Approximation of the density of a solution of a nonlinear SDE -- application to parabolic SPDEs
Let us consider a stochastic differential equation \[ dX_{t} = b[X_{t},m_{t}] dt + \sigma [X_{t}, m_{t}] dB_{t} \tag{1} \] in \(\mathbb R^{d}\), where \(B\) is an \(m\)-dimensional Wiener process, \(m_{t}\) denotes the law of the random variable \(X_{t}\), and the mappings \(b[y,\nu ]\), \(\sigma [y,\nu ]\) are linear in the measure \(\nu \), that is, \(b[y,\nu ] = \int _{\mathbb R^{d}} b(y,z) d\nu (z)\), \(\sigma [y,\nu ] = \int _{\mathbb R^{d}} \sigma (y,z) d\nu (z)\). It is known that under some regularity assumptions on the functions \(b\), \(\sigma \) the solution \(X_{t}\) has a density \(p_{t}\in {\mathcal C}^\infty (\mathbb R^{d})\) and (1) may be viewed as a limit problem as \(n\to \infty \) for the following system of weakly interacting diffusions \[ dX^{i,n}_{t} = b[X^{i,n}_{t},\mu ^{n}_{t}] dt + \sigma [X^{i,n}_{t},\mu ^{n}_{t}] dB^{i}_{t}, \quad 1\leq i\leq n, \] \(B^{1},\ldots ,B^{n}\) being independent Wiener processes in \(\mathbb R^{m}\) and \(\mu ^{n}_{t}\) denoting the empirical measure \(n^{-1}\sum ^{n}_{i=1} \delta _{X^{i,n}_{t}}\) [see e.g. \textit{A. S. Sznitman}, in: Calcul des probabilités. Lect. Notes Math. 1464, 165-251 (1991; Zbl 0732.60114), and the references therein]. Let \(V^\varepsilon \) be a sequence of mollifiers, \(V^\varepsilon (y) = \varepsilon ^{-d}V(\varepsilon ^{-1}y)\) for a symmetric \({\mathcal C}^\infty \)-density \(V\) on \(\mathbb R^{d}\), and define \((\mu ^{n}_{t}*V^\varepsilon)(y) = n^{-1}\sum ^{n} _{i=1} V^\varepsilon (X^{i,n}_{t}-y)\). Using probabilistic tools including coupling techniques and Malliavin calculus the author finds an estimate of \(\mu ^{n}_{t} * V^\varepsilon - p_{t}\) in a suitable Sobolev-type norm. The obtained results are then extended to a stochastic parabolic equation \[ \frac {\partial u}{\partial t}(t,x) = \frac {\partial ^{2}u} {\partial x^{2}}(t,x) + f[u(t,x),m(t,x)] + g[u(t,x),m(t,x)] \frac {\partial ^{2}W}{\partial t\partial x}, \quad 0<x<1, \tag{2} \] with Dirichlet or Neumann boundary data and \(u(0,\cdot) = u_{0} \in {\mathcal C}^{1}([0,1])\); \(W\) denotes a Brownian sheet and \(m(t,x)\) the law of \(u(t,x)\). In particular, existence of a density for the (unique) mild solution \(u(t,x)\) to (2) is established.
parabolic stochastic partial differential equations
scientific article; zbMATH DE number 1240034