Existence of solutions to weak parabolic equations for measures

Abstract

Let $A = (a^{ij})$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in the space of non-negative operators on $\mathbb{R}^d$ and let $b = (b^i)$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in $\mathbb{R}^d$. Let

\[Lu(t, x) = \partial_{t} u(t, x) + a^{ij}(t, x)\partial_{x_i}\partial_{x_j}u(t, x) + b^i(t, x)\partial_{x_i}u(t, x), \quad u \in C_0^\infty((0, 1)\times \mathbb{R}^d).\]

Under broad assumptions on $A$ and $b$, we construct a family $\mu = (\mu_t)_{t \in [0, 1)}$ of probability measures $\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem $L^{*}\mu = 0$ with initial condition $\mu_0 = \nu$, where $\nu$ is a probability measure on $\mathbb{R}^d$, in the following weak sense:

$$\[ \int_0^1\int_{\mathbb{R}^d} Lu(t, x)\, \mu_t(dx)\, dt=0, \quad u \in C_0^\infty((0, 1) \times \mathbb{R}^d), \] $$

and

$$\[ \lim\limits_{t \to 0} \int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx) = \int_{\mathbb{R}^d} \zeta(x)\, \nu(dx), \quad \zeta \in C_0^\infty(\mathbb{R}^d). \] $$

Such an equation is satisfied by transition probabilities of a diffusion process associated with $A$ and $b$ provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, $b$ may be locally unbounded or $A$ may be degenerate. An infinite-dimensional analogue is discussed as well. Main methods are $L^p$-analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time.