varphi \, + \,{{b}^{i}}{{\partial }_{{{{x}_{i}}}}}\varphi ,\quad L{\text{*}}\varphi \, = \,{{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}({{a}^{{ij}}}\varphi )\, - \,{{\partial }_{{{{x}_{i}}}}}({{b}^{i}}\varphi ).$$

Then Eq. (1) can be written in a shorter form $L{\text{*}}\varrho = 0$. A function $\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$ is called a solution to Eq. (1) if

$${{a}^{{ij}}}\varrho ,{{b}^{i}}\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$$

and for every function $\varphi \in C_{0}^{\infty }(\Omega )$ the equality

$$\int\limits_\Omega ^{} {L\varphi (x)\varrho (x){\kern 1pt} dx = 0} $$

holds. A nonnegative solution $\varrho$ to the Kolmogorov equation (1) with the unit integral is called a probability solution. Equations of type (1) are also called double divergence form equations.

An important motivation for the study of such equations is that they hold for invariant measures of diffusion processes.

In case of locally Lipschitz coefficients, the existence of a probability solution is given by the classical theorem of Hasminskii [2] under the condition of existence of a Lyapunov function. This theorem was generalized in [3–5], where either the diffusion coefficient is nondegenerate and locally Sobolev with the order of integrability higher than dimension along with the same local integrability of the drift coefficient or both the diffusion and drift coefficients are continuous. According to [6], in the first case the solution is locally Sobolev and its continuous version is locally separated from zero. It was proved in [7, 8] that in the case where the matrix $A = ({{a}^{{ij}}})$ is nondegenerate and satisfies Dini’s condition and the coefficients ${{b}^{i}}$ are bounded, the solution has a continuous version, and if the coefficients ${{a}^{{ij}}}$ are Hölder continuous, then the solution has a Hölder continuous version. These results have been generalized in [9] to the case of locally integrable ${{b}^{i}}$. Analogous results have been obtained in [10, 11] under the assumption that A satisfies the Dini mean oscillation condition (see the definition below), which is weaker than the classical Dini condition. We recall that a mapping satisfies Dini’s condition if for its modulus of continuity $\omega$ we have

$$\int\limits_0^1 {\frac{{\omega (t)}}{t}} dt < \infty .$$

Some interesting counter-examples were constructed in [12, 13], in particular, an example of a positive definite and continuous diffusion matrix A for which the equation ${{\partial }_{{{{x}_{i}}}}}{{\partial }_{{{{x}_{j}}}}}({{a}^{{ij}}}\varrho ) = 0$ has a locally unbounded solution. The Harnack inequality for double divergence form equations with the matrix A belonging to the Sobolev class with a sufficiently high order of integrability is a corollary of the Harnack inequality for divergence form elliptic equations (see [4, Ch. 3]).

In the case where the matrix A satisfies Dini’s condition, the Harnack inequality was obtained in [14] for $b = 0$; for a bounded drift $b$ it was established in [9] and the proof heavily used the boundedness of $b$. Another way of proving the Harnack inequality for $b = 0$ was suggested in [11]. In [6, 9] (see also [4, Ch. 1]) the integrability of solutions was investigated without Dini’s condition. In particular, it was shown that if $A \in VMO$ and the coefficient $b$ is locally integrable to some power $p > d$, then the solution belongs to all $L_{{{\text{loc}}}}^{p}(\Omega )$.

This paper contains the following new results: (i) the Harnack inequality for nonnegative solutions if the matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the coefficient $b$ is locally integrable to a power p > d, (ii) sufficient conditions for the local exponential integrability of $\varrho$, (iii) a generalization of the Hasminskii theorem on existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini’s condition and the drift is not locally bounded.

These results are based on a new result on transforms of Zvonkin type. The so-called Zvonkin’s transform introduced in [1] is an efficient method in the theory of diffusion processes for smoothing the drift coefficient. We apply Zvonkin’s transform not to diffusion processes, but to solutions of the Kolmogorov equation, moreover, we do not assume any connection of solutions with diffusion processes. It is shown that with the aid of a suitable change of coordinates an integrable drift can be transformed into a continuously differentiable drift such that the new diffusion matrix enables us to apply known results about regularity of solutions.

We assume that the following conditions are fulfilled.

For convenience, we assume that the coefficients ${{a}^{{ij}}}$ are defined on ${{\mathbb{R}}^{d}}$ and for some number $\eta > 0$ and all $x \in {{\mathbb{R}}^{d}}$ the following inequalities hold:

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\nu \cdot {\text{I}} \leqslant A(x) \leqslant {{\nu }^{{ - 1}}} \cdot {\text{I}}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(}}{{H}_{a}}{\text{)}}$$

The coefficients ${{a}^{{ij}}}$ belong to the class VMO, that is, there exists a continuous increasing function $\omega$ on $[0, + \infty )$ such that $\omega (0) = 0$ and

$$\mathop {\sup }\limits_{z \in {{\mathbb{R}}^{d}}} {{r}^{{ - 2d}}}\int\limits_{B(z,r)}^{} {\int\limits_{B(z,r)}^{} {{\text{|}}{{a}^{{ij}}}(x) - {{a}^{{ij}}}(y){\text{|}}{\kern 1pt} dxdy \leqslant \omega (r),\quad r > 0,} } $$

where $B(z,r)$ is the ball of radius $r$ centered at $z$,

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b \in L_{{{\text{loc}}}}^{{d + }}(\Omega ),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{(}}{{H}_{b}}{\text{)}}$$

which means that for every ball $B \subset \Omega$ there is a number $p = p(B) > d$ such that the restriction of |b| to B belongs to ${{L}^{p}}(B)$.

Now we construct the Zvonkin transform $\Phi$.

Let $B({{x}_{0}},4R) \subset \Omega$ and $\beta (x) = b(x)$ if $x \in B({{x}_{0}},4R)$ and $\beta (x) = 0$ if $x \notin B({{x}_{0}},4R)$. Then $\beta \in {{L}^{p}}({{\mathbb{R}}^{d}})$ and ${\text{||}}\beta {\text{|}}{{{\text{|}}}_{{{{L}^{p}}({{\mathbb{R}}^{d}})}}} = {\text{||}}b{\text{|}}{{{\text{|}}}_{{{{L}^{p}}(B({{x}_{0}},4R))}}}.$ Let $1 \leqslant k \leqslant d$. Let us consider on ${{\mathbb{R}}^{d}}$ the elliptic equation

$${\text{tr}}(A{{D}^{2}}u) + \langle \beta ,\nabla u\rangle - \lambda u = - {{\beta }^{k}},\quad \lambda > 0.$$

(2)

One can show that for every $\delta > 0$ there exists $\lambda > 0$ such that for every $k \leqslant d$ Eq. (2) has a solution ${{u}_{k}} \in {{C}^{1}}({{\mathbb{R}}^{d}}) \cap {{W}^{{p,2}}}({{\mathbb{R}}^{d}})$ for which

$$\mathop {\sup }\limits_{x \in {{\mathbb{R}}^{d}}} {\text{|}}\nabla {{u}_{k}}(x){\text{|}} \leqslant \delta ,\quad {\text{||}}{{u}_{k}}{\text{|}}{{{\text{|}}}_{{{{W}^{{2,p}}}({{\mathbb{R}}^{d}})}}} \leqslant M,$$

where the constant M depends only on $d$, $\nu$, $\omega$, and ${\text{||}}b{\text{|}}{{{\text{|}}}_{{{{L}^{p}}(B({{x}_{0}},4R))}}}$. Let $u = ({{u}^{1}}, \ldots ,{{u}^{d}})$ with the aforementioned solutions u^k to Eq. (2). Below $u{\kern 1pt} '$ and $\Phi {\kern 1pt} '$ denote the Jacobi matrices of the mappings u and $\Phi$. We take $\delta$ small enough such that for all $x \in {{\mathbb{R}}^{d}}$

$${\text{||}}u{\kern 1pt} '(x){\text{||}} \leqslant \frac{1}{2}\quad {\text{and}}\quad \frac{1}{2} \leqslant \det (I + u'(x)) \leqslant 2.$$

Set

$$\Phi (x) = x + u(x).$$

Proposition 1. (i) The mapping $\Phi$ is a diffeomorphism of ${{\mathbb{R}}^{d}}$ of class C¹, moreover, the functions ${{\partial }_{{{{x}_{i}}}}}{{\Phi }^{k}}$ are locally Hölder continuous. (ii) We have

$$\frac{1}{2}{\text{||}}x - y{\text{||}} \leqslant {\text{||}}\Phi (x) - \Phi (y){\text{||}} \leqslant 2{\text{||}}x - y{\text{||}}.$$

We recall that $B({{x}_{0}},4R) \subset \Omega$. Let $\Psi = {{\Phi }^{{ - 1}}}$ and ${{y}_{0}} = \Phi ({{x}_{0}})$ and consider the ball $B({{y}_{0}},2R)$. According to the inequalities in (ii) above, we have

$$B({{x}_{0}},R) \subset \Psi (B({{y}_{0}},2R)) \subset B({{x}_{0}},4R).$$

Proposition 2. Let $\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$ be a solution to Eq. (1). Then the function

$$\sigma (y) = {\text{|}}\det \Psi {\kern 1pt} '(y){\text{|}}\varrho (\Psi (y))$$

on $B({{y}_{0}},2R)$ satisfies equation $\mathcal{L}{\text{*}}\sigma = 0$, where

$$\mathcal{L}f(y) = {{q}^{{km}}}(y){{\partial }_{{{{y}_{k}}}}}{{\partial }_{{{{y}_{m}}}}}f(y) + {{h}^{k}}(y){{\partial }_{{{{y}_{k}}}}}f(y)$$

and the coefficients have the form

$$\begin{gathered} {{q}^{{km}}}(y) = {{a}^{{ij}}}(\Psi (y)){{\partial }_{{{{x}_{i}}}}}{{\Phi }^{k}}(\Psi (y)){{\partial }_{{{{x}_{j}}}}}{{\Phi }^{m}}(\Psi (y)), \\ {{h}^{k}}(y) = \lambda {{u}^{k}}(\Psi (y)). \\ \end{gathered} $$

Observe that the vector field $h(y) = \lambda u(\Psi (y))$ is continuously differentiable on the ball $B({{y}_{0}},2R)$. In addition, the derivative of $\Phi$ also satisfies the Hölder condition. Therefore, the function $\sigma$ on $B({{y}_{0}},2R)$ satisfies equation $\mathcal{L}{\text{*}}\sigma = 0$, in which the coefficients ${{q}^{{mk}}}$ of the second order terms form a nondegenerate matrix and belong to the class $VMO$ and the coefficients ${{h}^{k}}$ are continuous on $B({{y}_{0}},2R)$. This enables us to apply the results from [7–9, 11] to the function $\sigma$ and then to transfer them to $\varrho$. Let us give an example demonstrating a simple derivation of the known result of [9, Theorem 3.1] from the case of a nice drift.

Example 1. If conditions $({{{\text{H}}}_{a}})$ and $({{{\text{H}}}_{b}})$ are fulfilled and the matrix A satisfies Dini’s condition, then every solution $\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$ to Eq. (1) has a continuous version.

Proof. Let us verify the existence of a continuous version of $\varrho$ on the ball $B({{x}_{0}},R{\text{/}}2) \subset B({{x}_{0}},4R) \subset \Omega$. Let $\Phi$ be the diffeomorphism constructed above. By Proposition 2 the function $\sigma (y) = \varrho (\Psi (y)){\text{|}}\det \Psi {\kern 1pt} '(y){\text{|}}$ satisfies on $B({{y}_{0}},2R)$ an equation with some coefficients for which the hypotheses of [8, Theorem 1] are fulfilled, that is, the matrix $({{q}^{{mk}}})$ is nondegenerate and the functions q^mk, ${{h}^{k}}$ satisfy Dini’s condition. Hence $\sigma$ has a continuous version on $B({{y}_{0}},R)$. Since $\Phi$ is a diffeomorphism of class ${{C}^{1}}$, the mappings $\Phi$ and $\Psi$ take sets of measure zero to sets of measure zero and a modification of the function $\sigma$ on a set of measure zero yields a change of $\varrho$ on a set of measure zero. Therefore, the function $\varrho$ has a continuous version on $B({{x}_{0}},R{\text{/}}2)$.

We now present our new results on solutions to stationary Kolmogorov equations obtained by means of Zvonkin’s transform constructed above.

Following [10, 11], we say that a measurable function f on $\Omega$ satisfies the Dini mean oscillation condition if

$$\int\limits_0^1 {\frac{{w(r)}}{r}{\kern 1pt} dr < \infty ,} $$

where

$$w(r) = \mathop {\sup }\limits_{x \in \Omega } \frac{1}{{{\text{|}}\Omega (x,r){\text{|}}}}\int\limits_{\Omega (x,r)}^{} {{\text{|}}f(y) - {{f}_{\Omega }}(x,r){\text{|}}{\kern 1pt} dy,} $$

$$\begin{gathered} {{f}_{\Omega }}(x,r) = \frac{1}{{{\text{|}}\Omega (x,r){\text{|}}}}\int\limits_{\Omega (x,r)}^{} {f(y){\kern 1pt} dy,} \\ \Omega (x,r) = \Omega \cap B(x,r). \\ \end{gathered} $$

The classical Dini condition implies the Dini mean oscillation condition.

The next assertion generalizes the Harnack inequality to the case where the diffusion matrix satisfies the Dini mean oscillation condition and the drift coefficient is locally unbounded (and is merely integrable to some power larger than the dimension). In the known results, the drift coefficient is either zero or locally bounded.

Theorem 1. Suppose that condition $({{{\text{H}}}_{a}})$ is fulfilled, on every ball the matrix A satisfies the Dini mean oscillation condition with some function $\omega$, and ${{b}^{i}} \in L_{{{\text{loc}}}}^{{d + }}(\Omega )$. Suppose also that $\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$ is a solution to Eq. (1). Then the function $\varrho$ has a continuous version. Moreover, if $\varrho \geqslant 0$, then the continuous version of $\varrho$ satisfies the Harnack inequality: for every ball $B({{x}_{0}},R{\text{/}}2)\,\, \subset \,\,B({{x}_{0}},4R)\,\, \subset \,\,\Omega$ there exists a number C such that

$$\mathop {{\text{sup}}}\limits_{x \in B({{x}_{0}},R/2)} \varrho (x) \leqslant C\mathop {{\text{inf}}}\limits_{x \in B({{x}_{0}},R/2)} \varrho (x),$$

where C depends on $R$, $w$, $d$, $\nu$, $p$, and ${\text{||}}b{\text{|}}{{{\text{|}}}_{{{{L}^{p}}(B({{x}_{0}},4R))}}}$, and does not depend on $\varrho$. The modulus of continuity of $\varrho$ on $B({{x}_{0}},R{\text{/}}2)$ depends on the same objects.

By [9, Theorem 2.1], if a^ij and ${{b}^{i}}$ satisfy conditions $({{{\text{H}}}_{a}})$ and $({{{\text{H}}}_{b}})$ and ${{a}^{{ij}}} \in VMO$, then every solution $\rho \in L_{{{\text{loc}}}}^{1}(\Omega )$ is locally integrable to every power $p \geqslant 1$. If the functions a^ij satisfy the Dini mean oscillation condition, then the solution is locally bounded and even continuous. Let us consider an intermediate case where A belongs to a smaller class than $VMO$, but does not satisfy Dini’s condition. Suppose that a^ij and ${{b}^{i}}$ satisfy conditions $({{{\text{H}}}_{a}})$ and $({{{\text{H}}}_{b}})$ and

$${\text{||}}A(x) - A(y){\text{||}} \leqslant \omega ({\text{||}}x - y{\text{||}}),$$

where $\omega$ is an increasing continuous function on $[0, + \infty )$ with $\omega (0) = 0$. Assume also that for some ${{C}_{\omega }} > 0$ and all $t \geqslant 0$ we have

$$\omega (t) \geqslant {{C}_{\omega }}{{t}^{{1 - d/p}}}.$$

For example, the function $\omega (t) = {\text{|}}\ln t{{{\text{|}}}^{{ - 1}}}$ near zero is suitable, which does not satisfy Dini’s condition.

Theorem 2. Let $\varrho \in L_{{{\text{loc}}}}^{1}(\Omega )$ satisfy Eq. (1). Then, for every closed ball $B \subset \Omega$, the following assertions hold.

(i) If $\mathop {\lim }\limits_{t \to 0 + } \omega (t){\text{|}}\ln t{\text{|}} = 0$, then

$$\exp ({{\gamma }_{1}}{\text{|}}\varrho {{{\text{|}}}^{{{{\gamma }_{2}}}}}) \in {{L}^{1}}(B)\quad {\kern 1pt} for\;all\quad {{\gamma }_{1}},\;{{\gamma }_{2}} > 0{\kern 1pt} .$$

(ii) If the function $\omega (t){\text{|}}\ln t{\text{|}}$ is bounded on $(0,1]$, then exist numbers ${{\gamma }_{1}},{{\gamma }_{2}} > 0$ such that

$$\exp ({{\gamma }_{1}}{\text{|}}\varrho {{{\text{|}}}^{{{{\gamma }_{2}}}}}) \in {{L}^{1}}(B).$$

(iii) If for some $\beta \in (0,1)$ the function $\omega (t){\text{|}}\ln t{{{\text{|}}}^{\beta }}$ is bounded on $(0,1]$, then for some $\gamma > 0$

$$\exp (\gamma {\text{|}}\ln ({\text{|}}\varrho {\text{|}} + 1){{{\text{|}}}^{{\tfrac{1}{{1 - \beta }}}}}) \in {{L}^{1}}(B).$$

The next theorem generalizes assertion (i) in Theorem 2.4.1 in [4] to the case where the coefficients a^ij satisfy the Dini mean oscillation condition. It was assumed in the cited theorem that the functions a^ij belong locally to the Sobolev class ${{W}^{{p,1}}}$ with some $p > d$.

Theorem 3. Suppose that the coefficients a^ij and ${{b}^{i}}$ are defined on ${{\mathbb{R}}^{d}}$, ${{b}^{i}} \in L_{{{\text{loc}}}}^{{d + }}$, and for every ball $B$ we can find a number ${{\nu }_{B}} > 0$ and continuous nonnegative increasing function ${{w}_{B}}$ on [0, 1] such that ${{w}_{B}}(0)$ = 0, the function ${{w}_{B}}(t){\text{/}}t$ is integrable on [0, 1], ${{\nu }_{B}}\, \cdot \,I\, \leqslant \,A(x)\, \leqslant \,\nu _{B}^{{ - 1}}\, \cdot \,I$ for all x, and for all $r \in (0,1]$ one has

$$\mathop {\sup }\limits_{x \in B} \frac{1}{{{\text{|}}B(x,r){\text{|}}}}\int\limits_{B(x,r)}^{} {{\text{|}}{{a}^{{ij}}}(y) - a_{B}^{{ij}}(x,r){\text{|}}{\kern 1pt} dy \leqslant {{w}_{B}}(r).} $$

Then there is a continuous and positive solution $\varrho$ to Eq. (1)on ${{\mathbb{R}}^{d}}$.

This result gives the following generalization of the Hasminskii theorem.

Corollary 1. If in addition to the hypotheses of Theorem 3 there is a function V of class $W_{{{\text{loc}}}}^{{d,2}}({{\mathbb{R}}^{d}})$ along with numbers $C > 0$ and $R > 0$ for which

$$\mathop {\lim }\limits_{|x| \to + \infty } V(x) = + \infty ,\quad LV(x) \leqslant - C\quad if{\kern 1pt} \quad {\text{|}}x{\text{|}} \geqslant R,$$

then there exists a continuous positive probability solution $\varrho$ to Eq. (1)on ${{\mathbb{R}}^{d}}$.