Pathwise uniqueness in infinite dimension under weak structure condition
In this talk we consider a stochastic differential equation which evolves in a separable Hilbert space $H$ of the form \begin{align} \label{SDE} dX(t)=AX(t)dt+\Lambda B(X(t))dt+GdW(t), \quad t\in(0,T], \quad X(0)=x\in H, \end{align} where $A$ is the generator of a strongly continuous and analytic semigroup of linear and bounded operators $(e^{tA})_{t\geq0}$ on $H$, $B:H\to H$ is a bounded and $\theta$-H\"older continuous function for some suitable $\theta\in(0,1)$, $W$ is a $U$-cylindrical Wiener process, where $U$ is another separable Hilbert space, and $G:U\to H$ and $\Lambda:H\to H$ are linear and bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to equation \eqref{SDE} \begin{align} \label{mild_solution} X(t)=e^{tA}x+\int_0^t e^{(t-s)A}\Lambda B(X(s))ds+\int_0^te^{(t-s)A}GdW(s), \qquad t\in[0,T], \end{align} depends on the initial datum in a Lipschitz way, i.e., there exists a positive constant $C=C(T)$ such that, if $x_1,x_2\in H$ and there exist two mild solutions $X_1,X_2$ to \eqref{SDE} of the form \eqref{mild_solution}, with $x$ replaced by $x_1$ and $x_2$, respectively, defined on the same stochastic basis $(\Omega,\mathcal F,(\mathcal F_t)_{t\geq0},\mathbb P)$ with respect to the same $U$-cylindrical Wiener process $W$ on $(\Omega,\mathcal F,(\mathcal F_t)_{t\geq0}\mathbb P)$, then \begin{align*} \sup_{t\in[0,T]}\mathbb E[|X_1(t)-X_2(t)|_H]\leq C|x_1-x_2|_H. \end{align*} This implies that, for \eqref{SDE}, pathwise uniqueness holds true. Here, the presence of the operator $\Lambda$ plays a crucial role. The conditions assumed on the coefficients are in particular satisfied when $A$ is a realization of the damped wave operator in dimension $1$ with damping parameter $\alpha<\frac12$. This is a joint work with Davide Bignamini (Università degli studi dell'Insubria).
Area: CS7 - Advances in SPDEs (Giuseppina Guatteri and Federica Masiero)
Keywords: Stochastic partial differential equations, Multiplicative noise, Bismut formula, Malliavin calculus, semilinear Kolmogorov equations
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